Name: Sign Convention
Uploaded: 2019-07-19
Duration: 7 min 37 s
Description: All distances are measured from the optical centre of the lens. Distances measured in the direction opposite to the incident light's direction are considered...

Question

Establish the relation between object distance, image distance and focal length for a convex mirror.

Accepted Answer

Refer to the diagram below for the reflection taking place at the convex mirror.

The virtual image of the object O' is I' which is produced by meeting of the reflected rays produced from the ray O'A and OB.

The $∆ O O' C$ and $∆ I I' C$ are similar to each other.

Therefore, $\frac{O O'}{I I'} = \frac{O C}{I C} . . . . . . . . . . . . . . . . . (1)$

Similarly, $∆ A N F$ and $∆ I I' F$

$\frac{A N}{I I'} = \frac{N F}{I F}$

or $\frac{O O'}{I I'} = \frac{N F}{I F} . . . . . . . . . . . . . . . . . . . . . . (2)$

From equations $1$ and $2$ , we have, $\frac{O C}{I C} = \frac{N F}{I F} . . . . . . . . . . . . . . (3)$ .

For, small-aperture consideration, $N F ≃ P F$ , equation $3$ becomes,

$\frac{O C}{I C} = \frac{P F}{I F}$

The above equation becomes, $\frac{P O + P C}{P C - P I} = \frac{P F}{P F - P I} . . . . . . . . . . . (4)$

Putting the values according to the sign convention, $P O = - u, P C = + R, P I = + v a n d P F = + f$

The above equation becomes

$\frac{- u + R}{R - v} = \frac{f}{f - v} . . . . . . . . . . . . . . . . (5)$

Putting $R = 2 f$ and on rearranging the terms, we get,

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

Establish the relation between object distance, image distance and focal length for a convex mirror.

Important Questions on Reflection of Light : Spherical Mirrors

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