A light ray enters the first face of a glass prism surrounded by air. The prism has a refr… — JAMB UTME

A light ray enters the first face of a glass prism surrounded by air. The prism has a refracting angle of $75^\circ$ and a refractive index of $\sqrt{2}$ . Inside the prism, the ray strikes the second refracting face at the critical angle for total internal reflection. Calculate the angle of incidence, in degrees, at the first face.

Cevap: 45 degrees

Cevap

The angle of incidence at the first face is

45^\circ

To find the angle of incidence at the first face, first determine the critical angle

C

at the second glass-air surface using

\sin C = \frac{1}{n} = \frac{1}{\sqrt{2}}

, which yields

C = 45^\circ

. Since the ray strikes the second face at this critical angle,

r_2 = 45^\circ

. Next, using the geometric relationship for a prism

A = r_1 + r_2

, the angle of refraction at the first surface is

r_1 = A - r_2 = 75^\circ - 45^\circ = 30^\circ

. Finally, applying Snell's law at the first face gives

\sin i = n \sin r_1 = \sqrt{2} \sin 30^\circ = \frac{\sqrt{2}}{2}

. Taking the inverse sine yields

i = 45^\circ

Adım Adım Çözüm

Find the critical angle at the second face

Critical angle

C = 45^\circ

, so

r_2 = 45^\circ

Light travels from glass to air at the critical angle, so

\sin C = \frac{1}{n} = \frac{1}{\sqrt{2}}

Determine the angle of refraction at the first face

r_1 = 30^\circ

The apex angle of a prism satisfies

A = r_1 + r_2

, hence

r_1 = A - r_2 = 75^\circ - 45^\circ = 30^\circ

Apply Snell's Law at the entry boundary

\sin i = \frac{\sqrt{2}}{2}

Refraction at the first surface gives

\sin i = n \sin r_1 = \sqrt{2} \sin 30^\circ = \sqrt{2} \times 0.5 = \frac{\sqrt{2}}{2}

Solve for the incident angle

i

i = 45^\circ

\arcsin\left(\frac{\sqrt{2}}{2}\right) = 45^\circ

Anahtar Kavram

Refraction through a prism combined with total internal reflection critical angle condition

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