Question

Which Maxwell's equation represents Gauss's Law in Magnetostatics?

• A. $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$
• B. $\nabla \cdot \mathbf{B} = 0$
• C. $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
• D. $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$

Hint:

Remember: No magnetic monopoles $\Rightarrow \nabla \cdot \mathbf{B} = 0$.

Answer 1

The Correct Option is B

Concept: Maxwell's equations describe the behavior of electric and magnetic fields. Gauss's law for magnetism states that there are no magnetic monopoles.
Step 1: Understanding Gauss's Law in Magnetostatics.
It states that the net magnetic flux through a closed surface is zero.
Step 2: Mathematical expression.
This is represented as: \[ \nabla \cdot \mathbf{B} = 0 \]
Step 3: Physical meaning.
Magnetic field lines always form closed loops; they do not begin or end like electric field lines.
Step 4: Evaluating the options.

• $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$ $\rightarrow$ Gauss's law for electricity (incorrect)
• $\nabla \cdot \mathbf{B} = 0$ $\rightarrow$ Gauss's law for magnetism (correct)
• $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ $\rightarrow$ Faraday's law (incorrect)
• $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$ $\rightarrow$ Ampere's law (incorrect)

Step 5: Conclusion.
Thus, Gauss's Law in magnetostatics is $\nabla \cdot \mathbf{B} = 0$.