Question

A ray of light enters a glass block from air and comes out from the opposite surface. If the angle of refraction at the first surface is not the same as the angle of incidence at the second surface, then:
(a) What is the product of the ratio \(\frac{\sin i}{\sin r}\) at the first surface and at the second surface?
(b) State whether the opposite surfaces are parallel or not parallel.
(c) How did you reach the conclusion in (b) above?

Hint:

For a rectangular (parallel-sided) glass slab, the emergent ray is parallel to the incident ray. This only happens if \(r_1 = i_2\). If this condition is not met, the surfaces cannot be parallel.

Answer 1

Step 1: Understanding the Concept:
According to Snell's Law, the refractive index of glass with respect to air (\( _{a}\mu_{g} \)) is the ratio of the sine of the angle of incidence in air to the sine of the angle of refraction in glass.
Step 2: Key Formula or Approach:
Refractive index of glass with respect to air: \( _{a}\mu_{g} = \frac{\sin i_{1}}{\sin r_{1}} \)
Refractive index of air with respect to glass: \( _{g}\mu_{a} = \frac{\sin i_{2}}{\sin r_{2}} \)
By the Principle of Reversibility: \( _{a}\mu_{g} \times _{g}\mu_{a} = 1 \)
Step 3: Detailed Explanation:
At the first surface, the ratio \( \frac{\sin i}{\sin r} \) is equal to \( _{a}\mu_{g} \).
At the second surface, the ray travels from glass to air, so the ratio \( \frac{\sin i}{\sin r} \) is equal to \( _{g}\mu_{a} \).
The product of these two ratios is:
\[ _{a}\mu_{g} \times _{g}\mu_{a} = \frac{1}{_{g}\mu_{a}} \times _{g}\mu_{a} = 1 \]
This product is always 1 regardless of whether the surfaces are parallel or not.
Step 4: Final Answer:
The product of the ratios is 1.
(b)
Step 1: Understanding the Concept:
In a rectangular glass block with parallel opposite faces, the normals at the point of incidence on both faces are parallel to each other.
Step 2: Detailed Explanation:
If the surfaces were parallel, the ray inside the glass would act as a transversal between two parallel normals.
This would mean the angle of refraction at the first surface (\( r_{1} \)) and the angle of incidence at the second surface (\( i_{2} \)) would be alternate interior angles, making them equal (\( r_{1} = i_{2} \)).
The question states that \( r_{1} \) is not the same as \( i_{2} \).
Therefore, the normals are not parallel, which implies the surfaces are not parallel.
Step 3: Final Answer:
The opposite surfaces are not parallel.
(c)
Step 1: Understanding the Concept:
Geometrical properties of parallel lines and transversals are applied to the path of light through a medium.
Step 2: Detailed Explanation:
For parallel surfaces, the normal at the first surface is parallel to the normal at the second surface.
The refracted ray inside the medium acts as a transversal.
In such a case, the angle of refraction at the first surface (\( r \)) and the angle of incidence at the second surface (\( i' \)) must be equal as they are alternate interior angles.
Since the question provides that these two angles are not equal (\( r \neq i' \)), the condition for parallel surfaces is violated.
Step 3: Final Answer:
The conclusion is reached based on the fact that \( r \neq i' \), which only happens when the refracting surfaces are inclined at an angle to each other.