Question

Show that, if you place the object so that the image is at your near point instead of at infinity, the angular magnification is increased to the value
$\alpha =\frac{d}{f}+1$

Answer 1

Angular magnification $\alpha$ is given by formula

$\alpha =\frac{d}{u}$ where $u$ is the object distance from eyepiece. and $d$ is the near point.

when image appears to formed at near point i.e $v=-d$

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

$\frac{1}{f}=\frac{1}{-d}-\frac{1}{u}$

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

object distance =$\frac{1}{\frac{1}{d}+\frac{1}{f}}$ and - sign signifies left direction.

$\alpha =\frac{d}{u}⟹$ $\alpha =1+\frac{d}{f}$