Mirror Formula for Spherical Mirrors

A spherical mirror is a mirror that has the shape of a piece cut out of a spherical surface. The two types of spherical mirrors are convex mirror and concave mirror. Convex mirrors are the spherical mirrors in which inward surfaces are painted while concave mirrors are the spherical mirrors in which the outward surfaces are painted. Convex mirrors are also known as diverging mirrors as the rays diverge after falling on the convex mirror whereas concave mirrors are also known as converging mirrors since the rays tend to converge after falling on the concave mirror. 

The mirror formula tells us how the object distance and the image distance are related to the focal length of a spherical mirror.

Let us further learn more about the mirror formula for spherical mirrors. Before that, let us learn a few other important things about spherical mirrors.

History of Spherical Mirrors in Human Civilization

• In very ancient times, pools of water were the first surfaces that were used as mirrors by humans. In the case of manufacturing, the very first mirrors to be manufactured were nothing more than pieces of highly polished pieces of stones. 

• Chinese people, way back in 500 AD, first started the procedure of making mirrors with silver-mercury amalgams. 

• Various types of experiments were conducted by Claudius Ptolemy with curved polished iron mirrors.  

Terminologies

While studying spherical mirrors, a person needs to become aware of a few basic terminologies associated to spherical mirrors. They are as follows:

• Center of Curvature- Center of Curvature of a mirror is that point in the center of the mirror that passes through the curve of the mirror. This point has the same curvature and tangent at the point where it passes through.

• The radius of Curvature- The linear distance between the center of curvature and the pole of the mirror is known as the radius of curvature. 

• Principal axis- Principal axis is basically an imaginary line. This imaginary line passes through the optical center and the center of curvature of any spherical mirror or a lens. 

• Pole- The midpoint of the spherical mirror is known as its pole

• Aperture- In a lens or a mirror, there is a specific point from which the reflection of light actually happens. This point also happens to give the size of the mirror.

• Principal Focus- Focal point, also known as the principal axis is on the axis of the mirror or the lens. At this point, the rays of light parallel to the axis appear to converge or actually converge after refraction or reflection. 

• Focus- Focus is the point of the mirror or lens, any given point, where the light rays that are parallel to the principal axis will converge after reflection from the mirror. 

Spherical Mirror Formula

The spherical mirror formula explains to us the relationship between object distance, image distance, and the focal length of a spherical mirror. The object distance is the distance between the object and the pole of the mirror and is denoted by the letter uu while the image distance is the distance between the image and the pole of the mirror and is denoted by the letter vv. The distance of the principal focus from the pole of the mirror is known as the focal length. The expression which gives us the relation between these three quantities is called the spherical mirror formula, which is given by:

\[\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\]

The mirror formula is applicable for all spherical mirrors for any possible position of the object.

Spherical Mirror Formula Derivation

The verification of the mirror formula \[\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\] is shown below:

(Image to be added soon)

We will now derive the mirror equation between u, v, and f.

In the above figure, A′B′F and MPF are two right-angled similar triangles.

\[\Rightarrow \frac{B'A'}{PM}=\frac{B'F}{FP}\]

\[\Rightarrow \frac{B'A'}{BA}=\frac{B'F}{FP}\]

∴PM = AB -------------(1)

\[\Rightarrow \] ∠APB = ∠A′PB

∴ ΔA′B′P ∼ ΔAB′P

∴\[\frac{B'A'}{BA}=\frac{B'P}{BP}\]  -------------(2)

Substituting the equations (1)(1) and (2)(2), we get,

\[\Rightarrow \frac{B'F'}{FP}=\frac{B'P - FP}{FP}=\frac{B'P}{BP}\]

B'P = -v, FP = -f, BP = -u

By putting the values of u,v,u,v, and ff in the following equation, we get

\[\Rightarrow \frac{-v+f}{-f}=\frac{-v}{-u}\]

\[\Rightarrow \frac{v-f}{f}=\frac{v}{u}\]

On further solving, we get the result as:

\[\frac{1}{v}=\frac{1}{u}=\frac{1}{f}\]

Conclusion

Spherical mirrors are a very important part of physics. If you study this portion thoroughly, you can score a lot of marks. Many competitive exams tend to ask questions from this part.