A ray of light travels from a medium of refractive index `mu` to air. Its angle of incidence in the medium is `i`, meansured from the normal to the boundary , and its angle of deviation is `delta. delta` is plotted against `i`. Which of the following best represents the resulting curve ?

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the behavior of a ray of light as it travels from a medium with refractive index \( \mu \) to air, and how the angle of deviation \( \delta \) changes with the angle of incidence \( i \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a ray of light traveling from a denser medium (refractive index \( \mu \)) to air (refractive index = 1).
- The angle of incidence \( i \) is measured from the normal to the boundary.

2. **Applying Snell's Law**:
- According to Snell's Law, we have:
\[ \mu \sin(i) = 1 \sin(R) \]
- Here, \( R \) is the angle of refraction.

3. **Finding the Angle of Refraction**:
- Rearranging Snell's Law gives us:
\[ \sin(R) = \mu \sin(i) \]
- Therefore, we can express \( R \) as:
\[ R = \sin^{-1}(\mu \sin(i)) \]

4. **Calculating the Angle of Deviation**:
- The angle of deviation \( \delta \) is defined as:
\[ \delta = R - i \]
- Substituting the expression for \( R \):
\[ \delta = \sin^{-1}(\mu \sin(i)) - i \]

5. **Analyzing the Behavior of \( \delta \)**:
- For angles \( i < i_c \) (where \( i_c \) is the critical angle), \( \delta \) will be a nonlinear function of \( i \).
- As \( i \) approaches \( i_c \), \( \delta \) reaches its maximum value.

6. **Behavior Beyond the Critical Angle**:
- If \( i > i_c \), total internal reflection occurs, and the light does not refract into the air.
- In this case, the angle of deviation can be calculated as:
\[ \delta = 180^\circ - 2i \]
- This relationship is linear with a negative slope.

7. **Graphing \( \delta \) vs. \( i \)**:
- For \( i < i_c \), the graph of \( \delta \) vs. \( i \) is nonlinear and increases to a maximum.
- For \( i > i_c \), the graph becomes linear with a negative slope, indicating that as \( i \) increases, \( \delta \) decreases.

8. **Conclusion**:
- The resulting graph of \( \delta \) plotted against \( i \) will show a nonlinear increase up to the critical angle and then a linear decrease after the critical angle.

### Final Answer:
The best representation of the curve of \( \delta \) vs. \( i \) is a graph that shows a nonlinear increase followed by a linear decrease after the critical angle.

---