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 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:
Show Hint
- \( 10\mu_0 \)
- \( 5\mu_0 \)
- 0
- \( 7.5\mu_0 \)
The Correct Option is A
Solution and Explanation
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.
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