A wave has velocity u in medium P and velocity 2 u in medium Q . If the wave is incident in medium P at an angle of `30^(@)` then the angle of refraction will be

Options

A. `30^@`
B. `45^@`
C. `60^@`
D. `90^@`

Answer

To solve the problem, we will use Snell's Law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the velocities of the waves in the two media.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Velocity in medium P, \( v_1 = u \)
- Velocity in medium Q, \( v_2 = 2u \)
- Angle of incidence in medium P, \( \theta_1 = 30^\circ \)

2. **Apply Snell's Law:**
Snell's Law is given by:
\[ \frac{v_1}{v_2} = \frac{\sin \theta_1}{\sin \theta_2} \]
where \( \theta_2 \) is the angle of refraction we need to find.

3. **Substituting the Values:**
Substitute the values of \( v_1 \) and \( v_2 \):
\[ \frac{u}{2u} = \frac{\sin 30^\circ}{\sin \theta_2} \]
Simplifying the left side:
\[ \frac{1}{2} = \frac{\sin 30^\circ}{\sin \theta_2} \]

4. **Calculate \( \sin 30^\circ \):**
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
Therefore, substituting this value into the equation gives:
\[ \frac{1}{2} = \frac{\frac{1}{2}}{\sin \theta_2} \]

5. **Cross-Multiplying:**
Cross-multiply to solve for \( \sin \theta_2 \):
\[ 1 \cdot \sin \theta_2 = 2 \cdot \frac{1}{2} \]
This simplifies to:
\[ \sin \theta_2 = 1 \]

6. **Finding the Angle of Refraction:**
The angle whose sine is 1 is:
\[ \theta_2 = \sin^{-1}(1) = 90^\circ \]

### Conclusion:
The angle of refraction \( \theta_2 \) is \( 90^\circ \).