Using the Divergence Theorem, which of the following is Gauss' Law in differential form?

1)
∇
×
𝐵
→
=
1
𝜖
0
𝑞

2)
∇
×
𝐸
→
=
1
𝜖
0
𝑞

3)
∇
∙
𝐵
→
=
1
𝜖
0
𝑞

4)
∇
∙
𝐸
→
=
1
𝜖
0
𝑞

Question

Using the Divergence Theorem, which of the following is Gauss' Law in differential form?

1) 
∇
×
𝐵
→
=
1
𝜖
0
𝑞

2) 
∇
×
𝐸
→
=
1
𝜖
0
𝑞

3) 
∇
∙
𝐵
→
=
1
𝜖
0
𝑞

4) 
∇
∙
𝐸
→
=
1
𝜖
0
𝑞

BlackTom AI · Accepted Answer

Gauss’s law in integral form states that the electric flux through a closed surface equals the enclosed charge divided by ε0:
∮_S E · dA = Q_enc / ε0.

1) Relevant knowledge points and concepts
- Divergence Theorem: ∮_S F · dA = ∫_V (∇·F) dV, relating a surface integral to the volume integral of the divergence.
- Gauss’s law (integral form) above.
- Differential form of a field law: replace surface integral equality with an equality of integrands using the Divergence Theorem.
- Note: Q_enc can be written as ∫_V ρ dV, where ρ is the charge density (some answers use q to denote charge density).

2) Complete reasoning process step by step
Start with Gauss’s law (integral form):
∮_S E · dA = Q_enc / ε0.
Express Q_enc as a volume integral of charge density:
Q_enc = ∫_V ρ dV, so ∮_S E · dA = (1/ε0) ∫_V ρ dV.
Apply the Divergence Theorem to the left-hand side:
∫_V (∇·E) dV = (1/ε0) ∫_V ρ dV.

3) Intermediate steps (explicit)
∮_S E · dA = ∫_V (∇·E) dV
Q_enc/ε0 = (1/ε0) ∫_V ρ dV
Therefore ∫_V (∇·E) dV = ∫_V (ρ/ε0) dV.

4) Why the correct answer is correct
Because the equality of the two volume integrals must hold for any arbitrary volume V, their integrands must be equal pointwise:
∇·E = ρ/ε0.
This is Gauss’s law in differential form. In the question’s notation where q denotes the local charge density, this matches option 4: ∇·E = (1/ε0) q.

5) Why the other options are incorrect
- Option 1 (∇×B = 1/ε0 q): Curl of B is related to current density and time derivative of E (Ampère–Maxwell law), not directly to charge density.
- Option 2 (∇×E = 1/ε0 q): Curl of E equals −∂B/∂t (Faraday’s law), not charge density.
- Option 3 (∇·B = 1/ε0 q): Divergence of B is zero (no magnetic monopoles): ∇·B = 0, so it cannot equal charge/ε0.

6) Conclusion
Therefore the correct differential form of Gauss’s law is option 4, ∇·E = (1/ε0) q, which matches the provided answer.

Using the Divergence Theorem, which of the following is Gauss' Law in differential form? 1) ∇ × 𝐵 → = 1 𝜖 0 𝑞 2) ∇ × 𝐸 → = 1 𝜖 0 𝑞 3) ∇ ∙ 𝐵 → = 1 𝜖 0 𝑞 4) ∇ ∙ 𝐸 → = 1 𝜖 0 𝑞 单项选择题

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