Using Stokes' Theorem, which of the following is Faraday's Law induction in differential form?

1)
∇
∙
𝐵
→
=
∂
𝐸
→
∂
𝑡

2)
∇
×
𝐵
→
=
−
∂
𝐸
→
∂
𝑡

3)
∇
∙
𝐸
→
=
∂
𝐵
→
∂
𝑡

4)
∇
×
𝐸
→
=
−
∂
𝐵
→
∂
𝑡

Question

Using Stokes' Theorem, which of the following is Faraday's Law induction in differential form?

1) 
∇
∙
𝐵
→
=
∂
𝐸
→
∂
𝑡

2) 
∇
×
𝐵
→
=
−
∂
𝐸
→
∂
𝑡

3) 
∇
∙
𝐸
→
=
∂
𝐵
→
∂
𝑡

4) 
∇
×
𝐸
→
=
−
∂
𝐵
→
∂
𝑡

BlackTom AI · Accepted Answer

Stokes' theorem relates the circulation of a vector field around a closed curve to the surface integral of its curl: ∮_C E · dl = ∫_S (∇×E) · dA. Faraday's law in integral form states that the electromotive force around a closed loop equals the negative time rate of change of magnetic flux through any surface bounded by that loop:
∮_C E · dl = − d/dt ∫_S B · dA.

1) Relevant knowledge points and concepts
- Stokes' theorem: ∮_C F · dl = ∫_S (∇×F) · dA.
- Faraday's law (integral form) as given above.
- If two surface integrals of vector fields are equal for every surface S, their integrands must be equal pointwise.

2) Complete reasoning process step by step
Start with Faraday's integral law:
∮_C E · dl = − d/dt ∫_S B · dA.
Apply Stokes' theorem to the left-hand side:
∫_S (∇×E) · dA = − d/dt ∫_S B · dA.

3) Intermediate calculation steps
Exchange time derivative and surface integral (assuming smooth fields):
∫_S (∇×E) · dA = − ∫_S (∂B/∂t) · dA.
Bring terms together:
∫_S [ (∇×E) + ∂B/∂t ] · dA = 0.
Since this holds for any surface S, the integrand must vanish everywhere:
∇×E + ∂B/∂t = 0,
therefore
∇×E = − ∂B/∂t.

4) Why the correct answer is correct
This differential equation is the local (differential) form of Faraday's law, obtained directly by applying Stokes' theorem to the integral form and using arbitrariness of S. It expresses how a time-varying magnetic field generates a curl (circulation) of the electric field.

5) Why the other options are incorrect
- Option 1: ∇·B = ∂E/∂t is incorrect. The correct Gauss law for magnetism is ∇·B = 0 (no magnetic monopoles).
- Option 2: ∇×B = −∂E/∂t is incorrect in sign and content. Maxwell–Ampère law is ∇×B = μ0 J + μ0ε0 ∂E/∂t (positive displacement current term and source current term μ0 J).
- Option 3: ∇·E = ∂B/∂t is incorrect. Gauss's law for electric fields is ∇·E = ρ/ε0 (electric charge density), not related to time derivative of B.

6) Conclusion confirming the final answer
Therefore the correct differential form of Faraday's law is ∇×E = −∂B/∂t, which corresponds to option 4. This matches the provided answer.

Using Stokes' Theorem, which of the following is Faraday's Law induction in differential form? 1) ∇ ∙ 𝐵 → = ∂ 𝐸 → ∂ 𝑡 2) ∇ × 𝐵 → = − ∂ 𝐸 → ∂ 𝑡 3) ∇ ∙ 𝐸 → = ∂ 𝐵 → ∂ 𝑡 4) ∇ × 𝐸 → = − ∂ 𝐵 → ∂ 𝑡 单项选择题

类似问题

Using Stokes' Theorem, which of the following is Faraday's Law induction in differential form? 1) ∇ ∙ 𝐵 → = ∂ 𝐸 → ∂ 𝑡 2) ∇ × 𝐵 → = − ∂ 𝐸 → ∂ 𝑡 3) ∇ ∙ 𝐸 → = ∂ 𝐵 → ∂ 𝑡 4) ∇ × 𝐸 → = − ∂ 𝐵 → ∂ 𝑡

A circular coil with an area of 0.0500 m2 is wrapped with 500 turns of wire and is placed in a uniform magnetic field perpendicular to its plane. If the field changes in value from -0.100 T to +0.150 T in an interval of 0.500 s, what is the magnitude of the average voltage induced in the coil?

As a magnet is moved within a coil, the rate of change of the magnet field is proportional to the coil's induced (output) voltage. This describes which law?

A flat square coil of wire with 15 turns and an area of 0.40 m2 is placed with the plane of its area parallel to a magnetic field of 0.75 T. The coil is flipped so its plane is perpendicular to the magnetic field in a time of 0.050 s. What is the magnitude of the average induced emf in the coil?

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