In the case of reversible chemical reactions, the term Dynamic equilibrium is said to be achieved only when the rate of the backward reaction is equal to the rate of the forward reaction during the reaction process. The suffix ‘Dynamic’ is primarily used to describe that the chemical reaction is ongoing. Some example of dynamic equilibrium are given below.

Dynamic equilibrium can be achieved when the forward reaction and the reverse reaction are still occurring, however, the rate of occurrence of both the reactions are equal and unchanging i.e. they are in equilibrium. In other words, any chemical reaction is said to be in dynamic equilibrium when the rate of the forward reaction is equivalent to the rate of backward reaction simultaneously and the chemical reaction is reversible. Some example of dynamic equilibrium are:

NaCl reaction

Nitrogen Dioxide-Carbon monoxide reaction

Let’s take another example of dynamic equilibrium: Nitrogen Dioxide gas reacts with Carbon monoxide to form nitrogen monoxide gas and carbon dioxide as shown in the equation NO2(g) + CO(g) ⇌  NO(g) + CO2(g). This is a reversible chemical reaction. This reaction is said to be in dynamic equilibrium when the rate of reaction of nitrogen dioxide with carbon monoxide is equal to the rate of reaction of carbon dioxide and nitrogen monoxide.

Acetic acid reaction

Dynamic equilibrium can be seen in a single-phase system such as an acidic-basic ion equilibrium in an H₂O solution. For example, an aqueous solution of acetic acid dissociates into its H+(acidic) and basic ions while simultaneously being produced by the same ions. This reaction is given by the equation: CH₃COOH ⇌ CH₃COO + H⁺

Carbon dioxide reaction

We have observed examples of dynamic equilibrium in our daily life too. One such example is a sealed soda can or bottle. A sealed soda bottle contains carbon dioxide in both liquid form and the gaseous form (in the form of bubbles) along with the soda. The gaseous phase of carbon dioxide is in dynamic equilibrium with the liquid/aqueous phase of carbon dioxide. The liquid phase of carbon dioxide gets converted into its gaseous phase at the same rate as the gaseous phase of carbon dioxide dissolves into the liquid form of carbon dioxide. The equation of the mentioned reaction is given as CO2(g) ⇌ CO2(aq). 

Nitrogen and hydrogen reaction

Industrial ammonia synthesis by using Haber’s process is also an example of a reaction that can be in a dynamic equilibrium. The reaction is given by the equationN₂ (g) + 3H₂ (g) ⇌ 2NH₃ (g). In this nitrogen and hydrogen molecules combine to form ammonia and ammonia simultaneously disintegrates to form the nitrogen and hydrogen molecules.

Nitrogen dioxide reaction

Another example of dynamic equilibrium is the dimerization of nitrogen dioxide in the gaseous phase. The reaction is given by the equation 2NO ⇌ N₂O₄

Let us dig a bit deeper to understand the concept of dynamic equilibrium better.

What are the examples of reactions that can never be in dynamic equilibrium?

Irreversible reactions can never be in dynamic equilibrium. In these kinds of reactions, the reactants get converted into products but the vice-versa does not take place. Thus, any possibility of establishing dynamic equilibrium is eliminated.

One such example is the reaction of iron with water vapor to form rust. This is given by the equation:

 4 Fe(s) + 6 H₂O(l) + 3O₂ (g) → 4 Fe(OH)³ (s)

Rust cannot disintegrate back into iron and water vapor. We can see that because this is an irreversible reaction, the arrow from the reactants to products is pointed in only a single direction.

Another example of an irreversible reaction is the reaction of a fuel with atmospheric oxygen to form carbon dioxide and water vapor. This reaction is given by the equation

Fuel + O₂ → CO₂ + H₂O

This is called the combustion reaction. The products i.e. carbon dioxide and water vapor cannot react back to produce fuel and oxygen. Therefore, the reaction is one-sided.

There can be many examples of such irreversible reactions where the products cannot return back to the reactants. In all such reactions, establishing a dynamic equilibrium between the products and the reactants is not possible.

How to maintain dynamic equilibrium in a system?

It is difficult to maintain a dynamic equilibrium in a reaction. Every slight change in the temperature, pressure, or concentration of a reaction has the ability to shift or knock off the dynamic equilibrium.

For this reason, the soda bottle becomes flat or loses the bubbles after it is left open. Once the can is opened the carbon dioxide in the gaseous phase is able to react or interact with the atmospheric carbon dioxide and other gases. It is hence no longer a closed system. This releases the gaseous carbon dioxide that was present in the form of bubbles and knocks off the dynamic equilibrium that was previously established when the bottle was sealed.  

What are the conditions for Dynamic equilibrium?

The conditions necessary for dynamic equilibrium in a reaction is given as follows:

• The amount of reactants and products needs to be unchanged as that in the start of the reaction. It is not allowed to add reactants during the reaction externally.
• The reaction should be in a closed system so that no other influence or substances can get added up the reaction.
• The reaction has to be reversible in nature.
• It is important to maintain the physical parameters such as temperature, pressure, etc. at equilibrium throughout the reaction.

How to detect that a system is in dynamic equilibrium?

We can say that in dynamic equilibrium a system is in a steady state i.e. the variables in a chemical reaction do not change with time since the rate of the reversible and forward reactions are equal. 

• If we observe a reaction at dynamic equilibrium, then we will not be able to see any change and it would look like no reaction is taking place. However, the reaction taking place in the forwarding direction is compensated by the reaction taking place in the reverse direction simultaneously.

• If you are given a reaction you can make out if the reaction is in dynamic equilibrium or not by observing the amount or quantity of the reactants and products of the reaction. If you observe that the quantity of product exceeds the quantity of reactants or the quantity of reactants exceeds the quantity of products, then you can cancel out the possibility of dynamic equilibrium in the reaction.

• However, if you see that the amount of product and reactant remains the same as that at the starting of the reaction i.e. the amount of reactants and products remains unchanged throughout the reaction, then it may or may not be in dynamic equilibrium. Sometimes the changes in the amount of products and reactants are very minute that makes it difficult to be detected by naked eyes. At other times the reaction may be in static equilibrium.

What is Static equilibrium?

Static equilibrium in a reaction refers to the phase where the reaction is at a halt or there is no reaction taking place between the reactants or the products.

We can say that in the case of static equilibrium, the rate of forwarding reaction and the rate of backward reaction is both zero. The quantity of the products and the reactants stay unchanged. An example of static equilibrium is the process involving the formation of graphite from diamond and vice-versa. The reaction is given by the equation C(diamond) ⇌ C(graphite)

The stability of graphite is more than that of a diamond. One needs to heat graphite as high as 2000°C or more to trigger its activation energy and convert it into a diamond. In-room temperature, this conversion would require millions of years to complete. There would be an infinitesimal amount of conversion between the two substances in general conditions.  Hence, it can be said that at room temperature this reaction is at static equilibrium.

What are the differences between a dynamic and a static equilibrium?

We hope this post provided all the necessary information regarding the example of dynamic equilibrium.

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Content

 Skewness

    The curve which is the plotted observations represents the skewness if the shape of the curve is not symmetric, of the given set. In other words the lack of symmetry in the graph of the given information represents the skewness of the given set. Depending on the tail in the right or left the skewness is known as positively skewed or negatively skewed. The distribution depending on this skewness is known as positively skewed distribution or negatively skewed distribution

The mean, mode and median shows the nature of distribution so if the nature or shape of the curve is symmetric these measure of central tendencies are equal and for the skewed distributions these measure of central tendencies varies as either mean>median>mode or mean<median<mode.

Variance and Skewness

Measure of Skewness

To find the degree and the direction of the frequency distribution whether positive or negative the measure of skewness is very helpful even with the help of the graph we know the positive or negative nature of the skewness but the magnitude will not be exact in graphs hence these statistical measures gives the magnitude of lack of symmetry.

To be specific the measure of skewness must have

1. Unit free so that the different distributions can be comparable if the units are same or different.
2. Value of measure for symmetric distribution zero and positive or negative for positive or negative distributions accordingly.
3. The value of measure should vary if we move from negative skewness to positive skewness.

There are two types of measure of skewness

1. Absolute Measure of skewness
2. Relative Measure of skewness

Absolute Measure of skewness

In the symmetrical distribution the mean, mode and median are same so in absolute measure of skewness the difference of these central tendencies gives the extent of symmetry in the distribution and the nature as positive or negative skewed distribution but the absolute measure for different units is not useful while comparing two sets of information.

The Absolute skewness can be obtained using

1. Skewness(S_k)=Mean-Median
2. Skewness(S_k)=Mean-Mode
3. Skewness(S_k)=(Q₃-Q₂)-(Q₂-Q₁)

Relative Measure of skewness

Relative measure of skewness is used to compare the skewness in two or more distributions by eliminating the influence of variation, relative measure of skewness is known as coefficient of skewness, the following are the important relative measure of skewness.

1. Karl Pearson’s Coefficient of Skewness

This method is used most often to calculate skewness

this coefficient of skewness is positive for positive distribution, negative for negative distribution and zero for the symmetric distribution. This Karl Pearson’s coefficient usually lies between +1 and -1. If Mode is not defined then to calculate the Karl Pearson’s coefficient we use the formula as

If we use this relation then Karl Pearson’s coefficient lies between +3 and -3.

2. Bowleys’s Coefficient of Skewness|Quartile measure of skew ness

In Bowleys’s coefficient of skewness the quartile deviations were used to find the skewness so it is also known as quartile measure of skewness

or we can write it as

this value of coefficient is zero if the distribution is symmetric and the value for positive distribution is positive, for negative distribution is negative. The value of S_k lies between -1 and +1.

3. Kelly’s Coefficient of Skewness

In this measure of skewness the percentiles and deciles are used to calculate the skewness, the coefficient is

where these skewness involves the 90, 50 and 10 percentiles and using deciles we can write it as

in which 9,5 and 1 deciles were used.

4. β and γ Coefficient of Skewness| Measure of skew ness based on moments.

Using the central moments the measure of skewness the β coefficient of skewness can be define as

this coefficient of skewness gives value zero for the symmetric distribution but this coefficient not tells specifically for the direction either positive or negative, so this drawback can be removed by taking square root of beta as

this value gives the positive and negative value for the positive and negative distributions respectively.

Examples of skewness

1.  Using the following information find the coefficient of skewness

Solution: To find the coefficient of skewness we will use karl Pearson’s coefficient

the karl pearson coefficient of skewness is

the modal class is maximum frequent class 40-50 and the respective frequencies are

thus

so the coefficient of skewness will be

which shows the negative skewness.

2. Find the coefficient of skewness of the frequency distributed marks of 150 students in certain examination

Solution: To calculate the coefficient of skewness we require mean, mode, median and standard deviation for the given information so for calculating these we form the following table

now the measures will be

and

hence the coefficient of skewness for the distribution is

3. Find the mean, variance and coefficient of skewness of distribution whose first four moments about 5 are 2,20,40 and 50.

Solution: since the first four moments are given so

so we can write it

so the coefficient of skewness is

Positively skewed distribution definition|Right skewed distribution meaning

Any distribution in which the measure of central tendencies i.e mean, mode and median having positive values and the information in the distribution lacks the symmetry.

In other words the positively skewed distribution is the distribution in which the measure of central tendencies follows as mean>median>mode in the right side of the curve of the distribution.

If we sketch the information of the distribution the curve will be right tailed because of which positively skewed distribution is also known as right skewed distribution.

from above curve it is clear that the mode is the smallest measure in positively or right skewed distribution and the mean is the largest measure of central tendencies.

positively skewed distribution example|example of right skewed distribution

1. For a positively skewed or right skewed distribution if the coefficient of skewness is 0.64, find the mode and median of the distribution if mean and standard deviations are 59.2 and 13 respectively.

Solution: The given values are mean=59.2, s_k=0.64 and  σ=13 so using the relation

2. Find the standard deviation of the positively skewed distribution whose coefficient of skewness is 1.28 with mean 164 and mode 100?

Solution: In the same way using the given information and the formula for the coefficient of positively skewed distribution

so the standard deviation will be 50.

3. In the quarterlies deviations if the addition of first and third quarterlies is 200 with median 76 find the value of third quartile of the frequency distribution which is positively skewed with coefficient of skewness 1.2?

Solution: To find the third quartile we have to use the relation of coefficient of skewness and quarterlies, since the given information is

from the given relation we have

from these two equations we can write

so the value of the third quartile is 120.

4. Find the coefficient of skewness for the following information

Solution: here we will use Bowley’s measure of skewness using quartiles

As N^th/4=15^th observation of class is 102.5-107.5 , N^th/2=30^th observation of class is 107.5-112.5 and 3N^th/4=45^th observation of class is 112.5-117.5 so

and

and median is

thus

which is positively skewed distribution.

where is the mean in a positively skewed distribution

We know that the positively skewed distribution is right skewed distribution so the curve is right tailed the meaning of this most of the information will be nearer to the tail so the mean in a positively skewed distribution is nearer to the tail and since in positively or right skewed distribution mean>median>mode so mean will be after the median.

Right skewed distribution mean median mode|relationship between mean median and mode in positively skewed distribution

In the positively skewed or right skewed distribution the measure of central tendencies mean, median and mode are in the order mean>median>mode, as mode is the smallest one then median and the largest central tendency is the mean which for the right tailed curve is nearer to the tail of the curve for the information.

so the relationship between mean median and mode in positively skewed distribution is in the increasing order and with the help of the difference of these two central tendencies the coefficient of skewness can be calculated, so mean, median and mode gives the nature of skewness also.

positively skewed distribution graph|positively skewed distribution curve

The graph either in the form of smooth curve or in the form of histogram for the discrete information, the nature is right tailed as the mean of the information gather around the tail of the curve as skewness of distribution discusses the shape of the distribution. Since the large amount of data is in left of the curve and tail of the curve onto the right is longer.

some of the graphs of positively distributed information are as follows

from the above graphs it is clear that the curve has lacking the symmetry in any aspects .

positively skewed score distribution

In any distribution if the scores are in the positively skewed that is the score following the positively skewed distribution as mean>median>mode and the curve of the distribution score having right tailed curve in which score is affected by the large value.

This type of distribution is known as positively skewed score distribution. All the properties and rules for this distribution are the same from positively skewed or right skewed distribution.

positive skew frequency distribution

In positively skewed frequency distribution on average the frequency of the information are smaller as compared to the distribution so the positive skew frequency distribution is nothing but the positively skewed or right skewed distribution where the curve is right tailed curve.

positive vs negative skewed distribution|positively skewed distribution vs negatively skewed

FAQs

How do you know if a distribution is positively or negatively skewed

The skewness is positive if mean>median>mode and negative if mean<median<mode,

From the distribution curve also we can judge if the curve is right tailed it is positive and if the curve is left tailed it is negative

How do you determine positive skewness

By calculating the measure of coefficient of skewness if positive then skewness is positive or by plotting the curve of distribution if right tailed then positive or by checking mean>median>mode

What does a positive skew represent

The positive skewness represent that the score of the distribution lies nearer to large values and the curve is right tailed and the mean is the largest measure

How do you interpret a right skewed histogram

if the histogram is right skewed then the distribution is positively skewed distribution where mean>median>mode

In distributions that are skewed to the right what is the relationship of the mean median and mode

The relationship is mean>median>mode

Conclusion:

The skewness is important concept of statistics which gives the asymmetry or lack of symmetry present in the distribution of probability depending on the positive or negative value it is classified as positively skewed distribution or negatively skewed distribution, in the above article the brief concept with examples discussed   , if you require further reading go through

https://en.wikipedia.org/wiki/skewness

For more post on mathematics, please follow our Mathematics page

What is a Square Wave Generator : Circuit Diagram & Advantages

Table of contents

Square wave generator | square wave signal generator

What is a square wave generator?

A square wave generator is a non-sinusoidal waveform oscillator that is capable of generating square waves. The Schmitt trigger circuit is an implementation of square wave generators. Another name for the square wave generator is an Astable or a free-running multivibrator.

Square Wave Generator Circuit | square wave signal generator circuit

Square wave and triangular wave generator | Square and triangular wave generator using op amp

Square wave generator using op amp

A square wave generator using an operational amplifier is also called an astable multivibrator. When an operational amplifier is forced to operate in the saturation region, it generates square waves. The output of the op-amp swings between the positive and the negative saturation and produces square waves. That’s why the op-amp circuit here is also known as a free-running multivibrator.

Square wave generator working

The circuit of the op-amp contains a capacitor, resistors, and a voltage divider. The capacitor C and the resistor R are connected with the inverting terminal, as shown in figure 1. The non-inverting terminal is connected to a voltage divider network with resistors R₁ and R₂. A supply voltage is provided to the op-amp. Let us assume that the voltage across the non-inverting terminal is V₁ and across the inverting terminal is V₂. V_d is the differential voltage between the inverting and the non-inverting terminal. Initially, the capacitor has no charge. Therefore, we can take V₂ as zero.

We know, V_d = V₁-V₂

As initially, V₂=0, V_d = V₁

We know, V₁ is a function of output offset voltage, R₁, and R₂. The leakage results in the generation of the output offset voltage.

V_d can be positive or negative. It depends upon the polarity of the output offset voltage.

Let us assume initially, V_d is positive. So the capacitor has no charge, and the op-amp has maximum gain. So the positive differential voltage will drive the op-amp’s output voltage Vo towards the positive saturation voltage.

So, V₁=R₁/R₁+R₂V_sat

At this point, the capacitor starts charging towards the positive saturation voltage through the resistor R. It will increase its voltage from zero to a particular value. After reaching a value slightly greater than V₁, the op-amp will give a negative output voltage, and reach the negative saturation voltage. Then the equation becomes,

V_d = -V₁+V₂

-V₁=R₁/R₁+R₂(-V_sat)

As V₁ is negative now, the capacitor starts discharging towards negative saturation voltage up to a certain value. After reaching a value slightly less than V₁, the output voltage will move to positive saturation voltage again.

This total phenomenon repeatedly happens , generating the square waves(shown in figure 2). Therefore we get square waves that switch between +V_sat and -V_sat.

Therefore, V₁=R₁/R₁+R₂(V_sat)

The time period of the output of square wave, T=2RCln (2R₁+R₂/R₂)

Triangular wave generator using op amp

There are two parts of a triangular wave generator circuit. One part generates the square wave, and the second part converts the square wave into a triangular waveform. The first circuit consists of an op-amp and a voltage divider connected to the op-amp’s non-inverting terminal. The inverting terminal is grounded.

The output of this op-amp acts as input for the second part, which is an integrator circuit. That contains another operational amplifier whose inverting terminal is connected with a capacitor and a resistor, as shown in figure 3. The non-inverting terminal of the op-amp is made ground. Let’s say the first output is Vo₁ and the second output is Vo₂. Vo₂ is connected with the first op-amp as feedback.

The comparator S1 continuously compares the voltage of point A(figure 3) with ground voltage, i.e., zero. According to the positive and the negative value, the square wave is generated at Vo1. In the waveform, we see that when the voltage at point A is positive, S1 gives +Vsat as output. This output provides input for the second op-amp that produces a negative-going ramp Voltage Vr as output. Vr gives negative voltage up to a certain value. After some time, the voltage at A falls below zero, and S1 gives -Vsat as output.

At this stage, the value of V_r starts increasing towards the positive saturation voltage. When the value crosses +Vr, the output of the square wave goes up to +V_sat. This phenomenon goes on continuously, providing the square wave as well as the triangular wave ( shown in figure 4).

For this entire circuit, we notice that when V_r gets changed from positive to negative, a positive saturation voltage is developed. Similarly, when V_r gets changed from negative to positive, a negative saturation voltage is developed. Resistor R₃ is connected to Vo₁ while, resistor R₂ is connected to Vo₂. Therefore, the equation can be written as,

-V_r/R₂ = -(+V_sat/R₃)

V_r = -R₂/R₃(-V_sat)

The peak to peak output voltage V_pp=V_r-(-V_r)=2V_r=2R₂/R₃(V_sat)

Output at the integrator circuit is given by,

Here, V_o=V_pp and V_input= -V_sat

So, by putting the values we get,

Therefore,

So, frequency

Square wave generator formula

Time period of square wave generator

The time period of the square wave generator,

R = resistance

C = capacitance of the capacitor connected with the inverting terminal of the op-amp R₁ and R₂ = resistance of the voltage divider. 

Square wave generator frequency formula

The frequency of the square wave generator,

Variable frequency square wave generator

Most commonly, multivibrator circuits are used in generating square waves. RC or LR circuits can generate a periodic sequence of quasi-rectangular voltage pulses utilizing the saturation characteristic of the amplifier. The variable frequency square wave generator circuit consists of four major components- A linear amplifier and an inverter with a total gain of K, a clipper circuit with some specific input-output characteristics, and a differentiator comprising RC or LR network with the time constant ?. The time period of the obtained signal is

T=2?ln(2K-1)

This multivibrator circuit can produce uniform voltage pulses because of the symmetrical saturation characteristic of the clipper circuit. We can vary the oscillation frequency by varying either the time constant of the differentiator or the gain of the amplifier.

AVR square wave generator

It is possible to generate different waveforms using AVR microcontrollers by interfacing a Digital to Analog Converter(DAC). The DAC converts the microcontroller provided digital inputs into analog outputs, and thus generates different analog waveforms. The DAC output is actually the current equivalent of the input. So, we use 741 operational amplifier integrated circuit as a current to voltage converter.

The microcontroller gives low and high outputs in alternate fashion as an input to DAC after applying some delay. Then the DAC generates corresponding alternate analog outputs through the op-amp circuit to produce a square waveform.

High frequency square wave generator

High-frequency square wave generators produce accurate waveforms with minimum external hardware components. The output frequency can range from 0.1 Hz to 20 MHz. The duty cycle is also variable. The high-frequency square wave generators are used in-

• ‌Precision Function Generators
• ‌Voltage-Controlled Oscillators
• ‌Frequency Modulators
• ‌Pulse-Width Modulators
• ‌Phase Lock Loops
• ‌Frequency Synthesizer
• ‌FSK Generators

Time Period and Frequency Derivation of Square Wave Generator

According to the ideal op-amp conditions, the current through it is zero. Therefore, by applying Kirchhoff’s law, we can write,

The ratio R₁/R₁+R₂ is known as the feedback fraction and is denoted by β.

When V₁ reaches positive saturation voltage,

 V₀ = +V_sat,

V₁/β = +V_sat

Or, V₁ =  βV_sat

Similarly, when V₁ reaches negative saturation voltage,

 V₀ = -V_sat,

V₁/β = -V_sat

Or, V₁ = -βV_sat

By this time, the capacitor has charged to CV₁ = CβV₀; it again starts discharging. So, according to the general capacitor equation with an initial charge Q₀,

Q=CV(1-e^t/RC)+Q₀e^t/RC

We know, here V = -V₀ and Q₀=βCV₀

So,

Now, when Q goes to -CV₁ = -CβV₀, another switch occurs at t=T/2. At this time, 

Therefore,

Frequency

555 timer square wave generator circuit | 555 square wave generator circuit

Square wave generator using 555 IC | 555 square wave generator

555 square wave generator 50% duty cycle

The square wave generator can be constructed using the 555 timer integrated circuit. It is efficient for generating square pulses of lower frequency and adjustable duty cycle. The left part of the IC includes the Pins 1-4- Ground, Trigger, Output, and Reset. Pins 5-8 are on the right side. Pin 5, pin 6, pin7, and pin 8 are the control voltage, the threshold, the discharge, and the positive supply voltage respectively. The main circuitry consists of the 555 IC, two resistors, two capacitors, and a voltage source of 5-15 Volts. This circuit can further be optimized using a diode to produce a perfect square wave. The 555 timer can easily create square waves in astable mode.

The circuit diagram is shown in figure 5. Pin 2(Trigger) and pin 6(Threshold) are connected so that the circuit continuously triggers itself on each cycle. The capacitor C charges through both the resistors but discharges only through R₂ connected to pin 7(discharge). The timer starts when pin 2 voltage decreases below 1/3V_CC. If the 555 timer is triggered through pin 2, the pin 3 output becomes high. When this voltage climbs up to 2/3V_CC, the cycle finishes, and the pin 3 output becomes low. This phenomenon results in a square wave output.

The below equations determine the charging time or T_on and the discharging time or T_off:

T_on= 0.693(R₁+R₂)C

T_off= 0.693R₂C

So the total cycle time T = 0.693(R₁+R₂+R₂)C =0.693(R₁+2R₂)C

Therefore, frequency f = 1/T = 1.44/(R₁+2R₂)C

Duty cycle =T_on/T=R₁+R₂/R₁+2R₂

555 variable frequency square wave generator

To make a variable frequency square wave generator, we take a 555 timer IC. At first, we make pin 2 and pin 6 short-circuited. Then we connect a jumper wire between pin 8 and pin 4. We connect the circuit to positive V_cc. Pin 1 is connected to the ground. A capacitor of 10 nF is attached with pin 5. A variable capacitor is joined with pin 2. Pin 4 and pin 8 are made short-circuit. A 10 Kohm resistor is connected between pin 7 and pin 8. A 100 Kohm potentiometer is connected between pin 6 and pin 7. This circuit produces square waveforms. We can adjust the frequency with the help of the potentiometer.

ATtiny85 square wave generator

The ATtiny85 8-bit AVR microcontroller based on RISC CPU, has an 8 pin interface and 10 bit ADC converter. The timer in ATtiny85 sets up the Pulse width modulation mode and helps in varying the duty cycle so that the proper square wave is generated.

Square wave sound generator

Square waves are one of the four fundamental waves that create sound. The other three waves are the triangular wave, sine wave, and sawtooth wave. Together the waves can produce different sounds if we vary the amplitude and frequency. If we increase the voltage, i.e., the amplitude, the volume of the sound increases. If we increase the frequency, the pitch of the sound increases.

1khz square wave generation in 8051

We can program the 8051 microcontrollers to generate a square wave of the desired frequency. Here, the frequency of the signal is 1 kHz, so the time period is 1 millisecond. The 50% duty cycle is best for perfect square waves. So, T_on=T_off= 0.5 ms.

Circuit and connections: To make the circuit, we need the following components-

• ‌8051 microcontroller
• ‌Digital to analog converter
• Resistors and capacitors
• Operational amplifier

We connect the reset pin to the voltage source (Vcc) and the DAC data pins to port 1 of the 8051 microcontroller. The most significant bit has to be connected with the A₁ pin (pin 5) on the DAC and the least significant bit with the A₈ pin.

Logic: At first, we set any of the 8051 ports to logic 1 or high and then wait for some time to get a constant DC voltage. This time is known as delay. Now we set the same port to logic 0 or low and again wait for some time. The process continues in a loop until we turn off the microcontroller.

Square wave generator using IC 741 | square wave generator using op amp 741

The IC 741 square wave generator circuit is depicted in the figure above(figure 6). The operational amplifier in the circuit built using the general IC 741. Pin 2 of the IC is connected to the inverting terminal, and pin 3 is connected to the non-inverting terminal. Pin 7 and pin 4 are connected to the positive and negative supply voltage, respectively. The output is connected to pin 6. The capacitor, the resistor, and the voltage divider are connected, as shown in the figure.

The working principle of IC 741 circuit is similar to that of the general square wave generator. The capacitor keeps on charging and discharging between the positive and the negative saturation voltage. Thus it produces the square wave. 

The time period T=2RC ln (2R₁+R₂/R₂)

The frequency is the reciprocal of the time period, i.e., f=1/2RC ln (2R₁+R₂/R₂)

MATLAB code to generate square wave

The Matlab command to generate a square wave is given below-

clc
close all
clear  #clearing all previous data
t=1:0.01:50;  #defining X axis from 1 to 50 with step 0.01
Y=square (t,50);   #taking a variable Y for a square wave with 50% duty cycle
plot(Y,t);  #plotting the curve
xlabel('Time');  #labelling X-axis as Time
ylabel('Amplitude');  #labelling Y-axis as Amplitude
title('Square Wave'); #the title of the plot is Square Wave
axis([-2 1000 5 -5]);  #modifying the graph for visualization

Square wave generator astable multivibrator

Square wave generator using transistor | transistor square wave generator

Another technique of building a square wave generator (Astable Multivibrator) is using a BJT or bipolar junction transistor. The operation of this square wave generator or astable multivibrator depends upon the switching property of the BJT. When a BJT acts as a switch, it has two states- on and off. If we connect +V_cc in the collector terminal of the BJT when the input voltage V_i is less than 0.7 volt, the BJT is said to be in the off state. In the off state, the collector and the emitter terminal get disconnected from the circuit.

Therefore, the transistor behaves to be an open switch. So the I_c=0 (I_c is the collector current) and the voltage drop between the collector terminal and the emitter terminal(V_ce) is positive V_cc.

Now when Vi>0.7 volt, the BJT is in on state. We short the collector and the emitter terminal. Therefore, V_ce=0 and the current I_c will be the saturation current(Ic_sat).

The circuit diagram is shown in figure 7. Here, the transistors S₁ and S₂ look identical, but they have different doping properties. S₁ and S₂ have load resistors RL₁ and RL₂ and are biased through R₁ and R₂, respectively. The collector terminal of S₂ is connected to the base terminal of S₁ through the capacitor C₁, and the collector terminal of S₁ is connected to the base terminal of S₂ through the capacitor C₂. So, we can say that the astable multivibrators are made with two identical common-emitter configurations.

The output is obtained from any of the two collectors to the ground. Suppose we are taking Vc₂ as the output. So the entire circuit is connected to the supply voltage V_cc. The negative terminal of V_cc is grounded. When we close the switch K, both the transistors try to stay in the on state. But eventually, one of them stays in the on state and the other one in the off state. When S₁ is in the on state, the collector and the emitter terminal of S₁ get shorted. So, Vc₁=0. Meanwhile, S₂ is in the off state.

Therefore, the collector current Ic₂=0 and Vc₂=+V_cc. So for the T₁ time interval, the transistor Vc₁ remains in logic 1, and Vc₂ remains in logic 0. While S₂ is in the off state, the capacitor C₂ gets charged. Let us say the voltage across C₂ is Vc₂. So we connect the positive terminal of the capacitor to the base of S₂, and the negative terminal of the capacitor to the emitter of S₂. So the voltage Vc₂ is directly provided to the base and the emitter terminal of S₂.

As the capacitor is continuously charging, after some time, Vc₂ goes up above 0.7 volts. At this point, S₂ comes to the on state, and the voltage difference between the collector and the emitter terminal of S₂ equals zero. Now, S₁ acts in the on state, and the output voltage of S₁ is +V_cc. The capacitor C₁ starts charging, and when the voltage across the capacitor crosses 0.7 volts, S₁ again changes its state. So for the T₁ time interval, the transistor Vc₁ remains in logic 0, and Vc₂ remains in logic 1.

This phenomenon repeats automatically until the power supply is turned off. The continuous transition between V_cc and 0 generates the square wave.

Square wave generator using NAND Gate

The use of a NAND gate is one of the simplest ways to make a square wave generator. We need the following components to build the circuit are- two NAND gates, two resistors, and one capacitor. The circuit is shown in figure 8. The resistor-capacitor network is the timing element in this circuit. The G₁ NAND gate controls its output. The output of this RC network is fed back to G₁ through the resistor R₁ as input. This procedure occurs until the capacitor is fully charged.

When the voltage across C reaches the positive threshold of G₁, the NAND gates change states. Now the capacitor discharges up to the negative threshold of G₁, and again the gates change their states. This process occurs in a loop and produces a square waveform. The frequency of this waveform is calculated using, f=1/2.2RC

Square wave generator using Schmitt Trigger

The working of a Schmitt trigger square wave generator circuit is quite similar to the NAND gate implementation. The Schmitt trigger circuit is shown in figure 9. Here also, the RC network provides the timing. The inverter takes its output in the form of a feedback as one of the inputs.

Initially, the NOT gate input is less than the minimum threshold voltage. So the output state is High. Now the capacitor begins to charge through the resistor R₁. When the voltage across the capacitor touches the maximum threshold voltage, the output state again drops to low. This cycle repeats again and again and generates the square wave. The frequency of the square wave is found by f=1/1.2RC

Square wave generator verilog code | square wave generator using verilog

`timescale 1ns / 1ps
module square_wave_generator(
input clk,
input rst_n,
output square_wave
);
// Input clock is 100MHz
localparam CLK_FREQ = 100000000;
// Counter to toggle the clock
integer counter = 0;

reg square_wave_reg = 0;
assign square_wave = square_wave_reg;
always @(posedge clk) begin

if (rst_n) begin
counter <= 8'h00;
square_wave_reg <= 1'b0;
end

else begin

// If counter is zero, toggle square_wave_reg
if (counter == 8'h00) begin
square_wave_reg <= ~square_wave_reg;

// Generate 1Hz Frequency
counter <= CLK_FREQ/2 - 1; 
end

// Else count down
else
counter <= counter - 1;
end
end
endmodule

8051 C program to generate square wave

#include <reg51.h> // including 8051 register file
sbit pin = P1^0; // declaring a variable type SBIT
for P1.0
main()
{
P1 = 0x00; // clearing port
TMOD = 0x09; // initializing timer 0 as 16 bit timer
loop:TL0 = 0xAF; // loading value 15535 = 3CAFh so after
TH0 = 0x3C; // 50000 counts timer 0 will be
overflow
pin = 1; // sending high logic to P1.0
TR0 = 1; // starting timer
while(TF0 == 0) {} // waiting for first overflow for 50 ms
TL0 = 0xAF; // reloading count again
TH0 = 0x3C;
pin = 0; // sending 0 to P1.0
while(TF0 == 0) {} // waiting for 50 ms again
goto loop; // continuing with the loop
}

8253 square wave generator

8253 is a programmable interval timer. It has 3 16-bit counters and operates in six modes. Each of the counters has three modes as -CLK(input click frequency), OUT(output waveform), and GATE(to enable or disable the counter). Mode 3 is known as the square wave generator mode. In this operating mode, the out is high when the count is loaded. The count is then gradually decremented. When it comes down to zero, the out becomes low, and again the count starts loading. Thus a square wave is generated.

Adjustable square wave generator

An adjustable square wave generator can be built using a potentiometer in place of a general voltage divider. As the resistor value is changeable, we can adjust the parameters of the square wave output.

Advantages of square wave generator

A square wave generator has the following advantages-

• The circuit can be easily designed. It does not need any complex structure.
• It is cost-effective.
• Maintenance of the square wave generator is very easy.
• A square wave generator can produce signals with maximum frequencies.

Comparator square wave generator

Comparator circuits that are efficient in hysteresis are used to make square wave generators. Hysteresis refers to the action of providing positive feedback to the comparator. This hysteresis occurs for Schmitt trigger and Logic gate square wave generators, and almost perfect square waves are generated.

High voltage square wave generator

The high voltage square wave generator can be made using a MOSFET (metal-oxide-semiconductor field-effect transistor). This square wave generator device is effective in producing square waves of different amplitudes.

Square to sine wave generator | square wave to sine wave generator

The square wave to sine wave converter circuit makes use of multiple RC networks. It has three resistors and three capacitors. The three-stage RC filter first changes the square wave into a triangular wave and then converts it into the sine wave. The values of the resistor and the capacitor decide the frequency of the square wave.

Square wave to sine wave generator circuit

Digital square wave generator

Digital function generators are one of the most preferred ways of generating square pulses. It is called direct digital synthesis (DSS). The components required for DSS are a phase accumulator, a digital to analog converter, and a look-up table containing waveforms. DSS generates an arbitrary periodic waveform from a ramp signal and thus generates a digital ramp. This technique is accurate and highly stable.

1 mHz square wave generator circuit

The Schmitt trigger oscillator circuit is one of the most effective ways to generate a 1 mhz square wave. The circuit comprises a couple of Schmitt inverters, a variable resistor, some capacitors, and resistors. 

Square wave generator chip

741 Operational amplifier IC is the most popular chip for the generation of square waves. Besides this, 555 timer IC is also used to make square wave generator circuits.

Square wave generator application | application of square wave generator

The applications of a square wave generator are-

• ‌It is used to generate square waves and other circuits that produce triangular or sinusoidal waves from square waves.
• ‌Square wave generators are useful in controlling clock signals.
• ‌It is used in musical instruments to emulate various sounds.
• ‌Function generators, Cathode Ray Oscilloscopes, make use of square wave generators.

FAQs

How do you find the frequency of a square wave generator?

For a square wave generator, T=2RC ln (2R₁+R₂/R₂). The frequency of the wave is determined from this equation.

Therefore, frequency f=1/2RC ln (2R₁+R₂/R₂)

What is the triangular waveform generator?

A triangular waveform generator is an electronic waveform generator circuit.

A triangular waveform generator generates triangular waves. Generally, a square wave generator combined with an integrator circuit produces triangular waves.

How can you generate square wave and triangular wave?

An astable multivibrator circuit is considered one of the best practices to generate square waves. It involves an operational amplifier, a capacitor, a resistor, and one voltage divider network.

We can use the output square wave achieved from an astable multivibrator as the input of an integrator circuit in order to generate square waves. Also, we can use a Schmitt trigger feedback circuit with an integrator to get triangular waves.

What are the applications of a square wave generator?

A square waveform generator is widely used in electronics.

Some useful applications of a square wave generator are-

• Clock Signals
• ‌Emulation of sound from various instruments
• ‌Sine wave/triangular wave converter circuits
• ‌Transistor switching
• ‌Amplifier response checking
• ‌Control system operations

I want to make a variable duty cycle square wave generator where input voltage is 12V. What will be the requirement and how to make it?

A square wave generator, combined with diodes can help in varying the duty cycle.

The square wave generator circuit given below allows us to make changes in the duty cycle. Two diodes are connected in parallel here, but in opposite directions. One diode starts working when the output is high, the other one comes into operation when the output is low. When the output is high, the D₁ diode starts operating. Similarly when the output is low, D₂ operates. Thus, the circuit goes to logic high and low and generates a square waveform.

The time period T=2RC ln (2R₁+R₂/R₂)

How to generate a square wave using an op-amp?

We know, there are numerous ways to generate a square wave.

An operational amplifier when used with a capacitor, a resistor and a voltage divider, produces output as square wave. The square wave generation happens when the output switches between the positive and the negative saturation voltage continuously.

How can I generate a square wave from a triangular wave by using only a resistor and capacitor?

We know, a differentiator circuit gives square wave as output when it takes triangular wave input.

So, to generate a square wave from a triangular wave, we can keep the capacitor in series with the source and ground the resistor first. By this, we can make a high-pass filter. If the frequency of the triangular wave is lesser than the cut-off frequency of the high-pass filter, then the filter differentiates the triangular wave and produces a square wave.

What is the equation of the square wave?

A square wave can be represented in different forms.

The most common equation of a square wave is –

x(t)=sgn(sin 2πt/T)=sgn(sin(2πft))

y(t)=sgn(cos 2πt/T)=sgn(cos(2πft))

Where, T= Time period and f=frequency of the wave.

We can modify the equation according to the conditions given.

How to convert a triangular wave into a square wave?

Square wave is nothing but the integral of a triangular wave.

To convert a triangular wave into a square wave, we can use a differentiator amplifier circuit. This circuit comprises an op amp, a capacitor and a resistor.

What happens if a square wave passes through a capacitor?

Different waveform generators use capacitor in their circuitry.

If a square wave passes through a capacitor, it can generate different types of waveforms according to the other circuit parameters.

What is the application of an audio frequency sine and square wave generator?

Musical instruments make use of high quality waveform generators.

An audio frequency sine and square wave generator is used as an audio oscillator. The circuit consists of a wein bridge oscillator which provides the best audio frequency range.

What is the difference between pulse wave and square wave?

Square wave is nothing but a subset of the pulse wave.

A square wave is a special type of pulse wave where the positive halves of the cycle equal the negative halves. A pulse wave with 50% duty cycle is said to be a square wave.

How to generate a trapezoidal waveform from an op amp?

We can generate a trapezoidal waveform in three steps.

This method gives almost a trapezoidal shaped waveform.

• ‌Generating a square wave
• ‌Converting the square wave into a triangular wave using an integrator
• Using clipper circuit to limit the voltage without affecting the rest of the waveform.

What is the advantage of using a square waveform as an input signal?

A square waveform is a periodic waveform which is non-sinusoidal in nature. The amplitude of a square wave have fixed maxima and minima at a particular frequency.

The main advantages of using a square waveform as an input signal is-

• ‌It has a wide bandwidth of frequencies.
• ‌Easy and quick visualization in an oscilloscope is possible with square waves.
• ‌Square waveforms can indicate issues to be fixed.

Does the LC circuit convert square wave output voltage to pure sinusoidal output? If so, what is the operation behind it?

An LC circuit is a network consisting of single or multiple inductor and capacitor.

Yes, LC filter circuits efficiently convert square waves into sine waves. The filter circuit allows only the fundamental frequency of the square wave to pass and filter out other high frequency harmonics. Thus the square wave gets converted into a sine wave.

Why we will get square wave as output in comparator circuit?

A comparator circuit compares an AC sinusoidal signal with a DC reference signal.

The input signal upon becoming larger than the reference signal, yields a positive output. When it is less than the reference signal, the output is negative. In both the scenarios, the difference of the signals is so large that it is considered to be equivalent with the maximum possible output (±V_sat). So, it is assertive that the output continuously dangles between positive and negative saturation voltage. That’s why we get square waves as comparator output.

How do I generate a square wave for different duty cycles in 8051 using embedded C?

#include<reg51.h>
sbitpbit=PI^7;
void delay_on();
void delay_off();
void main()
{
TMOD=0x01;  //initializing timer 0 in mode 1
 while(1);        // repeating this
delay_on();   //800 microsecond delay
pbit=0;            //output pin low
delay_off();  //200 microsecond delay
}
}
//function for 800 microsecond delay
Void delay_on()
{
TH0=OxFD;
TR0=1;   //turning the timer 0 ON
while(!TF0);   //waiting for timer overflow
TR0=0;      //switching the timer 0 OFF
TF0=0;      //clearing the overflow flag
}
//function for 200 microsecond delay
Void delay_off()
{
TH0=OxFF;
TL0=0x48;
TR0=1;  
while(!TF0);   
TR0=0;     
TF0=0;     
}   //clearing TF0

How do we write an embedded C code to generate a square wave of 50 Hz?

#include<reg51.h>
void delay(int time)
{
int i,j;
for(i=0;i<time;i++)
for(j=0;j<922;j++);
}
void main()
{
while(1)
{
p1=255;
delay(10);
p1=0;
delay(10);
}
}

The answer is “Yes”, light can bend around corners.

When light passes around the edges of an object it tends to bend its path around the corners. This property of light is known as diffraction. The phenomenon of diffraction depends on the propagation of light. For studying this phenomenon, light is treated as a wave.  

Contents:

• What is diffraction of light?
• How can light bend around corners?
• Diffraction in Atmosphere
• Examples of diffraction seen in everyday life
• Does light travel in a straight line? If so, how?
• How is interference related to water waves?
• How can light bend around corners inside an optical fiber?
• What is the condition of maximum deviation of light in prism?
• Difference between diffraction and scattering?
• Is it possible for an incident ray to have an angle of more than 90 degrees?

What is diffraction of light?

DIffraction of light refers to the phenomena of bending of light waves around the corners of an obstructing object having a size comparable to the wavelength of light. The phenomenon of diffraction depends on the propagation of light. For studying this phenomenon, light is treated as a wave.

The degree or extent by which the light rays bend is dependent upon the size of the obstructing object and the wavelength of light. When the size of the object is much larger compared to the wavelength of light then the extent of bending is negligible and cannot be noticed properly. However, when the wavelength of light is comparable to the size of the obstructing object (such as a dust particle) then the extent of diffraction is high i.e. the light waves bend at larger angles. In such cases, we can observe the diffraction of light with the naked eye.

Let us learn more about how light bends around corners:

How can light bend around corners?

According to classical physics, the phenomenon of diffraction is experienced by a light wave because of the way it propagates. The phenomenon was described by Christiaan Huygens and Augustin-Jean Fresnel in the Huygens-Fresnel principle and the principle of superposition of waves. Lightwave propagation can be visually interpreted by taking every single particle in the medium of propagation as a point source that gives rise to the secondary wavefront of a spherical wave.

The displacement of the waves from every point source gets added up to form a secondary wave. Amplitudes and relative phases of every wave play an important role in determining the subsequent spherical wave formed. The amplitude of the resultant wave can take any value lying between 0 and the addition of the individual amplitudes of the point sources.

Therefore, a general diffraction pattern consists of a series of minima and maxima.

According to modern Quantum optics, every Photon that passes through a thin slit gives rise to its own wave function. This wave function depends on several physical factors such as the dimensions of the slit, the distance from the screen, and the initial conditions of the photon generation. 

The diffraction phenomenon can be qualitatively understood by taking into consideration the relative phases of the secondary waves fronts. The superposition of two half circles of waves results in constructive interference. When two half circles of waves cancel each other out, it results in destructive interference.

Diffraction in Atmosphere:

Light gets diffracted in the atmosphere by bending around the atmospheric particles. Usually, the light gets diffracted by the tiny water droplets suspended in the atmosphere. The bending of light can give rise to light fringes light, dark, or colored bands. The silver lining that can be observed around the edges of clouds or the coronas of the moon or the sun is also a result of the diffraction of light. 

Examples of diffraction seen in everyday life

Some examples of diffraction or bending of light can be seen often in our day to day life such as:

CD or DVD: In a CD or DVD disc we can often see the formation of a rainbow-like pattern. This rainbow-like pattern is formed due to the phenomenon of diffraction. Here, the CD or DVD acts as a diffraction grating. 

Hologram: A hologram is designed such as to produce a diffraction pattern. Such holograms are often seen in credit cards or book covers. 

Laser beam propagation: The change in the beam profile of a laser beam as determined by the phenomenon of diffraction that occurs when the laser beam propagates through a medium. The lowest recorded divergence due to the fraction is provided by a planar spatially coherent wavefront with a Gaussian beam profile. Generally, the larger the output beam, the slower is the divergence.

The extent of divergence of a laser beam can be reduced by first diverging the beam with the help of a convex lens and then converging or collimating the beam with the help of a second convex lens having a focal point coinciding with the focal point of the first convex lens. In this way, the resultant beam will have a larger diameter compared to the original beam and hence, the divergence would be reduced.

Diffraction limited imaging: Diffraction limits the resolving power of an imaging system. Due to distraction, the light beam is unable to focus at a single point. Instead, the formation of an error disk takes place which has a central bright spot with a concentric circle surrounding it. It is seen that with a larger aperture the lenses are able to resolve images more finely. 

Single-slit diffraction: The diffraction of a long slit with negligible width is taken. The slit is then illuminated with a point source of light. After passing through the slit the light gets diffracted into a series of circular wavefronts. The slit is wider than the wavelength of light then it can produce interference patterns in the space that lies below the slit.

The concept of bending of light might induce certain questions in people’s minds. Let us have a look at some of those questions:

Does light travel in a straight line? If so, how?

Light is an electromagnetic wave and therefore it travels in the form of a wave. However, the wavelength of light is very small. Hence, a light wave is approximately taken as a ray that travels in a straight line. The wave property of light can be observed only when it interacts with objects having a size comparable to the wavelength of light. For the objects in our day-to-day life, the interaction with light is taken as rays that travel in a straight line. For smaller objects, light bends around corners due to diffraction.

How is interference related to water waves?

The interference of light waves causes the optical effects resulting from the bending of light. We can visualize this fact by imagining the waves of light as water waves. Supposed you keep a wooden plank on a water surface to float, you will notice that the water waves would make the wooden plank bounce up and down in accordance with the incident water waves. These water waves further spread out in every direction and interfere with the neighboring water waves.

When the crests of two water waves merge it leads to the formation of an amplified wave i.e. constructive interference takes place. However, when the trough of a wave interferes with the crest of another wave, they cancel each other out resulting in a null amplitude that has no vertical displacement i.e. destructive interference. When the troughs of two separate waves interfere they form a more depressed trough.

This same pattern is observed in the case of light waves. When the light from the sun encounters droplets of water suspended in the atmosphere, the light waves interact with each other in a manner similar to that mentioned above in the case of water waves. In the case of light waves, constructive interference takes place when the peak amplitude of two light waves interact to produce a more amplified wave.

In other words, when two crests of light waves interact or interfere they form a brighter pattern. Destructive interference occurs when the trough of a light wave interferes with the crest of another wave. This destructive interference is observed by the formation of a darker pattern. 

How can light bend around corners inside an optical fiber?

Light rays get refracted after entering the optical fiber material.

The light waves propagate through The optical fiber core by getting refracted back and forth from the boundary or the interface between the core and the cladding. Light propagates through the optical fiber without passing or transmitting through the fiber by a phenomenon of total internal refraction. 

Total internal reflection can take place only when the angle of the incident light on the boundary of the optical fiber is greater than the critical angle of the fiber. When the angle is greater than the critical angle the light gets refracted into the optical fiber instead of leaking out through the cladding.

What is the condition of maximum deviation of light in prism?

The maximum deviation of light in a prism can be possible due to the following two conditions:

1. The maximum deviation of light can take place only if the angle incident on the prism is a right angle i.e. 90 degrees. This property is also known as grazing incidence due to the fact that the light rays almost “graze” along the surface of the prism.

2. The second condition for maximum deviation of light in a prism is that when an emerging ray gets reflected at 90 degrees or we can say that it grazes along the surface of the prism. This condition is similar to the condition mentioned above for the second surface.

Note: we should not confuse the maximum deviation angle with the angle of minimum deviation of a prism.

What is the difference between scattering and diffraction?

Scattering of light: Scattering of light occurs when light strikes small objects such as dust particles or gaseous molecules of water vapor, it tends to get deviated from its straight path of propagation. This phenomenon is termed the scattering of light. Scattering of light can be noticed or observed in several environmental phenomena. The blue color of the sky, the white color of clouds, the red color of the sky during sunset and sunrise, the Tyndall effect, etc. are some examples of scattering of light.

Traffic lights or danger signals are usually red in color because red scatters the least out of all wavelengths. The extent of scattering is inversely proportional to the fourth power of wavelength of light. The phenomenon of scattering can be observed as wave interactions and particle interactions both. The property of scattering is linked with wave interactions.

Diffraction of light: diffraction of light refers to the phenomenon by which light rays tend to bend around the corners of an object having a size comparable to the wavelength of light. Diffraction is observed only by treating light as a wave only. The property of diffraction is linked with wave propagation. The pattern interference pattern observed during single slit experiment, gratings, holograms excreta occur due to diffraction.

Is it possible for an incident ray to have an angle of more than 90 degrees?

The angle of incidence to a surface is defined as the angle made by the light ray from the normal to the point it touches. Therefore, the maximum angle that can be made with the normal to the surface is 90 degrees on either side.

We hope this post answered your queries regaing the phenomenon of diffraction.

This is a sequential post related to Co-ordinate Geometry, specially on Points. We already discussed few Topic earlier in the post “A Complete Guide to Coordinate Geometry”. In this post we will discuss the remaining topics.

Basic Formulae on Points in Co-ordinate Geometry in 2D:

All the basic formulae on points in Analytical Geometry are described here and for easy and quick learning at a glance about the formulae a ‘Formulae Table on Points’ with graphical explanation is presented below.

Two points distance formulae | Analytical Geometry:

Distance is a measurement to find how far objects, places etc. are from each other. It has a numerical value with units. In Co-ordinate Geometry or Analytical geometry in 2D, there is a formula which is derived from Pythagorean theorem ,to calculate the distance between two points. we can write it as ‘Distance’ d =√ [(x₂-x₁)²+(y₂-y₁)² ] , where  (x₁,y₁) and (x₂,y₂) are two points on the xy-plane. A brief graphical explanation is followed by ‘Formulae Table on Points topic no 1’ below.

A distance of a point from origin | Co-ordinate Geometry:

If we starts our journey with Origin in xy-plane and end up with any point of that plane, the distance between origin and the point can also be find by a formula, ‘Distance’ OP=√ (x² + y²), which is also a reduced form of “Two points distance formula” with one point at (0,0). A brief graphical explanation is followed by ‘Formulae Table on Points topic no 2’ below.

Points section formulae |Coordinate Geometry :

If a point divides a line segment joining two given points at some ratio, we can use section formulae to find the coordinates of that point while the ratio at which the line segment is divided by, is given and vice versa. There is a possibility that the line segment can be divided either internally or externally by the point. When the point lies on the line segment between the two given points, Internal section formulae is used i.e.

and

And when the point lies on the external part of the line segment joining the two given points, external section formulae is used i.e.

Where (x , y) is supposed to be the required coordinates of the point. These are very necessary formulae to find the centroid, incenters ,circumcenter of a triangle as well as the center of mass of systems, equilibrium points etc. in physics. Must watch the short view of different types of section formulae with graphs given below in the ‘Formulae Table on Points topic no 3; case-I and case-II’.

Mid Point Formula| coordinate Geometry:

It is a easy formulae derived from Internal points section formulae described above. While we need to find the midpoint of a line segment i.e coordinate of the point which is equidistant from the two given points on the line segment i.e the ratio gets 1:1 form, then this formula is required. The formula is in the form of

If a point divides a line segment joining two given points at some ratio, we can use section formulae to find the coordinates of that point while the ratio at which the line segment is divided by, is given and vice versa. There is a possibility that the line segment can be divided either internally or externally by the point. When the point lies on the line segment between the two given points, Internal section formulae is used i.e.

and

And when the point lies on the external part of the line segment joining the two given points, external section formulae is used i.e.

        and

Where (x , y) is supposed to be the required coordinates of the point. These are very necessary formulae to find the centroid, incenters ,circumcenter of a triangle as well as the center of mass of systems, equilibrium points etc. in physics. Must watch the short view of different types of section formulae with graphs given below in the ‘Formulae Table on Points topic no 3; case-I and case-II’.

Mid Point Formula| coordinate Geometry:

It is a easy formulae derived from Internal points section formulae described above. While we need to find the midpoint of a line segment i.e coordinate of the point which is equidistant from the two given points on the line segment i.e the ratio gets 1:1 form, then this formula is required. The formula is in the form of

and

Go through the “Formulae Table on Points topic no 3- case-III’ below to get the graphical idea on this.

Area of a triangle in Coordinate Geometry:

A triangle has three side and three vertices on the plane or in 2 dimensional field. Area of the triangle is the internal space surrounded by these three sides. The basic formula of area calculation of a triangle is (1/2 X Base X Height). In Analytical Geometry ,if the coordinates of all the three vertices are given, the area of the triangle can easily be calculated by the formula, Area of the Triangle   =|½[x₁ (y₂–  y₃ )+x₂ (y₃–  y₂)+x₃ (y₂-y  ₁)]| ,actually this can be derived from the basic formula of area of a triangle using two points distance formula in coordinate geometry. Both the cases are graphically described in the ‘Formulae Table on Points topic 4’ below.

Collinearity of points ( Three points) |Coordinate Geometry:

Collinear means ‘being on the same line’. In geometry, if three points lie on one single line in the plane, they never can form a triangle with area other than zero i.e. if the formula of area of triangle is substituted by the coordinates of the three collinear points, the result for area of the imaginary triangle formed by those points will end up with zero only. So the formula becomes like ½[x₁ (y₂–  y₃ )+x₂ (y₃–  y₂)+x₃ (y₂-y  ₁)] =0 For more clear idea with graphical representation, go through the “Formulae Table on Points topic no 5” below.

Centroid of a triangle| Formula :

The three medians* of a triangle always intersect at a point, located in the interior of the triangle and divides the median in the ratio 2:1 from any vertex to the midpoint of the opposite side. This point is called the centroid of the triangle. The formula to find the coordinates of the centroid is

and

In the “Formulae Table on Points topic no 6” below, the above subject is described graphically for better understanding and for a quick view.

Incenter of a triangle|Formula:

It is the center of the triangle’s largest incircle which fits inside the triangle. It is also the point of intersection of the three bisectors of the interior angles of the triangle. The formula, used to find the incenter of a triangle is     

and

In the “Formulae Table on Points topic no 6” below, the above subject is described graphically for better understanding and for a quick view.

For easy graphical explanation the below “Formulae Table on Points topic no 7” is needed to see.

Shifting of Origin formula| Coordinate Geometry:

We have already learned in the previous post “A Complete Guide to Coordinate Geometry” that the origin lies on the point (0,0) which is the point of intersections of the axes in the plane. we can move the origin in all the quadrants of plane in respect with the origin , which will give new set of axes through it.

For a points in the above said plane , its coordinates will change along with the new origin and axes and that can be calculated by the formula, new coordinates of a point P(x₁,y₁) are x₁ = x- a ; y₁ = y-  b where the coordinates of the new origin is (a,b). To have clear understanding on this topic its preferable to see the graphical representation below in the “Formulae Table on Points topic no 8” .

Formulae table on Points in Coordinate Geometry in 2D:

﹡Circumcentre of a triangle :

It is the point of intersection of three perpendicular bisectors of the side of a triangle. It is also the center of a triangle’s circumcircle which only touches the vertices of the triangle.

﹡Medians:

Median is the line segment joining the vertex of triangle to the midpoint or the point, bisecting the opposite side of the vertex. Every triangle has three medians which always intersect each other at the centroid of the same triangle.                                                         

Solved Problems on Points in Coordinate Geometry in 2D.

For better learning about points in 2D, one basic example is solved here step by step and for practice on your own there are more problems with answers on each formula. There must be challenging problems with solution in the next articles just after getting a basic and clear idea on the topic of points in coordinate Geometry 2D.

Basic Examples on the Formulae “The distance between two points”

Problems 1:  Calculate the distance between the two given points (1,2) and (6,-3).

Solution: We already know, the formula of the distance  between two points  (x₁,y₁) and (x₂,y₂)  is d =√ [(x₂-x₁)²+(y₂-y₁)² ] …(1)                                                                                                                    

(See the formulae table above)   Here, we can assume that (x₁,y₁) ≌ (1,2) and (x₂,y₂) ≌ (6,-3) i.e x₁=1, y₁=2 and x₂=6, y₂ =-3 , If we put all these values in the equation (1),we get the required distance.

Therefore, the distance between the two points (1,2) and (6,-3) is

=√ [(6-1)²+(-3-2)² ]   units

= √ [(5)²+(-5)² ]  units

=√ [25+25 ] units

=√ [50 ]        units

=√ [2×5² ]       units

= 5√2   units (Ans.)

Note: Distance is always followed by some units.

More answered problems (Basic) are given below for further practice using the procedure described in above Problem 1:-

Problem 2: Find the distance between the two points (2,8) and (5,10).               

Ans. √13 units

Problem 3: Find the distance between the two points (-3,-7) and (1,-10).           

Ans. 5 units

Problem 4: Find the distance between the two points (2,0) and (-3,4).               

 Ans. √41 units

Problem 5: Find the distance between the two points (2,-4) and (0,0).                

Ans. 2√5 units

Problem 6: Find the distance between the two points (10,100) and (-10,100,). 

                                                                                                                               Ans. 20 units

Problem 7: Find the distance between the two points (√5,1) and (2√5,1).          

Ans. √5 units

Problem 8: Find the distance between the two points (2√7,2) and (3√7,-1).       

Ans. 4 units

Problem 9: Find the distance between the two points (2+√10, 0) and (2-√10, 0).   

                                                                                                                              Ans. 2√10 units

Problem 10: Find the distance between the two points (2+3i, 0) and (2-3i, 10).  { i=√-1 }

                                                                                                                                 Ans. 8 units

Problem 11: Find the distance between the two points (2+i, -5) and (2-i, -7).  { i=√-1 }

                                                                                                                                  Ans. 0 units

Problem 12: Find the distance between the two points (7+4i,2i) and (7-4i, 2i).  { i=√-1 }

                                                                                                                                   Ans. 8i units

Problem 13: Find the distance between the two points (√3+i, 3) and (2√3+i, 5). { i=√-1 }  

                                                                                                                                Ans. √7 units

Problem 14: Find the distance between the two points (5+√2, 3+i) and (2+√2, 7+2i). { i=√-1 } 

                                                                                                                           Ans. 2√(6+2i) units 

Basic Examples on the Formulae “The distance of a point from the origin”

Problems 15: Find the distance of a point (3,4) from the origin.

Solution:                                                                                                

 We have the formula of the distance of a point from the origin,  OP=√ (x² + y²) (See the formulae table above) So here we can assume (x,y) ≌ (3,4) i.e x=3 and y=4                                                                                            

Therefore, putting these values of x and y in the above equation, we get the required distance 

=√ (3² + 4²)    units

=√ (9 + 16)    units

=√ (25)         units

= 5  units

Note: Distance is always followed by some units.

Note: Distance of a point from the origin is actually the distance between the point and the point of origin i.e (0,0)

More answered problems are given below for further practice using the procedure described in above

Problem 15:-

Problem 16: Find the distance of a point (1,8) from the origin.                              

Ans. √65 units

Problem 17: Find the distance of a point (0,7) from the origin.                              

Ans. 7 units

Problem 18: Find the distance of a point (-3,-4) from the origin.                            

Ans. 5 units

Problem 19: Find the distance of a point (10,0) from the origin.                             

Ans. 10 units

Problem 20: Find the distance of a point (0,0) from the origin.                               

Ans. 0 units

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Basic Examples on other Formulae of points described above and few challenging questions on this topic in coordinate Geometry, are followed by next posts.