Question

Derive the expression for the angular magnification of a simple microscope.

Answer 1

Hint : When the object is at a distance closer than the distance of the least distinct vision, we can use a microscope to view the object clearly. The angular magnification of a microscope is defined as the ratio of the angle subtended by the image after refraction from the microscope to the angle subtended by the object at an unaided eye/without the apparatus.

Complete step by step answer
The angular magnification is defined as follows:
$\text{M = }{\displaystyle \frac{\text{angle subtended by eye using instrument}}{\text{angle subtended at unaided eye}}}$

When looking at a small object, as shown above, the angle subtended by the object at the unaided eye is very small and can be approximately written as
$\alpha ={\displaystyle \frac{h}{d}}$

When a lens is placed in between the object and the eye as is done in a simple microscope shown above, the angle subtended by the object will be
$\beta ={\displaystyle \frac{H}{d}}$
Since the image will be formed at the least distance of distinct vision $D$, i.e. $d=D$, the angular magnification can be defined as:
$M={\displaystyle \frac{\alpha }{\beta }}={\displaystyle \frac{H/D}{h/D}}$
$\Rightarrow M={\displaystyle \frac{H}{h}}$
Now from the lens-maker formula, we can write
${\displaystyle \frac{1}{v}}-{\displaystyle \frac{1}{u}}={\displaystyle \frac{1}{f}}$
We can rearrange the above equation to write,
$${\displaystyle \frac{v}{f}}=1+{\displaystyle \frac{v}{u}}$$
Since the ratio of the image to the object position is also equal to magnification and we want the image to form at the least distance of distinct vision $D$ which is typically $25cm$, we can say $v/u=M$ and $v=D$ and hence write
$M={\displaystyle \frac{D}{f}}-1$
This is the magnification power for a simple microscope to form an image at the least distance of distinct vision of the observer.

Note
If the object is placed at infinity, the angle $\beta$ subtended by the object is the same with or without the lens as. Also, the image formed by the lens will be formed at a distance equal to the focal length of the lens. So $d=f$ and we can write
$\beta ={\displaystyle \frac{h}{f}}$
So,
$M={\displaystyle \frac{\alpha }{\beta }}={\displaystyle \frac{h/f}{h/D}}$
$\Rightarrow M={\displaystyle \frac{D}{f}}$