When a monochromatic light ray is incident on a medium of refracive index`mu`with angle of incidence `theta_(1)`,the angle of refraction is `theta_(r).`if `theta_(i)` is changed slightly by`Deltatheta_(i)`,then the corresponding change in `theta_(r)`will be-

Options

A. `Deltatheta_i`
B. `muDeltatheta_i`
C. `1/mu.(costheta_i)/(costheta_r).Deltatheta_1`
D. `mu.(costheta_i)/(costheta_r).Deltatheta_i`

Answer

To solve the problem, we need to relate the change in the angle of refraction \( \Delta \theta_r \) to the change in the angle of incidence \( \Delta \theta_i \) using Snell's Law.

### Step-by-Step Solution:

1. **Start with Snell's Law**:
Snell's Law states that:
\[ \mu = \frac{\sin \theta_i}{\sin \theta_r} \]
where \( \mu \) is the refractive index, \( \theta_i \) is the angle of incidence, and \( \theta_r \) is the angle of refraction.

2. **Rearranging Snell's Law**:
From Snell's Law, we can express \( \sin \theta_r \) in terms of \( \theta_i \):
\[ \sin \theta_r = \frac{1}{\mu} \sin \theta_i \]

3. **Differentiate with respect to \( \theta_i \)**:
We differentiate both sides with respect to \( \theta_i \):
\[ \cos \theta_r \frac{d\theta_r}{d\theta_i} = \frac{1}{\mu} \cos \theta_i \]

4. **Express \( d\theta_r \) in terms of \( d\theta_i \)**:
Rearranging the above equation gives:
\[ \frac{d\theta_r}{d\theta_i} = \frac{1}{\mu} \frac{\cos \theta_i}{\cos \theta_r} \]

5. **Relate changes \( \Delta \theta_r \) and \( \Delta \theta_i \)**:
If we denote small changes in angles as \( \Delta \theta_i \) and \( \Delta \theta_r \), we can write:
\[ \Delta \theta_r = \frac{1}{\mu} \frac{\cos \theta_i}{\cos \theta_r} \Delta \theta_i \]

6. **Final Expression**:
Thus, the change in the angle of refraction \( \Delta \theta_r \) corresponding to a change in the angle of incidence \( \Delta \theta_i \) is given by:
\[ \Delta \theta_r = \frac{1}{\mu} \frac{\cos \theta_i}{\cos \theta_r} \Delta \theta_i \]

### Conclusion:
The final expression for the change in the angle of refraction \( \Delta \theta_r \) is:
\[ \Delta \theta_r = \frac{1}{\mu} \frac{\cos \theta_i}{\cos \theta_r} \Delta \theta_i \]