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Difficulty: MediumReflection of Light at Plane and Curved Mirrors

Two plane mirrors are inclined to each other at an angle of 60∘60^\circ. A ray of light strikes the first mirror at an angle of incidence of 40∘40^\circ and subsequently undergoes reflection at the second mirror. What is the total angle of deviation, in degrees, produced in the ray after the two successive reflections?

Answer: 240 degrees

Answer

The total deviation produced in the light ray after successive reflections from both inclined plane mirrors is 240∘240^\circ.
When a light ray undergoes successive reflections at two plane mirrors inclined at an angle θ\theta, the total deviation DD experienced by the ray is given by D=360∘−2θD = 360^\circ - 2\theta. Substituting θ=60∘\theta = 60^\circ yields D=360∘−2(60∘)=240∘D = 360^\circ - 2(60^\circ) = 240^\circ. Step-by-step ray tracing confirms d1=180∘−2(40∘)=100∘d_1 = 180^\circ - 2(40^\circ) = 100^\circ and d2=180∘−2(20∘)=140∘d_2 = 180^\circ - 2(20^\circ) = 140^\circ, giving a total deviation of 100∘+140∘=240∘100^\circ + 140^\circ = 240^\circ.

Step-by-Step Solution

1
Calculate the deviation at the first mirror
d1=100∘d_1 = 100^\circ
The angle of deviation for a single reflection at a plane mirror is d=180∘−2id = 180^\circ - 2i. For i1=40∘i_1 = 40^\circ, d1=180∘−80∘=100∘d_1 = 180^\circ - 80^\circ = 100^\circ.
2
Find the angle of incidence on the second mirror using geometry
i2=20∘i_2 = 20^\circ
The glancing angle on the first mirror is g1=90∘−40∘=50∘g_1 = 90^\circ - 40^\circ = 50^\circ. In the triangle formed by the ray path and the two mirrors inclined at 60∘60^\circ, the glancing angle on the second mirror is g2=180∘−(60∘+50∘)=70∘g_2 = 180^\circ - (60^\circ + 50^\circ) = 70^\circ. Thus, the angle of incidence on the second mirror is i2=90∘−70∘=20∘i_2 = 90^\circ - 70^\circ = 20^\circ.
3
Calculate the deviation at the second mirror and total deviation
d2=140∘d_2 = 140^\circ, D=240∘D = 240^\circ
The deviation at the second mirror is d2=180∘−2(20∘)=140∘d_2 = 180^\circ - 2(20^\circ) = 140^\circ. Since both reflections rotate the ray in the same sense, the total deviation is D=d1+d2=100∘+140∘=240∘D = d_1 + d_2 = 100^\circ + 140^\circ = 240^\circ.

Key Concept

Total deviation produced by reflection from two inclined plane mirrors is independent of the initial angle of incidence and is given by D=360∘−2θD = 360^\circ - 2\theta, where θ\theta is the angle of inclination between the mirrors.
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Topics

Reflection of Light at Plane and Curved Mirrors

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