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Mirror Formula for Spherical Mirrors

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What is The Mirror Formula for Spherical Mirrors?

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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.


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 $u$ while the image distance is the distance between the image and the pole of the mirror and is denoted by the letter $v$. 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:

$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$

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 $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ is shown below:

(Image to be added soon)

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

In the above figure, $A^{\prime}B^{\prime}F$ and $MPF$ are two right-angled similar triangles. 

$\Rightarrow \dfrac{B^{\prime}A^{\prime}}{PM} = \dfrac{B^{\prime}F}{FP}$

$\Rightarrow \dfrac{B^{\prime}A^{\prime}}{BA} = \dfrac{B^{\prime}F}{FP}$

$\therefore PM = AB$  ---------$(1)$

$\Rightarrow \angle{APB} = \angle{A^{\prime}PB}$

$\therefore \Delta A^{\prime}B^{\prime}P \sim \Delta AB^{\prime}P$

$\therefore \dfrac{B^{\prime}A^{\prime}}{BA} = \dfrac{B{\prime}P}{BP}$  -------------$(2)$

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

$\Rightarrow \dfrac{B^{\prime}F}{FP} = \dfrac{B^{\prime}P - FP}{FP} = \dfrac{B^{\prime}P}{BP}$

$B^{\prime}P = -v, FP = -f, BP = -u$

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

$\Rightarrow \dfrac{-v+f}{-f} = \dfrac{-v}{-u}$

$\Rightarrow \dfrac{v-f}{f} = \dfrac{v}{u}$

On further solving, we get the result as:

$\dfrac{1}{v} = \dfrac{1}{u} = \dfrac{1}{f}$


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.


FAQ (Frequently Asked Questions)

1. What is meant by object distance, image distance, and focal length? Also, give the mirror formula.

The distance between the object from the pole of a spherical mirror is called object distance and the distance between the image formed from the pole of the mirror is called the image distance. The focal length is the distance between the principal focus and the pole of the spherical mirror.

The mirror formula gives us the relation between the object distance, image distance, and the focal length which is given by the formula:

1/v + 1/u = 1/f

Here, v denotes the image distance, u denotes the object distance and f denotes the focal length of the spherical mirror.

2. Define the terms Center of Curvature, Radius of Curvature, Principal axis, Pole, and Aperture.

The Center of Curvature is the point in the center of the mirror that passes through the curve of the mirror and has the same tangent and curvature at the same time.

The Radius of Curvature is the linear distance between the pole of the mirror and the center of curvature.

The Principal axis is an imaginary line passing through the optical center and the center of curvature of any lens or spherical mirror.

Pole is the midpoint of the spherical mirror.

Aperture is a point from which the reflection of light takes place. It gives us the size of the mirror.

3. State some uses of a convex mirror and a concave mirror.

A convex mirror provides a wide-angle view and they produce an upright image. Wide-angle images are used in rearview mirrors, and for viewing large areas like parking lots and intersections. Convex mirrors are also present in side mirrors on cars.

A concave mirror is used in many applications. They reflect light inward to one focal point and are hence mostly used to focus light. It forms an upright and enlarged image and thus, it is used for makeup application or shaving. They are used in solar furnaces, salons, parlours and ophthalmoscopes.

They are also used in flashlights, headlights, and astronomical telescopes.