A conducting circular loop of radiius `r` carries a constant current `i`. It is placed in a uniform magnetic field `vec(B)_(0)` such that `vec(B)_(0)` is perpendicular to the plane of the loop . The magnetic force acting on the loop is

Options

A. `ir B_(0)`
B. `2pi irB_(0)`
C. zero
D. `pi irB_(0)`

Answer

To solve the problem of finding the magnetic force acting on a conducting circular loop carrying a constant current in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a circular loop of radius \( r \) carrying a constant current \( I \).
- The magnetic field \( \vec{B}_0 \) is uniform and perpendicular to the plane of the loop.

2. **Identify the Force on a Differential Element**:
- Consider a small segment of the loop, denoted as \( dL \), where the current \( I \) flows.
- The magnetic force \( dF \) on this differential segment in a magnetic field is given by the formula:
\[ dF = I \, dL \times \vec{B} \]
- Here, \( dL \) is the length of the differential segment and \( \vec{B} \) is the magnetic field.

3. **Determine the Direction of the Force**:
- The direction of the force can be determined using the right-hand rule.
- If we point the fingers of our right hand in the direction of the current (along \( dL \)) and curl them towards the magnetic field \( \vec{B} \), the thumb will point in the direction of the force \( dF \).

4. **Calculate the Total Force**:
- Since the loop is circular, we can consider the forces acting on opposite segments of the loop.
- For a segment at one point on the loop, there is an equal and opposite segment directly across the loop.
- The forces on these two segments will be equal in magnitude but opposite in direction, leading to cancellation.

5. **Conclusion**:
- Since every pair of diametrically opposite segments experiences equal and opposite forces, the net force acting on the entire loop is zero:
\[ F_{\text{net}} = 0 \]

### Final Answer:
The magnetic force acting on the conducting circular loop is **zero**.

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