The `(tau-theta)` graph for a coil is

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Answer

To determine the torque versus angle (τ-θ) graph for a coil carrying current in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Torque Formula**:
The torque (τ) experienced by a current-carrying coil in a magnetic field is given by the equation:
\[ \tau = \mathbf{m} \times \mathbf{B} \]
where \(\mathbf{m}\) is the magnetic moment and \(\mathbf{B}\) is the magnetic field.

2. **Magnetic Moment Calculation**:
The magnetic moment (m) for a coil can be expressed as:
\[ m = n \cdot I \cdot A \]
where:
- \(n\) = number of turns in the coil,
- \(I\) = current flowing through the coil,
- \(A\) = area of the coil.

3. **Torque Expression**:
Substituting the magnetic moment into the torque formula, we have:
\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
Here, \(B\) is the magnetic field strength and \(\theta\) is the angle between the magnetic moment and the magnetic field.

4. **Proportionality to Sine Function**:
From the equation, we can see that the torque is directly proportional to \(\sin(\theta)\):
\[ \tau \propto \sin(\theta) \]

5. **Graph Characteristics**:
- At \(\theta = 0^\circ\), \(\sin(0) = 0\) → \(\tau = 0\)
- At \(\theta = 90^\circ\), \(\sin(90) = 1\) → \(\tau\) is maximum.
- At \(\theta = 180^\circ\), \(\sin(180) = 0\) → \(\tau = 0\)

6. **Graph Shape**:
The graph of torque (τ) versus angle (θ) will be sinusoidal. It will start at zero, reach a maximum at \(90^\circ\), and return to zero at \(180^\circ\).

7. **Conclusion**:
The correct representation of the τ-θ graph for a coil carrying current in a magnetic field is a sinusoidal curve.

### Final Answer:
The correct graph for the torque versus angle (τ-θ) for a coil is a sinusoidal graph that starts at zero, reaches a maximum at \(90^\circ\), and returns to zero at \(180^\circ\).

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