Question

Which of the following statements is/are CORRECT according to the laws of electromagnetic induction in the Earth?

• A. Current density in a region of finite conductivity is solenoidal
• B. Magnetic vector potential is irrotational
• C. Magnetic field is solenoidal
• D. Curl of electric field is negative time rate of change of curl of magnetic vector potential

Hint:

Apply Maxwell's equations: \(\nabla\cdot\mathbf{B}=0\), \(\mathbf{B}=\nabla\times\mathbf{A}\), and Faraday's law.

Answer 1

The Correct Option is A, C, D

Step 1: Set up the governing equations.
Electromagnetic induction inside the Earth is described with Maxwell's equations in the quasi-static, low frequency form used in geophysics, where displacement current is ignored next to conduction current. The key relations are Gauss's law for magnetism, Faraday's law, and the definition of the magnetic vector potential \(\mathbf{A}\), where
\[ \mathbf{B} = \nabla \times \mathbf{A} \]

Step 2: Check option (A), current density is solenoidal.
In a region of finite, non-zero, conductivity, charge does not keep piling up as the field varies in the quasi-static regime used for EM induction, so the continuity equation gives
\[ \nabla \cdot \mathbf{J} = 0 \]
A vector field with zero divergence is called solenoidal. So the current density here is indeed solenoidal, and option (A) is correct.

Step 3: Check option (B), magnetic vector potential is irrotational.
A field is irrotational only if its curl is zero. But by definition, the curl of the magnetic vector potential gives the magnetic field itself:
\[ \nabla \times \mathbf{A} = \mathbf{B} \neq 0 \]
Since the curl of \(\mathbf{A}\) is not zero in general, \(\mathbf{A}\) is not irrotational. Option (B) is incorrect.

Step 4: Check option (C), magnetic field is solenoidal.
Gauss's law for magnetism states that magnetic field lines never start or end, meaning there are no magnetic monopoles:
\[ \nabla \cdot \mathbf{B} = 0 \]
This is exactly the definition of a solenoidal field, so option (C) is correct.

Step 5: Check option (D), curl of E versus curl of A.
Faraday's law of induction states
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Substituting \(\mathbf{B} = \nabla \times \mathbf{A}\) gives
\[ \nabla \times \mathbf{E} = -\frac{\partial}{\partial t}(\nabla \times \mathbf{A}) \]
which says the curl of the electric field equals the negative time rate of change of the curl of the magnetic vector potential. This matches option (D) exactly, so it is correct.

Step 6: Final answer.
Statements (A), (C) and (D) all follow directly from Maxwell's equations applied to EM induction in the Earth, while (B) goes against the very definition of the vector potential.
\[ \boxed{\text{(A), (C) and (D)}} \]