The continuity equation follows by taking the divergence of the Ampère-Maxwell law and using Gauss' law. This yields ∇·J = -∂ρv/∂t.
Formula / Key Relation
∇×H = J + ∂D/∂t
Take divergence:
0 = ∇·J + ∂(∇·D)/∂t
Using ∇·D = ρv:
∇·J = -∂ρv/∂t
Detailed Explanation
Concept & ReasoningAmpère-Maxwell law states ∇×H = J + ∂D/∂t. Taking divergence of both sides gives zero on the left because divergence of a curl is always zero. On the right, ∇·J + ∂(∇·D)/∂t = 0. Gauss' law gives ∇·D = ρv. Substituting produces ∇·J + ∂ρv/∂t = 0, the differential continuity equation expressing conservation of charge. Thus the two field laws used are Ampère-Maxwell and Gauss.How to Solve It in the ExamContinuity equation = divergence of Ampère-Maxwell + Gauss law.Important Exam PointRemember the vector identity ∇·(∇×H)=0.Related Revision PathThis question sits in the revision path Electromagnetic Fields → Maxwell Equations…
Concept & Reasoning
Ampère-Maxwell law states ∇×H = J + ∂D/∂t. Taking divergence of both sides gives zero on the left because divergence of a curl is always zero. On the right, ∇·J + ∂(∇·D)/∂t = 0. Gauss' law gives ∇·D = ρv. Substituting produces ∇·J + ∂ρv/∂t = 0, the differential continuity equation expressing conservation of charge. Thus the two field laws used are Ampère-Maxwell and Gauss.
How to Solve It in the Exam
Continuity equation = divergence of Ampère-Maxwell + Gauss law.
Important Exam Point
Remember the vector identity ∇·(∇×H)=0.
Related Revision Path
This question sits in the revision path Electromagnetic Fields → Maxwell Equations → Continuity Equation .
Quick Trick
Continuity equation = divergence of Ampère-Maxwell + Gauss law.
Common Mistake
Ohm's law is not required to derive charge continuity.
Exam Tip
Remember the vector identity ∇·(∇×H)=0.
SEO + Knowledge GraphRelated PYQs & Exam ContextExam path, knowledge path and deeper practice links.
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