Question

Which of the following Maxwell’s equations represents Faraday’s law of electromagnetic induction?

• A. \( \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \)
• 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:

The negative sign in Faraday's Law represents {Lenz's Law}, which states that the induced effect opposes the cause that produced it.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Faraday's Law states that a time-varying magnetic field induces an electromotive force (EMF), which in turn creates a circulating electric field.
Step 2: Key Formula or Approach:
The integral form is \( \oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \).
Using Stokes' Theorem, this is converted into the differential form used in Maxwell's equations.
Step 3: Detailed Explanation:
Let's look at the options:
- (A) \( \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \) is Gauss's Law for Electricity.
- (B) \( \nabla \cdot \mathbf{B} = 0 \) is Gauss's Law for Magnetism (no monopoles).
- (C) \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \) relates the curl of the electric field to the time rate of change of the magnetic field. This is the differential form of Faraday's Law.
- (D) \( \nabla \times \mathbf{B} = \mu_0 \mathbf{J} \) (ignoring displacement current) is Ampere's Circuital Law.
Step 4: Final Answer:
The equation \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \) represents Faraday’s law.