An arbitrary shaped closed coil is made of a wire of length `L` and a current `I` ampere is flowing in it. If the plane of the coil is perpendicular to megnetic field `vecB`, the force on the coil is

Options

A. Zero
B. iBL
C. 2iBL
D. `(1)/(2)iBL`

Answer

To find the force on an arbitrary shaped closed coil made of a wire of length \( L \) with a current \( I \) flowing through it, when the plane of the coil is perpendicular to a magnetic field \( \vec{B} \), we can follow these steps:

### Step 1: Understanding the Force on a Current-Carrying Wire
The force \( \vec{F} \) on a current-carrying wire in a magnetic field is given by the formula:
\[ \vec{F} = I \vec{L} \times \vec{B} \]
where \( \vec{L} \) is the length vector of the wire segment in the direction of the current.

### Step 2: Analyzing the Closed Coil
For a closed coil, the total length of the wire is \( L \). However, since the coil is closed, the starting point and the ending point of the wire are the same. Thus, the effective length vector \( \vec{L}_{\text{effective}} \) for the entire coil is zero because it forms a closed loop.

### Step 3: Calculating the Total Force
Since the effective length vector \( \vec{L}_{\text{effective}} \) is zero, we can substitute this into the force equation:
\[ \vec{F} = I \vec{L}_{\text{effective}} \times \vec{B} = I \cdot 0 \times \vec{B} = 0 \]

### Conclusion
The total force on the closed coil when the plane of the coil is perpendicular to the magnetic field \( \vec{B} \) is:
\[ \vec{F} = 0 \]

### Summary
The force on an arbitrary shaped closed coil with a current \( I \) in a magnetic field \( \vec{B} \) is zero due to the closed nature of the coil.

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