Question

Prove that for any spherical mirror 1/f=1/v+1/u,where the symbols have their usual meaning.

Answer 1

Answer:

f= focal length....

v = image distance....

u= object distance....

Answer 2

Answer:

Aim - To derive the  $\frac{1}{f} = \frac{1}{v} +\frac{1}{u}$  formula for the spherical mirror.

Let us consider a concave mirror as shown in the below image, where PA’ is -v, PC is -R, and PA is -u.

Equation 1, $\frac{AB}{A'B'} =\frac{AC}{A'C}$   [because ΔABC and ΔA’B’C are similar]

Equation 2, $\frac{AB}{A'B'} = \frac{PA}{PA'}$  [because ΔABP and ΔA’B’P are similar]

From equations 1 and 2,

$\frac{PA}{PA'} = \frac{AC}{A'C}$

$\frac{-u}{-v} = \frac{PA-PC}{PC-PA'}$

$\frac{u}{v}=\frac{-u-(-R)}{-R-(-v)}$

$\frac{u}{v} = \frac{R-u}{v-R}$

$uv-uR = vR-uv$

$2uv = R(u+v)$

$\frac{2}{R} =\frac{u+v}{uv}$

$\frac{2}{2f} =\frac{u+v}{uv}$   [because R = 2f]

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

Hence, the formula  $\frac{1}{f} = \frac{1}{v} +\frac{1}{u}$  is derived.

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