Question

A current \( I = 10A \) flows in an infinitely long wire along the axis of a hemisphere. The value of \( \int (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{s} \) over the hemispherical surface as shown in the figure is: 

• A. \( 10\mu_0 \)
• B. \( 5\mu_0 \)
• C. 0
• D. \( 7.5\mu_0 \)

Hint:

In electromagnetism, symmetry often simplifies complex integrals, leading to results like zero flux.

Answer 1

The Correct Option is C

Step 1: Understanding the integral.
The integral \( \int (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{s} \) represents the flux of the vector product of the velocity and magnetic field over the hemispherical surface. Since the current flows along the axis of the hemisphere, the cross-product of \( \mathbf{v} \) and \( \mathbf{B} \) will result in a zero flux through the surface due to symmetry. The flux across the hemispherical surface will cancel out.
Step 2: Conclusion.
The correct answer is option (C) because the flux integral evaluates to zero due to symmetry.