Question

Draw a neat labelled ray diagram showing the formation of an image at the least distance of distinct vision D by a simple microscope. When the final image is at D. derive an expression for its magnifying power at D.

Answer 1

The diagram is as shown:

Mathematically, magnifying power is given by M = \(\frac{β}{α}\) ........(i)

Now, from the triangles B'CA' and B'CA'', we have

tan β = \(\frac{A'B'}{D}\) and tan α = \(\frac{A''B'}{D}\) = \(\frac{AB}{D}\)

Since the angles are small, therefore, the tangents can be  replaced with the angles, hence

β  = \(\frac{A'B'}{D}\) and α = \(\frac{AB}{D}\)

Substituting in equation (i), we have M

Now, that two triangles A'B'C and ABC are similar, therefore, we have \(\frac{A'B'}{AB}\) = \(\frac{v}{u}\) ....(3)

Substituting in equation (2), we have M = \(\frac{v}{u}\) ....(4)

By lens formula, we have \(\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\).......(5)

Multiplying both sides with v, we have \(\frac{v}{v}-\frac{v}{u}=\frac{1}{f}\) or 1- M = \(\frac{v}{f}\)

But v = - D, therefore, the above equation becomes M = 1+\(\frac{D}{f}\)  ...(6)

This gives an expression for the magnifying power of a simple microscope.