Question

For electric and magnetic fields, \(\vec{E}\) and \(\vec{B}\), due to a charge density \(\rho(\vec{r},t)\) and a current density \(\vec{J}(\vec{r},t)\), which of the following relations is/are always correct?

• A. \(\nabla \times \vec{E} = 0\)
• B. \(\nabla \cdot \vec{B} = 0\)
• C. \(\nabla \cdot \vec{J} + \dfrac{\partial \rho}{\partial t} = 0\)
• D. \(\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})\)

Hint:

The only equation not always valid is \(\nabla \times \vec{E} = 0\), which holds only in static conditions.

Answer 1

The Correct Option is B, D

Step 1: Recall Maxwell's equations.
1. \(\nabla \cdot \vec{E} = \dfrac{\rho}{\varepsilon_0}\) 2. \(\nabla \cdot \vec{B} = 0\) 3. \(\nabla \times \vec{E} = -\dfrac{\partial \vec{B}}{\partial t}\) 4. \(\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \dfrac{\partial \vec{E}}{\partial t}\)

Step 2: Analyze options.
- (A) Incorrect because \(\nabla \times \vec{E} = 0\) only for electrostatics, not always true. - (B) Correct since \(\nabla \cdot \vec{B} = 0\) holds universally (no magnetic monopoles). - (C) Correct — continuity equation derived from charge conservation: \(\nabla \cdot \vec{J} + \dfrac{\partial \rho}{\partial t} = 0\). - (D) Correct — Lorentz force law, always valid.

Step 3: Conclusion.
Hence, (B), (C), and (D) are always correct.