Two wires A and B of same length and of the same material have the respective radii `r_(1)` and `r_(2)`. Their one end is fixed with a rigid support, and at other end equal twisting couple is applied. Then the ratio of the angle of twist at the end of A and the angle of twist at the end of B will be

Options

A. `(r_(1)^(2))/(r_(2^(2)))`
B. `(r_(2)^(2))/(r_(2)^(2))`
C. `(r_(2)^(4))/(r_(1)^(4))`
D. `(r_(1)^(4))/(r_(2)^(4))`

Answer

To solve the problem, we need to find the ratio of the angle of twist at the ends of two wires A and B, given that they have the same length and are made of the same material, but have different radii.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Two wires A and B have the same length (L).
- Both wires are made of the same material, which means they have the same modulus of rigidity (N).
- The radii of the wires are \( r_1 \) for wire A and \( r_2 \) for wire B.
- An equal twisting couple (C) is applied to both wires.

2. **Use the Formula for Angle of Twist**:
The angle of twist \( \theta \) in a wire subjected to a twisting couple is given by the formula:
\[ C = \frac{\pi N r^4 \theta}{2L} \]
where:
- \( C \) is the twisting couple,
- \( N \) is the modulus of rigidity,
- \( r \) is the radius of the wire,
- \( \theta \) is the angle of twist,
- \( L \) is the length of the wire.

3. **Set Up the Equations for Both Wires**:
For wire A:
\[ C = \frac{\pi N r_1^4 \theta_A}{2L} \]
For wire B:
\[ C = \frac{\pi N r_2^4 \theta_B}{2L} \]

4. **Equate the Two Expressions for C**:
Since the twisting couple \( C \) is the same for both wires, we can set the two equations equal to each other:
\[ \frac{\pi N r_1^4 \theta_A}{2L} = \frac{\pi N r_2^4 \theta_B}{2L} \]

5. **Simplify the Equation**:
Cancel out the common terms (\( \pi \), \( N \), and \( 2L \)):
\[ r_1^4 \theta_A = r_2^4 \theta_B \]

6. **Rearrange to Find the Ratio of Angles**:
Rearranging gives:
\[ \frac{\theta_A}{\theta_B} = \frac{r_2^4}{r_1^4} \]

7. **Final Ratio**:
Therefore, the ratio of the angle of twist at the end of A to the angle of twist at the end of B is:
\[ \frac{\theta_A}{\theta_B} = \left(\frac{r_2}{r_1}\right)^4 \]

### Conclusion:
The ratio of the angle of twist at the end of wire A to the angle of twist at the end of wire B is given by:
\[ \frac{\theta_A}{\theta_B} = \left(\frac{r_2}{r_1}\right)^4 \]