A current loop in a magnetic field

Options

A. can be in equilibrium in one orientation
B. can be in equilibrium in two orientations both the equilibrium states are unstable
C. can be in equilibrium in two orientations, one stable while the other is unstable
D. experiences a torque whether the field is uniform or non-uniform in all orientations

Answer

To solve the problem of a current loop in a magnetic field and its equilibrium states, we can break it down into the following steps:

### Step 1: Understanding the Current Loop in a Magnetic Field
A current loop placed in a magnetic field experiences a torque due to the interaction between the magnetic moment of the loop and the magnetic field. The magnetic moment (M) is a vector quantity that depends on the current (I) flowing through the loop and the area (A) of the loop.

### Step 2: Torque on the Current Loop
The torque (τ) experienced by the loop can be expressed using the formula:
\[ \tau = \mathbf{M} \times \mathbf{B} \]
Where:
- \( \tau \) is the torque,
- \( \mathbf{M} \) is the magnetic moment,
- \( \mathbf{B} \) is the magnetic field.

The magnitude of the torque can also be expressed as:
\[ \tau = MB \sin \theta \]
Where:
- \( \theta \) is the angle between the magnetic moment and the magnetic field.

### Step 3: Condition for Equilibrium
For the current loop to be in equilibrium, the net torque must be zero:
\[ \tau = 0 \]
This implies:
\[ MB \sin \theta = 0 \]

### Step 4: Solving for Angles
From the equation \( MB \sin \theta = 0 \), we can deduce that:
1. \( M \) and \( B \) cannot be zero (as they represent physical quantities).
2. Therefore, \( \sin \theta = 0 \).

The sine function is zero at:
- \( \theta = 0^\circ \) (stable equilibrium)
- \( \theta = 180^\circ \) (unstable equilibrium)

### Step 5: Analyzing the Equilibrium States
- At \( \theta = 0^\circ \): The magnetic moment is aligned with the magnetic field, leading to a stable equilibrium. Any small displacement will result in a restoring torque.
- At \( \theta = 180^\circ \): The magnetic moment is opposite to the magnetic field, leading to an unstable equilibrium. Any small displacement will result in a torque that moves the loop further away from this position.

### Conclusion
Thus, we conclude that a current loop in a magnetic field can be in two orientations: one stable (0°) and one unstable (180°).

### Final Answer
The current loop can be in two orientations: one stable and one unstable.

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