A circular flexible loop of wire of radius r carrying a current I is placed in a uniform magnetic field B . If B is doubled, then tension in the loop

Options

A. remains unchanged
B. is doubled
C. is halved
D. becomes 4 times

Answer

To solve the problem, we need to analyze the forces acting on a circular flexible loop of wire carrying a current \( I \) in a uniform magnetic field \( B \). The key here is to understand how the tension in the loop changes when the magnetic field is doubled.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a circular loop of radius \( r \) carrying a current \( I \).
- The loop is placed in a uniform magnetic field \( B \).

2. **Forces on the Loop**:
- The magnetic force acting on a current-carrying conductor in a magnetic field is given by the formula:
\[ F = I \cdot L \cdot B \cdot \sin(\theta) \]
where \( L \) is the length of the conductor, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the direction of the current and the magnetic field.

3. **Analyzing the Circular Loop**:
- The loop can be divided into two halves: the upper half and the lower half.
- For the upper half, the force acting upwards due to the magnetic field is:
\[ F_{\text{upper}} = I \cdot (2r) \cdot B \cdot \sin(90^\circ) = 2IrB \]
- For the lower half, the force acting downwards is:
\[ F_{\text{lower}} = I \cdot (2r) \cdot B \cdot \sin(90^\circ) = 2IrB \]
- Since the forces are equal and opposite, they balance each other out, leading to no net force on the loop.

4. **Tension in the Loop**:
- The tension in the loop is what keeps it in equilibrium against the magnetic forces. This tension is directly related to the magnetic field strength \( B \).

5. **Effect of Doubling the Magnetic Field**:
- If the magnetic field \( B \) is doubled (i.e., \( B \rightarrow 2B \)), the forces acting on the upper and lower halves of the loop will also double:
\[ F_{\text{upper}}' = 2I \cdot (2r) \cdot (2B) = 4IrB \]
\[ F_{\text{lower}}' = 2I \cdot (2r) \cdot (2B) = 4IrB \]
- The new tension in the loop will also be affected by this change in the magnetic field.

6. **Conclusion**:
- Since the tension is directly proportional to the magnetic field strength, if \( B \) is doubled, the tension in the loop will also double.
- Therefore, the new tension \( T' \) can be expressed as:
\[ T' = 2T \]
- Where \( T \) is the original tension in the loop.

### Final Answer:
The tension in the loop will double when the magnetic field is doubled.