In this article we are going to discuss 9 facts related to magnetic field and distance.

**Let’s come to the relation between magnetic field and distance. Basically there comes a very important law which is known as inverse square law. Magnetic monopoles and dipoles show the same relationship between the magnetic field and distance that is- magnetic field strength usually increases when the distance between the field and source decreases.**

Whether it is a magnetic monopole or dipole magnetic field always varies inversely with the cube of the distance. Let’s assume that the magnetic field is B and the distance is R,then the relation between them is: B ∝ 1/R³ In a very simple way we can describe it with the help of an example.

Let’s take a magnet. Its field at distance 1 m is B. So its magnetic field at 3 m distance is 1/27 th of the previous value B. Hence the value of the magnetic field at 3 m distance will fall off to B/27[as the value of magnetic field falls off as the cube of the distance]. For some special magnets magnetic field falls off more faster than this,but for most magnets magnetic field varies inversely with the cube of distance.

**How does distance affect magnetic force?**

**In simple words it can be said that if two same poles of two magnets are brought closer to each other the force of repulsion between them will become more and more stronger. Similarly force of attraction between the two poles will become stronger when two opposite poles of two magnets are brought closer to each other.**

What is the reason behind it? The reason is magnetic force varies inversely with distance between the two magnets. Whenever they are brought closer to each other the magnetic force becomes very strong as the distance is less. Conversely whenever they are brought far from each other the magnetic force becomes very less as the distance is increased.

Magnetic force is quite similar to coulomb force. We know that in case of coulomb force the force varies inversely with the square of the distance,similarly magnetic force also varies inversely with the square of the distance. If F is the magnetic force and R is the distance from the source then F ∝ 1/R².

The magnetic force will become F1 ∝ 1/(2R)² when the distance between the two magnets becomes twice the previous value i.e, 2R can be concluded here.

⇒F1 ∝ 1/4R²

It means that the magnetic force has become ¼ th of the previous value. When the distance between the two magnets becomes thrice the previous distance R i.e, 3R the magnetic force will become F2∝ 1/(3R)².

⇒F2 ∝ 1/9R²

It means that the magnetic force has become 1/9 th of the previous value.

If the distance has become five times the previous value R i.e, 5R the magnetic force will become F3 ∝ 1/(5R)².

⇒ F3 ∝ 1/25R²

It means that the magnetic force has become 1/25 th of the previous value.

**What is the relationship between magnetic field and distance?**

**The magnetic field and distance possess an inverse relation in between them. So whenever we go far from a magnet, necessarily there will be a change in the magnetic effect or more specifically magnetic field,it decreases. Similarly when we go towards the magnet,the magnetic field increases.**

If we take an example of a wire then the strength of the magnetic field will show a fall with the increase in the distance. In this article we will give some examples of how magnetic fields vary with distance for different current carrying conductors. They are-

**Infinitely long wire****Solenoid****Toroid****Circular loop or current carrying coil**

**1. Infinitely long wire**

**In case of an infinitely long current carrying wire, if the magnetic is B at a distance R from the long current carrying wire then B = µ₀I/2πR where µ0 is the magnetic permeability in free space whose value is 4π x 10⁻⁷ H.m.It means that magnetic field for a infinitely long current carrying wire is inversely proportional to the distance R, B ∝ 1/R.**

#### **2. Solenoid**

** In case of a long current carrying solenoid the strength of the magnetic field does not depend on the distance of the solenoid from the axis. It only depends upon the current that is flowing through the solenoid(I) and the number of turns(N) around the solenoid. Magnetic field,B = µ _{0}nI. It means that magnetic field B is proportional to the current I, B = µ_{0}nI where n= N/L,L is the length of the solenoid.**

We used to calculate the value of the magnetic field of a solenoid with the help of Ampere’s circuital law,i.e, ∲B.dl = µ_{0}NI

Or, B∲dl = µ_{0}NI

Or, B.L = µ_{0}NI where ∲dl = L = length of the solenoid

Or, B = µ_{0}NI/L = µ_{0}nI where n=N/L

#### **3. Toroid**

**In case of a toroid as the number of turns around the toroid that are outside is zero hence the mathematical formula for magnetic field becomes B = µ _{0}nI = 0. Similarly the number of turns around the toroid that are inside is also zero hence the magnetic field of a toroid inside it is also zero. Distance has no effect on the field in this case also.**

** 4. Circular loop or current carrying coil**

**The magnetic field of a current carrying circular coil depends upon the distance of the coil from the axis(x) and also on the value of the radius of the circular coil(R). The mathematical expression of the magnetic field B is, B = µ _{0}NI/2 x R²/(√(R + x))³**

**How to calculate magnetic field from distance?**

**First we can take the example of a toroid. We already know that using Ampere’s circuital law we can calculate the value of magnetic field of a toroid inside and outside of it. According to the Ampere’s circuital law, ∲B.dl = µ0NI**

Or, B∲dl = µ_{0}NI

Or, B.(2πR) = µ_{0}NI

Or, B = µ_{0}NI/2πR where ∲dl = 2πR and R is the radius of the toroid. Inside the toroid and outside the toroid the number of turns is zero,hence the magnetic field is also zero.

For a long solenoid,we used to calculate the value of the magnetic field of a solenoid with the help of Ampere’s circuital law,i.e, ∲B.dl = µ0NI

Or, B∲dl = µ_{0}NI

Or, B.L = µ_{0}NI where ∲dl = L = length of the solenoid

Or, B = µ_{0}NI/L = µ_{0}nI where n(the number of turns per unit length of the solenoid)=N/L ,L is the length of the solenoid N is the number of turns of the solenoid.

**Is the magnetic field inversely proportional to distance?**

**Here we will be discussing why,how and when magnetic field becomes inversely proportional to distance. The cases where magnetic field is inversely proportional to the distance will be derived here.**

We should take here the example of an infinitely long current carrying wire. Let us imagine an amperian loop i.e,a circle around the wire. The radius of the circle is R and the distance between the imaginary circle and the infinitely long wire is r. We will calculate the magnetic field for two regions. One is r > R and the other is r < R.

Case I

**(r > R)**

According to the Ampere’s circuital law,∲B.dl = µ_{0}I

Here B∲dl= µ_{0}I

Or, B.(2πr) = µ_{0}I where ∲dl = 2πr

Or, B = µ_{0}I/2πr where I is the current enclosed by the loop. Therefore, B ∝ 1/r (r > R) it means that magnetic field is inversely proportional to the distance r.

Case II

( r < R)

Here the amperian loop is taken inside. Hence the value of the current enclosed has become Ien. Therefore, I_{en} = I.(πr²/πR²)= Ir²/R²

According to the ampere’s circuital law ∲B.dl = µ_{0}I_{en}

Or, B.(2πr) = µ_{0}.Ir²/R²

Or, B = µ_{0}.Ir/2πR² (r<R)

In this case magnetic field B is proportional to the distance r i.e, B ∝ r.

**How much does a magnetic field decrease with distance?**

**We will take an example of a current carrying coil here whose radius is R to show how much does a magnetic field decrease with distance. magnetic field of this current carrying coil is B = µ _{0}NI/2 x R²/(√(R² + x²))³ where x is the distance between the coil and the point whose magnetic field is going to be calculated.**

If the distance x is zero then the magnetic field becomes B = µ_{0}NI/2R i.e, magnetic field is inversely proportional to the distance R. at the center of the coil. Now using this formula we will be able to show how much a magnetic field decreases with distance. B ∝ 1/R when the distance R becomes twice the previous value that is 2R,then the magnetic field B becomes ½ of its previous value that is B/2.

Similarly when R becomes four times the previous value that is 4R,then the magnetic field B becomes ¼ of its previous value that is B/4.

If the distance R becomes half of its previous value that is R/2,then the magnetic field B becomes twice of its previous value that is 2B and If the distance R becomes one third of its previous value that is R/3,then the magnetic field B becomes thrice of its previous value that is 3B.

**How does the magnetic field change as the separation distance between the coils increases?**

**Let us take a pair of coils. One between the two is stationary and the other is moving. The magnetic flux which is linked with the stationary coil will be decreased if the distance between the moving coil and the stationary coil increases. Similarly the magnetic flux which is linked with the stationary coil is going to be increased if the distance between the moving coil and the stationary coil is decreased.**

As we know that magnetic flux,Φ = B.A where B is the magnetic field of the coil. Hence Φ ∝ B. it means that when the distance between the two coils increases the magnetic field also decreases whereas the magnetic field increases when the distance between the two coils decreases.

This magnetic flux Φ is also related to the mutual inductance M of the pair of coils. Φ= M.I where I is the current flowing through the coils. So when the distance between the two coils increases mutual inductance will decrease and when the distance between the two coils decreases mutual inductance will increase as mutual inductance is proportional to the magnetic flux.

**How to find current with magnetic field and distance?**

**Let us take one example of an infinitely long current carrying conductor. The mathematical expression of the magnetic field is B = µ _{0}I/2πR. The value of the current that is passing through this conductor can be calculated from this mathematical formula. **

**I = 2πR.B/µ _{0}.**

Let’s calculate the current value of a mathematical problem.

**The magnetic field strength of an infinitely long wire is 4 x 10 ^{-4} T. What will be the value of the current if this field is perpendicular to the distance of 0.08 m?[ µ_{0}= 4π x 10^{-7} T.m/A].**

Answer

B = 4 x 10^{-4} T

R = 0.08 m

µ_{0}= 4π x 10^{-7} T.m/A

I =?

current,I = 2πR.B/µ₀ = (2 x π x 0.08 x 4 x 10^{-4})/4π x 10^{-7}

= 160 A

In this way we can calculate the value of the current using the mathematical expressions of magnetic fields of solenoids,toroids,currents carrying circular loops etc.

**Conclusion**

In this article we have discussed the relation between the magnetic field and distance as well as the relation between magnetic force and distance. Besides how much a magnetic field changes with distance is also discussed.