# Spherical Mirror: 15 Interesting Facts To Know

## Spherical Mirror definition :

Certain mirrors are designed in such a way that their reflecting surface is curved. Spherical mirrors are a variant of curved mirror that has their reflecting surfaces shaped like a portion of a sphere.

## Curved Spherical Mirror

This is a spherical mirror with a curved reflecting surface.

## Different types of spherical mirrors | Two types of spherical mirror:

The curved mirrors generally have their surfaces shaped in two designs:

• Convex surface that has an outward bulge.
• Concave surface that has an inward recession.

## Example of Spherical Mirror

There are two types spherical mirrors, concave and convex, Apart from spherical surfaces, curved mirror are also found in some other shapes such as parabolic mirrors and other curved surfaces.

## Concave Spherical Mirror Definition:

A concave mirror is also best-known as a converging mirror as it is capable of converging light rays to a point. A concave spherical mirror is designed such that the reflecting surface is depressed inwards i.e. the incident light has to travel a bit more to reach the center of the mirror and less to reach the margin of the mirror and light rays get converged inwards way to the focal point of the concave mirror. These mirrors are especially used for focusing light rays.

The imaging pattern of concave mirrors are different from that of convex mirrors. In concave mirrors the image form differs based on the distance in between the spherical concave mirror and the position of the object. The incident light falling on different points on the mirror is reflected by the mirror at different angles because the normal at each point on the mirror is different. The Concave mirror is capable to produce both real and virtual images, for this type of mirror, focus (F) and the center of curvature (C) are lies outside the mirror and are termed as “real points”.

## Application of Spherical mirror (Concave):

• Reflecting telescopes are designed by using one or more concave mirrors.
• These mirrors help in producing magnified images of objects and are hence, used as make-up or shaving mirrors.
• In certain illumination applications, the light bulb is placed at the focal point of the concave mirror and the light directed from the focal point then gets reflected outwards after hitting the mirror. This application is seen in torches, headlamps and spotlights.
• At times, concave mirrors are also used for concentrating solar energy by converging the incident sunlight from a large area to a tiny point.
• Lasers also use concave mirrors for building optical cavities. that is necessary for lasing action.
• Due to the ability of concave mirrors to form magnified images, it is used by dentists as dental mirrors.
• Modern aircraft transporters also incorporate concave spherical mirrors in the mirror landing aid arrangement.

## Image Formation by Concave Mirror

Concave mirrors capable to create both real and virtual image depending upon the distance of the object from the mirror position.

1. When the object is placed between the mirror and the focal point (F), the image formed is virtual, erect, and magnified and the image will be formed behind the mirror.
2. When the object is placed at the focal point, the reflected light rays follow parallel propagation and are said to meet at infinity. Therefore, the image is said to form at infinity with large magnification. The image can be both real or virtual depending on whether the object approaches the focus from the mirror or from the center of curvature.
3.  When the object is located in-between the center of curvature (C) and the focal point (F), the image created is real, invert, and magnified in nature, and this image is usually formed beyond the center of curvature.
4. If the object is located at the center of curvature of the mirror, then the formed image is real, inverted, and has the same size as the object and will be created on the center of curvature itself.
5. If the object is placed far from the center of curvature of the mirror, then the formed image is real, inverted, and diminished, and the image will be formed in-between the center of curvature and the focus.
6. If the object is located at infinity,  then the formed image is real, inverted and point-sized and will be created on the focus.

## Convex Spherical Mirror Definition:

A convex mirror is also known as a diverging mirror as it is capable of diverging or spreading out light rays from a point. A convex spherical mirror is designed such that the reflecting surface bulges outwards i.e. the incident light has to travel a bit more to reach the margin of the mirror and less to reach the center of the mirror. The light rays appear to get diverged outwards from the focal point of the convex mirror. These mirrors are especially used for diverging light rays.

Convex mirrors can produce virtual images only. The focus and the center of curvature of the convex mirror has been stays inside the mirror and are generally as termed “imaginary points”. The image produced is seen to be present inside the mirror and cannot be projected on a screen and the size of formed-image is always lesser than object’s size, but the formed-image size will increase if the object distance decrease from mirror location. The incident light falling on different points on the mirror is reflected by the mirror at different angles because the normal at each point on the mirror is different.

## Application of Spherical mirror (Convex):

• The rear-view mirror used in vehicles are typically convex mirrors as these mirrors provide a wider field view, and erect images. However, these images can be deceptive and the objects may appear to be farther than they are actually.
• The wider field of view of these mirrors make it suitable as safety mirrors used in building hallways, hospitals, offices, hotels, schools, malls, apartment complexes, etc. These mirrors show whether there is any kind of obstruction present in the path ahead.
• Convex mirrors are also mounted on a pole in roads, driveways, alleys, and bridges when there is any kind of obstruction, sharp turn, narrow roads, etc. These mirrors help to prevent accidents that happen when the driver is unable to see the turns or an incoming car from the opposite side properly.
• Automated teller machines or ATMs also have convex mirrors fitted on its top left or right corners so that the users are able to know what is going on behind them. These mirrors form smaller images and therefore, provide a much larger area of observation.

## Image formation by convex mirror

Convex mirror can produce virtual images only i.e. the light rays coming from the object doesn’t actually pass through the image formed. However, if we extend the light rays, they appear to passing thru the image and this type of mirror forms a smaller or diminished, and erect images.

When, the distance in-between the object and the mirror decrease, then the size of the mirror will increase. At the point where the object touches the mirror, the size of the image is almost equal to the object size.  When the extended light rays appear to pass through the focus, the image formed is point sized and the object is said to be placed at infinity.

## Define Aperture of spherical mirror:

It is defined as the portion of the mirror that is available for interaction with the light rays, from this idea of aperture size of the mirror can be predicted.

## Define pole of a spherical mirror:

This is characterized by the center of the mirror’s total reflecting surface.

## Define principal focus of a spherical mirror | Focal Point of Spherical Mirror | Define Focus of a Spherical Mirror

This is on the axis of a spherical mirror where after reflection of rays of light parallel to the axis will converge or start to converge.

## Define focal length of a spherical mirror:

It is defined as the distance from the pole of the mirror to the point on the principal axis where the light rays incoming from infinity meets or appears to meet after reflection because of spherical mirror.

## Define Centre of Curvature of a spherical mirror:

This denotes the center of the sphere of that the spherical mirror is a part-of, it is usually denoted by the ‘C’.

## Define principal axis of a spherical mirror:

This refers that specific line which passing thru the center of curvature-C, pole-P, and focus-F of the spherical mirror.

## Define radius of curvature of spherical mirror:

This defined as the distance in-between the pole of the mirror and the center of curvature and It is usually twice of the focal length of the mirror (2F).

## Define Marginal rays:

Marginal rays are defined as the rays that strikes the spherical mirror after making the maximum angle from the principal axis. For geometrical optics calculations, marginal rays are often neglected.

## Define Paraxial rays:

Paraxial rays are defined as the light rays that strikes the spherical mirror after making an angle less than or equal to 14° with the principal axis. For geometrical optics calculations, only paraxial rays are taken into consideration.

## Spherical Mirror Equation | Concave Spherical Mirror equation

In usual term, we consider that for a spherical mirror

f – focal length.

do – distance of the object from the spherical mirror location.

di – distance of the image from the spherical mirror location.

## Spherical Mirror formula

Now, according to Gaussian optics, the equation of spherical mirror, correlating object distance, image distance, and focal length is given by:

Here, the rays are taken to be paraxial and the aperture is considered to be small. The light is incident on the mirror from the left side and the right side of the mirror is silvered.

## Derivation of Spherical Mirror formula

• The pole of the spherical mirror marks the starting point of every distance measured.
• The focal length f and the radius of curvature 2f of the spherical mirror is taken to be negative for convex mirrors and +ve for concave mirrors. Similarly, do and di is taken as -ve when the object is located in fronts of the mirror and the image created is real and di is +ve when the image is virtual. In other words, we can say that the righthand side of the mirror is take as +ve, and the lefthand side of the mirror is taken as -ve. As the object is always positioned on the lefthand direction, the distance do is always -ve.
• Erect images are considered to be +ve and inverted images are considered to be -ve. In other words, the distance calculated below(downward direction) the principal axis is taken as +ve and the distance computed above(upward direction) the principal axis is taken as +ve.

## For convex mirrors:

The focal length, focal point, center of curvature, radius of curvature, and image distance is always positive as all these points lie on the righthand side of the mirror position. The image size is also +ve as a convex mirror forms a virtual image that is erect in nature i.e. the image lies above the principal axis and the object distance is always -ve as the object is always placed at the lefthand side of the mirror position.

## For concave mirrors:

The focal length, focal point, center of curvature, and radius of curvature are always negative as all these points lie on the left hand side of the mirror. The image distance and image size may be negative or positive depending upon where the object is placed. di is positive when the object lies between the focal point and the pole, and the image formed is virtual and erect. For all other cases, the image size is negative.

## linear Magnification of spherical mirrors:

This is expressed by the ratio of the height of the image to the height of the object and for this case, So if hi is the height of the formed image and ho is the height of the actual object, then the magnification is given by the equation:

## Magnification formula for Spherical mirror

Magnification of spherical mirrors ( m )= hi/h0

The +ve magnification(m) denotes erect image formation and -ve magnification(m) denotes inverted image formation and if the magnification(m) is less than one, then the formed image is diminished in nature, and if the magnification(m) is more than one, then the formed image is magnified in nature.

## What are the aberrations of spherical mirrors?

Spherical mirrors suffer from five major types of aberrations:

## Spherical Aberration Mirror:

Spherical aberration refers to the imaging errors caused when marginal or off-axis rays are deflected either more or less compared to paraxial or on-axis rays. Due to this the focal points of the marginal rays and the paraxial rays do not coincide.

## Chromatic aberration:

Chromatic aberration refers to the imaging errors caused when the light rays having different wavelengths gets reflected at different angles, resulting in a different focal point for each wavelength.

## Comatic aberration:

Comatic aberration or coma refers to the imaging errors caused when the off-axis points sources like stars appear to be distorted. The off-axis points often get elongated and appears to form a tail (coma) similar to the shape of a comet.

## Astigmatism:

Astigmatism refers to the imaging errors caused when the light rays propagating in two different orthogonal planes have different focal points.

## Distortion:

Distortion refers to the imaging errors caused when there is a deviation from the general rectilinear propagation of light. In this, the straight lines may appear to be slightly bulged or shrinked in the middle.

## What are Parabolic mirrors?

Parabolic mirrors as the name suggests have a circular parabolic reflecting surface used for collecting and directing light rays. The parabolic mirror collects all the incident light rays (including marginal rays) and directs them towards its focus after reflection. Conversely, light rays coming from the focal point gets reflected forming a parallel collimated beam along the principal axis. The application of parabolic mirrors is seen in reflecting telescopes, flashlights, solar furnaces, stage spotlights, car headlights, and searchlights.

## Difference Between Parabolic and Spherical Mirror | Parabolic vs Spherical mirror

Parabolic mirrors are free of spherical and chromatic aberrations as regardless of where the light rays fall, the reflected rays will always pass through the same focus. This is dissimilar to spherical mirrors, where spherical aberration causes a different focus for marginal and paraxial rays.

## Advantages of parabolic mirrors over spherical mirrors:

• In spherical mirrors, the aperture size needs to be reduced to limit the marginal rays.
• In parabolic mirror, the marginal rays do not cause any issue, therefore, the aperture size can be increased.
• Larger aperture means collection of more light and improved image formation.

## Ray tracing of Spherical mirrors

• Step 1: We need to draw a ray from the top vertex of the given object and stretch it to the pole of the mirror forming an angle with the principal axis.
• Step 2: We need to draw the reflected ray on the opposite side of the optical axis at an angle equal to the angle of incidence from the pole of the mirror’s.
• Step 3: We can draw a second ray from the vertex of the object to the mirror surface,that propagating parallel to the principal-axis and reflected ray should be drawn passing thru the focal point.
• Step 4: We need to mark the intersection point of both the reflected rays.
• Step 5: We need to draw a straight line from the intersection point to the principal axis to represent the image formed.

## Q. Who discovered Spherical Mirrors ?

The invention of mirror is credited to popular chemist Justus von Liebig, though it has been started by mathematician Ibn al-Haytham, carried out many experiments using cylindrical and spherical geometries. A spherical mirror is designed by cutting out a piece of a sphere( coated with silver-mercury amalgam) from inside or outside surface.

## Q. What are toroidal mirrors?

A toroidal mirror as the name suggests has a portion of a torus with two radii of curvature, as its surface. Such toroidal mirrors are easier to build than parabolic or ellipsoidal mirrors but having issue related to spherical aberration and comas. However, these mirrors are capable of limiting errors rising due to astigmatism. These mirrors are also quite cheaper compared to ellipsoidal or paraboloid mirrors having the same surface quality. These mirrors find their application in Yolo telescopes and optical monochromators. Toroidal mirrors are preferred in these instruments as the light source is not placed on the principal axis of the mirror here.

## Q. What are the applications of spherical mirrors in daily life ? | What are the uses of spherical mirrors ?

Various applications of spherical mirror has been listed in Applications of Convex mirrors and Application of Concave Mirrors sections.

## Q. If the longitudinal magnification of a spherical mirror is m then what is its lateral magnification?

Ans. Longitudinal magnification is express as the ratio of the height of the image to the height of the object. Lateral magnification is given by the ratio of image distance to object distance, given that the medium is same on both sides of the mirror

## Q. What is relation between speed of object and speed of image for a spherical mirror?

Ans. The speed of the image formed by a spherical mirror is given by the product of the negative of the square of magnification with the speed of object.

## Q. What is the relation between the radius of curvature and the aperture of a spherical mirror?

Ans. Diameter of the Aperture. <=  2*Radius of curvature of the Mirror.

## Q. A concave mirror forms an image of height 20cm, The height of the object is 2cm If the longitudinal magnification of a spherical mirror is m then what is its lateral magnification ?

Ans. Here, Image Height hi = 20 cm, Height of the Object ho = 2 cm

We know,

M = hi/ho = 20cm/2cm =10 (Answer)

## Q. A concave mirror produces a real image with hi = 4 cm, of an object with ho = 1cm. The object is located at 20cm from the pole. then compute the image distance.

Ans. Here, Image Height hi = -4 cm, Height of the Object ho = 1 cm, Distance = -20cm

Now, m=hi/ho = -di/do

• -4cm/ 1cm= -di/do
• -4 = -di/ -20cm
• di = -80 cm (Answer)

## Q. An arrow has a height of 2.5 cm and is kept at a distance of 25 cm from a convex mirror with f=20cm and  the calculate the position and size of the produced image.

Solution :

ho = 2.5 cm, f = 20 cm, do= -25cm

Now we know,

1/ do +1/ di = 1/ f

• 1/20cm = -1/25cm + 1/v
• 9/100cm = 1/v
• V = 11.11 cm

Now

m=hi/ho = -di/do

• hi/2.5 = -11.11/-25
• hi = 1.11 cm (Answer)