Question

Choose the correct answer from the options given below.
A. \( \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} \) represents Ampere's circuital law.
B. \( \vec{\nabla} \cdot \vec{B} = 0 \) represents magnetic monopole doesn't exist experimentally.
C. \( \vec{\nabla} \times \vec{E} = 0 \) represents the conservative nature of the electric field.
D. \( \vec{\nabla} \times \vec{A} = 0 \) represents the solenoidal condition for a vector field.

• A. (A), (B) and (D) only.
• B. (A), (B) and (C) only.
• C. (A), (B), (C) and (D).
• D. (A), (C) and (D) only.

Hint:

The curl of a magnetic field \( \vec{\nabla} \times \vec{B} \) is proportional to the current density, and the curl of an electric field \( \vec{\nabla} \times \vec{E} \) is zero in static conditions.

Answer 1

The Correct Option is C

Step 1: Understanding the terms.
(A) \( \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} \) represents Ampere's law, where \( \mu_0 \) is the permeability of free space and \( \vec{J} \) is the current density.
(B) \( \vec{\nabla} \cdot \vec{B} = 0 \) implies that there are no magnetic monopoles.
(C) \( \vec{\nabla} \times \vec{E} = 0 \) represents the conservative nature of the electric field, which is true in electrostatics.
(D) \( \vec{\nabla} \times \vec{A} = 0 \) represents a condition for a solenoidal vector field.

Step 2: Conclusion.
All of the statements (A), (B), (C), and (D) are correct.