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Difficulty: MediumRefraction of Light, Total Internal Reflection, and Prisms

A ray of light traveling in air strikes the flat surface of a transparent glass slab at an angle of incidence of 60∘60^\circ. If the refractive index of the glass slab relative to air is 3\sqrt{3}, what is the angle of refraction inside the glass slab in degrees?

Answer: 30 degrees

Answer

The angle of refraction inside the glass slab is 30∘30^\circ.
According to Snell's Law, n=sin⁡isin⁡rn = \frac{\sin i}{\sin r}. Substituting n=3n = \sqrt{3} and i=60∘i = 60^\circ gives 3=sin⁡60∘sin⁡r\sqrt{3} = \frac{\sin 60^\circ}{\sin r}. Since sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}, rearranging gives sin⁡r=3/23=0.5\sin r = \frac{\sqrt{3}/2}{\sqrt{3}} = 0.5. Taking the inverse sine of 0.50.5 gives r=30∘r = 30^\circ.

Step-by-Step Solution

1
Identify Snell's law formula relating the angle of incidence and angle of refraction.
n=sin⁡isin⁡rn = \frac{\sin i}{\sin r}
Snell's law describes how light bends when crossing the boundary between two optical media.
2
Substitute the given values i=60∘i = 60^\circ and n=3n = \sqrt{3} into the equation.
3=sin⁡60∘sin⁡r\sqrt{3} = \frac{\sin 60^\circ}{\sin r}
Plugging in the known parameters allows us to isolate the unknown sine of the angle of refraction.
3
Substitute sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2} and rearrange for sin⁡r\sin r.
\sin r = \frac{\sqrt{3}/2}{\sqrt{3}} = 0.5
Canceling 3\sqrt{3} from both sides yields a simple numerical value for sin⁡r\sin r.
4
Take the inverse sine of 0.50.5 to find rr.
r=arcsin⁡(0.5)=30∘r = \arcsin(0.5) = 30^\circ
The angle whose sine is 0.50.5 is 30∘30^\circ.

Key Concept

Snell's Law of Refraction
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Topics

Refraction of Light, Total Internal Reflection, and Prisms

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