A small circular flexible loop of wire of radius `r` carries a current `I`. It is placed in a uniform magnetic field `B`. The tension in the loop will be doubled if (Choose the incorrect option)

Options

A. `I` is doubled
B. `B` is doubled
C. `r` is doubled
D. both `B` and `I` are doubled

Explanation

To solve the problem, we need to analyze how the tension in a circular loop carrying current in a magnetic field is affected by changes in current (I), magnetic field strength (B), and radius (r).

### Step-by-Step Solution:

1. **Understanding the Force on the Loop**:
- The magnetic force \( F_m \) acting on a small segment of the loop can be expressed as:
\[ F_m = I \cdot L \cdot B \cdot \sin(\theta) \]
- For a small angle \( \theta \), we can approximate \( \sin(\theta) \approx \theta \).

2. **Length of the Segment**:
- For a small segment of the loop, the length \( L \) can be represented as:
\[ L = r \cdot d\theta \]
- Here, \( r \) is the radius of the loop and \( d\theta \) is the small angle in radians.

3. **Magnetic Force Calculation**:
- Substituting \( L \) into the magnetic force equation gives:
\[ F_m = I \cdot (r \cdot d\theta) \cdot B \cdot \sin(\theta) \approx I \cdot (r \cdot d\theta) \cdot B \]

4. **Tension in the Loop**:
- The vertical component of the tension \( T_y \) must balance this magnetic force. Therefore, we can express the total tension in the loop as:
\[ T_y = I \cdot r \cdot B \]

5. **Effect of Doubling Variables**:
- If we double the current \( I \), the new tension becomes:
\[ T_y' = 2I \cdot r \cdot B = 2T_y \]
- If we double the magnetic field \( B \), the new tension becomes:
\[ T_y' = I \cdot r \cdot (2B) = 2T_y \]
- If we double the radius \( r \), the new tension becomes:
\[ T_y' = I \cdot (2r) \cdot B = 2T_y \]

6. **Combining Changes**:
- If we double both \( I \) and \( B \) simultaneously:
\[ T_y' = (2I) \cdot r \cdot (2B) = 4I \cdot r \cdot B = 4T_y \]
- This shows that the tension quadruples when both current and magnetic field are doubled.

7. **Conclusion**:
- The tension in the loop will be doubled if we double \( I \), \( B \), or \( r \) individually. However, if we double both \( I \) and \( B \) together, the tension will be quadrupled, not doubled. Therefore, the incorrect option is the one that states the tension will be doubled when both \( I \) and \( B \) are doubled.

### Incorrect Option:
- The incorrect option is the one that states that the tension will be doubled if both the current \( I \) and the magnetic field \( B \) are doubled.