Question

Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are μ_r, μ_y and μ_v respectively and those for the flint glass are μ'_r, μ'_y and μ'_v respectively. Find the ratio A'/A for which (a) there is no net angular dispersion, and (b) there is no net deviation in the yellow ray.

Answer 1

For the crown glass, we have:-

Refractive index for red rays = μ_r

Refractive index for yellow rays = μ_y

Refractive index for violet rays = μ_v

For the flint glass, we have:-

Refractive index for red rays = μ'_r

Refractive index for yellow rays = μ'_y

Refractive index for violet rays = μ'_v

Let δ_cy and δ_fy be the angles of deviation produced by the crown and flint prisms for the yellow light.

Total deviation produced by the prism combination for yellow rays:-

δ_y = δ_cy − δ_fy
    = 2δ_cy − δ_fy
    =2(μ_cy + 1)A − (μ_fy − 1)A'

Angular dispersion produced by the combination is given by

δ_v − δ_r = [(μ_vc − 1)A − (μ_vf − 1)A' + (μ_vc − 1)A] − \[\left[ \left( \mu_{rc} - 1 \right)A - \left( \mu_{rf} - 1 \right)A' + \left( \mu_{rc} - 1 \right)A \right]\]

Here,

μ_vc = Refractive index for the violet colour of the crown glass

μ_vf = Refractive index for the violet colour of the flint glass

\[\mu_{rc}\] = Refractive index for the red colour of the crown glass

\[\mu_{rf}\] = Refractive index for the red colour of the flint glass

On solving, we get

δ_v − δ_r = 2(μ_vc −1)A − (μ_vf − 1)A'

(a) For zero angular dispersion, we have:

δ_t − δ_t = 0 = 2(μ_vc −1)A − (μ_vf − 1)A'

\[\Rightarrow   \frac{A'}{A} = \frac{2( \mu_{vf} - 1)}{( \mu_{vc} - 1)}\]

\[ = \frac{2( \mu_r - \mu_r )}{( \mu_r - \mu)}\]

(b) For zero deviation in the yellow ray, δ_y = 0.

⇒ 2(μ_cy − 1)A = (μ_fy − 1)A

\[\Rightarrow   \frac{A'}{A} = \frac{2( \mu_{cy} - 1)}{( \mu_{fy} - 1)}\]

\[ = \frac{2( \mu_y - 1)}{(\mu '_y - 1)}\]