Question

If $\vec{B$ is the magnetic field in a region free of currents, then which of the following statements is/are correct? }

• A. $\vec{B} = -\nabla \phi$, where $\phi$ is the scalar potential
• B. $\vec{B}$ is rotational
• C. $\nabla \times \vec{B} = 0$
• D. $\nabla \cdot \vec{B} = 0$

Hint:

In regions free of currents, magnetic fields are both divergence-free and curl-free. This allows them to be expressed as the gradient of a scalar magnetic potential.

Answer 1

The Correct Option is A, C, D

Step 1: Recall Maxwell's equations in magnetostatics (no currents, no time-varying fields). \[ \nabla \cdot \vec{B} = 0, \qquad \nabla \times \vec{B} = \mu_0 \vec{J} = 0 \quad (\text{since } \vec{J} = 0) \] Thus, both divergence and curl of $\vec{B}$ vanish in this region. Step 2: Check Option (A).
If $\nabla \times \vec{B} = 0$, the field is irrotational. An irrotational field can be expressed as the gradient of a scalar potential: \[ \vec{B} = - \nabla \phi \] Hence, Option (A) is correct. Step 3: Check Option (B).
Saying $\vec{B}$ is "rotational" means $\nabla \times \vec{B} \neq 0$. But in the given case, $\nabla \times \vec{B} = 0$. So, Option (B) is incorrect. Step 4: Check Option (C).
As derived from Ampère's law without current, \[ \nabla \times \vec{B} = 0 \] Thus, Option (C) is correct. Step 5: Check Option (D).
From Gauss's law for magnetism, \[ \nabla \cdot \vec{B} = 0 \] Hence, Option (D) is correct. Final Answer: Correct statements are (A), (C), and (D). \[ \boxed{\text{Correct Answer: (A), (C), (D)}} \]