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: Recall the physical idea behind Faraday’s law. 
Faraday’s law explains how a changing magnetic field gives rise to an electric field.
When the magnetic flux through a surface varies with time, it induces an electromotive force (EMF), leading to a circulating electric field.

Step 2: Express the law mathematically.
In integral form, Faraday’s law is written as:

\[ \oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \]

By applying Stokes’ theorem, this integral relation is transformed into a local (differential) form that appears in Maxwell’s equations.

Step 3: Compare the given equations.
(A) $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$ describes how electric charges produce electric fields. This is Gauss’s law for electricity.

(B) $\nabla \cdot \mathbf{B} = 0$ states that magnetic monopoles do not exist. This is Gauss’s law for magnetism.

(C) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ directly links the curl of the electric field to a time-varying magnetic field. This is exactly the differential form of Faraday’s law.

(D) $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$ (without the displacement current term) represents Ampère’s circuital law.

Step 4: Final conclusion.
The mathematical expression corresponding to Faraday’s law of electromagnetic induction is:

\[ \boxed{\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}} \]