A small, conducting circular loop is placed inside a long solenoid carrying a current. The plane of the loop contains the axis of the solenoid. If the current in the solenoid is varied, the current induced in the loop is

Options

A. clockwise
B. anti-clockwise
C. zero
D. clockwise or anti-clockwise depending on whether the resistance is increased or decreased

Explanation

To solve the problem, we need to analyze the situation where a small conducting circular loop is placed inside a long solenoid that carries a current. The key aspect of this problem is to understand how the changing current in the solenoid affects the magnetic flux through the loop and subsequently induces a current in the loop.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a long solenoid carrying a current \( I \).
- A small conducting circular loop is placed inside this solenoid, and its plane contains the axis of the solenoid.

2. **Magnetic Field Inside the Solenoid**:
- The magnetic field \( B \) inside a long solenoid is given by the formula:
\[ B = \mu_0 n I \]
where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length of the solenoid, and \( I \) is the current through the solenoid.

3. **Magnetic Flux Through the Loop**:
- The magnetic flux \( \Phi \) through the loop is given by:
\[ \Phi = B \cdot A \]
where \( A \) is the area of the loop. Since the loop is placed such that its plane contains the axis of the solenoid, the angle between the magnetic field and the area vector is \( 0^\circ \) (they are parallel), and thus:
\[ \Phi = B \cdot A = \mu_0 n I \cdot A \]

4. **Induced Current in the Loop**:
- According to Faraday's law of electromagnetic induction, the induced electromotive force (emf) \( \mathcal{E} \) in the loop is given by the rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
- If the current \( I \) in the solenoid is varied, the magnetic flux \( \Phi \) through the loop will also change, leading to an induced emf.

5. **Determine the Induced Current**:
- The induced current \( I_{\text{induced}} \) in the loop can be found using Ohm's law:
\[ I_{\text{induced}} = \frac{\mathcal{E}}{R} \]
where \( R \) is the resistance of the loop. Since the emf is induced due to the changing current in the solenoid, there will be a current induced in the loop as long as the current in the solenoid is changing.

6. **Conclusion**:
- Therefore, if the current in the solenoid is varied, the induced current in the loop will be non-zero. The direction of the induced current will depend on the direction of the change in current in the solenoid (Lenz's law).

### Final Answer:
The current induced in the loop is non-zero when the current in the solenoid is varied.