Question

Derive the formula of linear magnification produced by spherical mirror in terms of focal length, image distance and object distance using mirror formula.​

Answer 1

Let , u be the distance of object from  the mirror.

       v be the distance  of image from the mirror.

        f be the focal length.

Now  We know the  Mirror Formula,

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

Multiply both side by v ,

$\frac{v}{f}=\frac{v}{u}+\frac{v}{v}\\\\1+\frac{u}{v}=\frac{v}{f}\\\\m=\frac{(-v)}{u} \\\\\frac{(-v)}{u}=1-\frac{v}{f}\\\\=\frac{f-v}{f}$

If we multiply both side by u,

$\frac{u}{f}=\frac{u}{u}+\frac{u}{v} \\\\m=\frac{(-v)}{u}\\\\\frac{(u)}{v}= \frac{u-f}{f}\\\\\frac{(-v)}{u}=\frac{f}{f-u}$

Thus finally our formula is,

$m=\frac{(-v)}{u}\\=\frac{f-v}{f}\\=\frac{f}{f-u}$

Answer 2

$\huge\underline\mathfrak\red{Derivation}$

Refer to the attachment!

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We must remember the mirror formula to solve this derivation.

that is,

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

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