A current carrying loop is free to turn in a uniform magnetic field. The loop will then come into equilibrium vhen its plane is inclined at

Options

A. `0^(@)` to the direction of the field
B. `45^(@)` to the direction of the field
C. `90^(@)` to the direction of the field
D. `135^(@)` to the direction of the field

Answer

To solve the problem of determining the angle at which a current-carrying loop comes into equilibrium in a uniform magnetic field, we can follow these steps:

### Step 1: Understand the Concept of Torque
A current-carrying loop behaves like a magnetic dipole. When placed in a magnetic field, it experiences a torque given by the formula:
\[ \tau = \mathbf{M} \times \mathbf{B} \]
where \(\tau\) is the torque, \(\mathbf{M}\) is the magnetic moment of the loop, and \(\mathbf{B}\) is the magnetic field.

### Step 2: Condition for Equilibrium
For the loop to be in equilibrium, the net torque acting on it must be zero:
\[ \tau = 0 \]
This occurs when the angle \(\theta\) between the magnetic moment vector \(\mathbf{M}\) and the magnetic field vector \(\mathbf{B}\) is such that:
\[ \sin \theta = 0 \]
This implies that \(\theta\) can be either \(0^\circ\) or \(180^\circ\).

### Step 3: Determine the Angle with Respect to the Plane of the Loop
The angle \(\theta\) we found is the angle between the magnetic moment vector \(\mathbf{M}\) and the magnetic field vector \(\mathbf{B}\). However, the question asks for the angle between the plane of the loop and the magnetic field.

- When \(\theta = 0^\circ\), the magnetic moment is aligned with the magnetic field, meaning the plane of the loop is perpendicular to the magnetic field. This means the angle between the plane of the loop and the magnetic field is \(90^\circ\).
- When \(\theta = 180^\circ\), the magnetic moment is anti-aligned with the magnetic field, which also means the angle between the plane of the loop and the magnetic field remains \(90^\circ\).

### Step 4: Conclusion
Thus, the loop will come into equilibrium when its plane is inclined at \(90^\circ\) to the direction of the magnetic field.

### Final Answer
The correct answer is that the loop will come into equilibrium when its plane is inclined at **90 degrees to the direction of the magnetic field**.

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