Question

A. A ray of light incident on face AB of an equilateral glass prism, shows minimum deviation of 30o. Calculate the speed of light through the prism.
B. Find the angle of incidence at face AB so that the emergent ray grazes along the face AC.

Answer 1

The speed of prism will be 1/√2times the speed of light , i.e (3x10⁸)/1.414 = 2.121x10⁸m/sec.

Explanation:

Solution (A)

Given, the angle of minimum deviation Dm = 30°

and the prism is equilateral so , A = 60°.

Then from the refractive index of prism

μ = $\frac{Sin(\frac{A+Dm}{2}) }{\frac{SinA}{2} }$

μ = $\frac{Sin\frac{90}{2} }{\frac{Sin60}{2} }$

μ = $\frac{\frac{1}{\sqrt{2} } }{\frac{1}{2} }$

μ = √2

We know that μ = V1/V2

Hence the speed of light in prism will be 1/√2times the speed of light i.e (3x10⁸)/1.414 = 2.121x10⁸m/sec.

Solution (B):- From Snell's law , we have  $\frac{sini}{sinr}$ = μ₁₂

For the emergent ray to graze at the face AC, the angle of refraction should be 90°, so applying Snell's law at face AC we get

$\frac{Sini_{AC} }{Sinr_{AC} }$ = μ₁₂

$\frac{Sini_{AC} }{Sin90} }$ = $\frac{1}{\sqrt{2} }$

or $i_{AC}$ = 45°

From the figure we can see that the angle of Face AB is 15°°

so again applying Snell's law then we get

$Sini_{AB} = (Sinr_{AB})x(u_{12})$

or

$i_{AB} = sin^{-1}(\sqrt{sin15}$

PLZ find the attached file for figure used.

Answer 2

The speed of light through the prism is 2.121 × 10⁸ m/s

The angle of incidence at face AB is sin⁻¹ (√sin 15°).

Explanation:

A. Speed of light:

When the light falls on the prism at minimum deviation, the refracted light becomes parallel to its base.

Angle of minimum deviation = D = 30°

The angle at which the prism is equilateral = 60°

The refractive index of the prism is given by the formula:

$\mu=\frac{\sin \left(\frac{A+D_{m}}{2}\right)}{\sin \frac{A}{2}}$

On substituting the values, we get,

μ = sin (90/2)/sin (60/2)

μ = sin 45/sin 30

μ = (1/√2)/(1/2)

∴ μ = √2

The refractive index is given by the formula:

μ = v₁/v₂

√2 = (3 × 10⁸)/v₂

v₂ = (3 × 10⁸)/√2

∴ v₂ = 2.121 × 10⁸ m/s

B. Angle of incidence:

From Snell's law, the formula is:

(sin i)/(sin r) = μ₁₂

When the angle of refraction is 90°, then we get,

(sin iac)/(sin 90°) = 1/√2

(sin iac) = 1/√2

∴ iac = sin⁻¹ (1/√2) = 45°

When the angle of refraction is 15°, then we get,

∴ iab = sin⁻¹ (√sin 15°)