Will there be any tension in a current carrying loop placed in uniform magnetic field ? If yes, will it be equal to BiR ?

Answer

To determine whether there is any tension in a current-carrying loop placed in a uniform magnetic field and if that tension is equal to BiR, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Consider a circular loop of radius \( R \) carrying a current \( I \).
- The loop is placed in a uniform magnetic field \( B \) directed inward.

2. **Analyzing a Small Section of the Loop**:
- Focus on a small section of the loop that subtends an angle \( \theta \) at the center of the loop.
- The length of this small section can be expressed as \( dL = R \cdot d\theta \).

3. **Calculating the Magnetic Force on the Section**:
- The magnetic force \( dF \) on this small section due to the magnetic field can be calculated using the formula:
\[ dF = I \cdot dL \cdot B \cdot \sin(\theta) \]
- Since the magnetic field is perpendicular to the current in this case, we can simplify this to:
\[ dF = I \cdot dL \cdot B \]

4. **Direction of the Force**:
- Using the right-hand rule, the direction of the force \( dF \) will be upward, while the tension \( T \) in the loop acts downward.

5. **Balancing Forces**:
- The upward force due to the magnetic field must be balanced by the downward tension in the loop.
- For a small angle \( \theta \), we can consider the vertical component of the tension acting on both sides of the small section:
\[ 2T \cdot \sin\left(\frac{\theta}{2}\right) = dF \]

6. **Using Small Angle Approximation**:
- For small angles, \( \sin\left(\frac{\theta}{2}\right) \approx \frac{\theta}{2} \).
- Therefore, the equation becomes:
\[ 2T \cdot \frac{\theta}{2} = I \cdot (R \cdot d\theta) \cdot B \]
- Simplifying gives:
\[ T \cdot \theta = I \cdot R \cdot B \cdot d\theta \]

7. **Final Expression for Tension**:
- Rearranging the equation, we find:
\[ T = I \cdot R \cdot B \]

8. **Conclusion**:
- Yes, there is tension in the current-carrying loop placed in a uniform magnetic field, and it is indeed equal to \( T = BIR \).