Ray Optics and Optical Test - 8

A plano-convex lens behaves as a concave mirror if its one surface (curved) is silvered. The rays refracted from plane surface are reflected from curved surface and again refract from plane surface. Therefore, in this lens two refractions and one reflection occur. Let the focal length of silvered lens is F and focal length of mirror is f(m)
1/F = 1/f + 1/f +1/f(m)
Plano-convex lens silvered on plane side has f(m) = infinity
1/F = 2/f + 1/ infinity
1/F = 2/f
f = 28××2 = 56 cm
Plano-convex lens silvered on convex side has f(m) = R/2
1/F = 2/f + 2/ R
1/10 = 2/56 + 2/R
R = 280/9 cm
Now
1/f = (μ–1)/R
if we put all value 
μ = 1.55

The curved silvered surface will behave as a concave lens of focal length fm = R/2 = −22/2 = −11cm = −0.11m
And hence
P_M = the power =−1/fm = −1/−0.11 = 1/0.11 D
Further as focal length of lens is 20 cm i.e. 0.2m its power is
P_L=1/f2 = 1/0.2 D
Now as in image formation, light after passing through the lens will be reflected back by curved mirror through lens again
P=P_L+P_M+P_L
=2P_L+P_M
P=2/0.2 + 1/0.11 = 210/11 D
So, focal length of equivalent mirror.
F = −1/P = 11/210 m = −110/21 cm
i.e. silvered lens behaves as a concave mirror of focal length (110/21)cm
So far object at a distance 10 cm in front of it.
1/v+1/−10 = 21/110
i.e. v = −11cm
The image will be 11 cm in front of silvered lens and will be real.

In normal adjustment, m = fo/fe = 6. Therefore fo = 6 fe
further, fo+fe = 14cm
or 7fe = 14cm or fe = 2cm
hence fo = 12cm

Angular magnification or magnifying power of compound microscope is defined as ratio of angle made at eye by image formed at infinity to the angle made by object, if placed at distance of distinct vision from an unaided eye.
Magnification = LD / fo . fe
where, L-length of the tube of microscope
As, m ∝ 1/fo
and m ∝ 1 / fe
∴ both eye piece and objective must be of smaller focal lengths, so, that magnification is higher.

magnifying power of telescope is directly proportional to fo/fe.
Hence fo should be large and fe should be small.

Combined power, P = P₁ + P₂ = +2 D + (-1 D) = +1 D
focal length of combination, f = 1/P = 1/1 = 1m = =100 cm
Since focal length is positive, it will behave as converging lens.