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:

Use Ampère’s law to calculate the flux when current flows along the axis of a symmetrical surface like a hemisphere.

Answer 1

The Correct Option is A

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. The current in the wire generates a magnetic field that passes through the surface of the hemisphere. Since the field is symmetric and the magnitude of the current is given as \( I = 10A \), the flux through the surface is calculated using Ampère’s law, which gives a flux of \( 10\mu_0 \).
Step 2: Conclusion.
The correct answer is option (A) because the flux integral evaluates to \( 10\mu_0 \) as per the calculation using Ampère’s law for a current flowing along the axis of the hemisphere.