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. clock wise
B. anticlockwise
C. zero
D. clockwise or anticlockwise depending on whether the resistance is increased or decreased.

Answer

To solve the problem of determining the induced current in a small conducting circular loop placed inside a long solenoid with a varying current, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a long solenoid that carries a current \( I \).
- Inside this solenoid, there is a small conducting circular loop whose 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 flowing through the solenoid.

3. **Magnetic Flux Through the Loop**:
- The magnetic flux \( \Phi \) through the loop is defined as:
\[ \Phi = B \cdot A \]
where \( A \) is the area of the loop.
- Since the magnetic field \( B \) is uniform inside the solenoid, the magnetic flux through the loop will also be uniform as long as the current \( I \) in the solenoid is constant.

4. **Variation of Current in the Solenoid**:
- If the current \( I \) in the solenoid is varied, the magnetic field \( B \) will also change as it is directly proportional to \( I \).
- This change in magnetic field will lead to a change in magnetic flux \( \Phi \) through the loop.

5. **Applying Faraday's Law of Electromagnetic Induction**:
- According to Faraday's law, the induced electromotive force (emf) \( \mathcal{E} \) in the loop is given by:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
- If the magnetic flux \( \Phi \) is changing (which it is, due to the varying current in the solenoid), then the induced emf will not be zero.

6. **Induced Current in the Loop**:
- The induced current \( I_{\text{induced}} \) in the loop can be calculated using Ohm's law:
\[ I_{\text{induced}} = \frac{\mathcal{E}}{R} \]
where \( R \) is the resistance of the loop.
- Since there is a non-zero induced emf due to the changing magnetic flux, the induced current will also be non-zero.

### Conclusion:
The induced current in the small conducting circular loop placed inside the solenoid will be **non-zero** as long as the current in the solenoid is varied.