Inconsistency in ampere circuital law and modification by Maxwell
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The entirety of Electrodynamics at the time of Maxwell was as follows:
∇⃗ ⋅E⃗ =ρϵ0∇→⋅E→=ρϵ0
∇⃗ ⋅B⃗ =0⃗ ∇→⋅B→=0→
∇⃗ ×E⃗ =−∂B⃗ ∂t∇→×E→=−∂B→∂t
∇⃗ ×B⃗ =μ0J⃗ ∇→×B→=μ0J→
The first two are the Gauss’s law for electric and magnetic fields.The third one is known as Faraday’s law. The fourth one is the Ampere’s law. Coupled with this is the continuity equation, ∇⃗ ⋅J⃗ +∂ρ∂t=0∇→⋅J→+∂ρ∂t=0 (5) which is the mathematical representation of conservation of charge.
Now, we know that in general, the divergence of a curl is zero.
Taking the divergence of Faraday’s law (3), we see that
∇⃗ ⋅∇⃗ ×E⃗ =−∇⃗ ⋅∂B⃗ ∂t∇→⋅∇→×E→=−∇→⋅∂B→∂t
which implies
∇⃗ ⋅∇⃗ ×E⃗ =−∂∂t∇⃗ ⋅B⃗ ∇→⋅∇→×E→=−∂∂t∇→⋅B→.
The right-hand side is zero by virtue of (2) and is consistent with the fact that divergence of a curl is zero.
But, taking the divergence of Ampere’s law (4) leads to mathematical inconsistency.
∇⃗ ⋅∇⃗ ×B⃗ =μ0∇⃗ ⋅J⃗ ∇→⋅∇→×B→=μ0∇→⋅J→
which implies,
0=μ0∇⃗ ⋅J⃗ 0=μ0∇→⋅J→.
The right-hand side is zero only for the special case when J⃗ J→ is independent of position as in electrostatics and magnetostatics. But in general, ∇⃗ ⋅J⃗ ∇→⋅J→ is not zero,
∇⃗ ⋅J⃗ =−∂ρ∂t∇→⋅J→=−∂ρ∂t.
This is the inconsistency associated with Ampere’s law.
Maxwell set out to address this problem and looked for ways to resolve this using the other equations.
Note that,
∇⃗ ⋅J⃗ =−∂ρ∂t=−∂∂tϵ0∇⋅E⃗ =−∇⋅(ϵ0∂ρ∂t)∇→⋅J→=−∂ρ∂t=−∂∂tϵ0∇⋅E→=−∇⋅(ϵ0∂ρ∂t), by using (1).
This implies that
∇⃗ ⋅(J⃗ +ϵ0∂∂tE⃗ )=0∇→⋅(J→+ϵ0∂∂tE→)=0.
Therefore the modified Ampere’s law can now be written as
∇⃗ ×B⃗ =μ0J⃗ +μ0ϵ0∂∂tE⃗ ∇→×B→=μ0J→+μ0ϵ0∂∂tE→
The extra term added by Maxwell is known as the Displacement current
∇⃗ ⋅E⃗ =ρϵ0∇→⋅E→=ρϵ0
∇⃗ ⋅B⃗ =0⃗ ∇→⋅B→=0→
∇⃗ ×E⃗ =−∂B⃗ ∂t∇→×E→=−∂B→∂t
∇⃗ ×B⃗ =μ0J⃗ ∇→×B→=μ0J→
The first two are the Gauss’s law for electric and magnetic fields.The third one is known as Faraday’s law. The fourth one is the Ampere’s law. Coupled with this is the continuity equation, ∇⃗ ⋅J⃗ +∂ρ∂t=0∇→⋅J→+∂ρ∂t=0 (5) which is the mathematical representation of conservation of charge.
Now, we know that in general, the divergence of a curl is zero.
Taking the divergence of Faraday’s law (3), we see that
∇⃗ ⋅∇⃗ ×E⃗ =−∇⃗ ⋅∂B⃗ ∂t∇→⋅∇→×E→=−∇→⋅∂B→∂t
which implies
∇⃗ ⋅∇⃗ ×E⃗ =−∂∂t∇⃗ ⋅B⃗ ∇→⋅∇→×E→=−∂∂t∇→⋅B→.
The right-hand side is zero by virtue of (2) and is consistent with the fact that divergence of a curl is zero.
But, taking the divergence of Ampere’s law (4) leads to mathematical inconsistency.
∇⃗ ⋅∇⃗ ×B⃗ =μ0∇⃗ ⋅J⃗ ∇→⋅∇→×B→=μ0∇→⋅J→
which implies,
0=μ0∇⃗ ⋅J⃗ 0=μ0∇→⋅J→.
The right-hand side is zero only for the special case when J⃗ J→ is independent of position as in electrostatics and magnetostatics. But in general, ∇⃗ ⋅J⃗ ∇→⋅J→ is not zero,
∇⃗ ⋅J⃗ =−∂ρ∂t∇→⋅J→=−∂ρ∂t.
This is the inconsistency associated with Ampere’s law.
Maxwell set out to address this problem and looked for ways to resolve this using the other equations.
Note that,
∇⃗ ⋅J⃗ =−∂ρ∂t=−∂∂tϵ0∇⋅E⃗ =−∇⋅(ϵ0∂ρ∂t)∇→⋅J→=−∂ρ∂t=−∂∂tϵ0∇⋅E→=−∇⋅(ϵ0∂ρ∂t), by using (1).
This implies that
∇⃗ ⋅(J⃗ +ϵ0∂∂tE⃗ )=0∇→⋅(J→+ϵ0∂∂tE→)=0.
Therefore the modified Ampere’s law can now be written as
∇⃗ ×B⃗ =μ0J⃗ +μ0ϵ0∂∂tE⃗ ∇→×B→=μ0J→+μ0ϵ0∂∂tE→
The extra term added by Maxwell is known as the Displacement current
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Answer:
Maxwell's equations are a set of four differential equations that form the theoretical basis for describing classical electromagnetism: Gauss's law: Electric charges produce an electric field. The electric flux across a closed surface is proportional to the charge enclosed.
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