The rate of change of torque `'tau'` with deflection `theta` is maximum for a magnet suspended freely in a uniform magnetic field of induction `B` when `theta` is equal to

Options

A. (a)`theta = 0^(@)`
B. (b)`theta = 45^(@)`
C. (c)`theta = 60^(@)`
D. (d)`theta = 90^(@)`

Answer

To solve the problem, we need to determine the angle `theta` at which the rate of change of torque `tau` with respect to deflection `theta` is maximum for a magnet suspended in a uniform magnetic field of induction `B`.

### Step-by-Step Solution:

1. **Understand the Torque Equation**:
The torque (`tau`) experienced by a magnetic moment (`M`) in a magnetic field (`B`) is given by the equation:
\[ \tau = M \cdot B \cdot \sin(\theta) \]
where `theta` is the angle between the magnetic moment and the magnetic field.

2. **Differentiate Torque with Respect to Theta**:
To find the rate of change of torque with respect to deflection, we need to differentiate `tau` with respect to `theta`:
\[ \frac{d\tau}{d\theta} = M \cdot B \cdot \frac{d}{d\theta}(\sin(\theta)) \]
Using the derivative of sine, we have:
\[ \frac{d\tau}{d\theta} = M \cdot B \cdot \cos(\theta) \]

3. **Maximize the Rate of Change**:
We want to find the value of `theta` that maximizes this derivative. The expression for the rate of change of torque is:
\[ \frac{d\tau}{d\theta} = M \cdot B \cdot \cos(\theta) \]
The maximum value of `\cos(\theta)` occurs when `\cos(\theta) = 1`.

4. **Determine the Angle**:
The cosine function reaches its maximum value of 1 at:
\[ \theta = 0^\circ \]

5. **Conclusion**:
Therefore, the rate of change of torque with respect to deflection is maximum when:
\[ \theta = 0^\circ \]

### Final Answer:
The angle `theta` at which the rate of change of torque is maximum is:
\[ \theta = 0^\circ \]