Question

a short linear object of length l lies along the axis of a concave mirror at a distance u from the pole of the mirror. what is the size of the image

Answer 1

The size of the image is $\bold{b\left(\frac{f}{u-f}\right)^{2}}$.

Solution:

Now, we need to use the Lens formula,

$\bold{\frac{1}{v}+\frac{1}{u}=\frac{1}{f} \longrightarrow (1)}$

Where,

v = Size of the image

u = Size of the object

On differentiating equation (1), we get,

$\bold{-\frac{d v}{v^{2}}-\frac{du}{u^{2}}=0}$

$\bold{\frac{d v}{d u}=-\frac{v^{2}}{u^{2}}=-\left(\frac{v}{u}\right)^{2}}$

$\bold{|d v|=\left-|\frac{v^{2}}{u^{2}}\right||d u| \longrightarrow (2)}$

$\bold{|d v|}$ = Size of image

$\bold{|d u|}$ = Size of object = b

$\bold{\frac{v^{2}}{w^{2}}=-\left(\frac{f}{u-f}\right)^{2}}$

On substituting equation (2) in above equation, we get,

Therefore, size of the image $\bold{=b\left(\frac{f}{u-f}\right)^{2}}$