Maxwell's Displacement Current; Maxwell Equations
Maxwell’s Displacement current; Maxwell Equations
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1 Chiranjibi Ghimire Topics
Laws of Electric & Magnetic Field Displacement Current and its Derivation Static equation and Faraday’s law Maxwell’s Equations with modification of Ampere’s law
2 Chiranjibi Ghimire 3 Chiranjibi Ghimire Displacement Current
In Electromagnetism, displacement current is a quantity appearing in Maxwell's equations that is defined in terms of rate of change of electric displacement field. If the current carrying wire possess certain symmetry, the magnetic field can be obtained by using Ampere's law
∫B•ds =μo ⋅Ienclosed The equation states that line integral of magnetic field around the arbitrary closed loop is equal to µ0Ienc . Where Ienc is the conduction current passing through surface by closed path.
4 Chiranjibi Ghimire Derivation of Displacement Current
5 Chiranjibi Ghimire Cont’d…
6 Chiranjibi Ghimire Cont’d…
7 Chiranjibi Ghimire Cont’d…
8 Chiranjibi Ghimire Displacement Current
Maxwell realized that Ampere’s law is not valid when the current is discontinuous as is true of the current through a parallel plate capacitor: He concluded that when the charge within an enclosed surface is changing it is necessary to add to Ampere’s law another current called the displacement current
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9 Chiranjibi Ghimire Maxwell’s Equation
10 Chiranjibi Ghimire 11 Chiranjibi Ghimire Maxwell’s Equation
Differential form in Vacuum
12 Chiranjibi Ghimire Static Equation and Faraday’s Law TheThe twotwo fundamentalfundamental equationsequations ofof electrostaticselectrostatics areare shownshown below:below:
∇⋅E = ρtotal / ε0 Coulomb's Law in Differential Form Coulomb's law is the statement that electric charges create diverging electric fields. ∇×E = 0 Irrotational Electric Fields when Static This means that if everything is static, then the electric fields have no curl. TheThe twotwo fundamentalfundamental equationsequations ofof magnetostaticsmagnetostatics areare shownshown below:below: ∇⋅B = 0 No Magnetic Monopoles Electric charges give rise to diverging electric fields, magnetic charges would give rise to diverging magnetic fields. But there are no magnetic charges (no magnetic monopoles). So there is no divergence to the magnetic fields. 13 Chiranjibi Ghimire Cont’d…
∇×B = µ0Jtotal Ampere's Law for Steady Currents This equation states that steadily moving electric charges give rise to curling magnetic fields. These four equations completely specify all electromagnetic fields when everything is static in time. But what happens if something changes in time? Faraday was the first to show that these equations are not complete if we want to include time-varying effects. He showed that changing magnetic fields give rise to curling electric fields. The irrotational E field equation of electrostatics became Faraday's law in electrodynamics. The second equation now stood as : ∇×E= −∂ B/∂ t Faraday's Law of Induction 14 Chiranjibi Ghimire Cont’d But these four equations are now logically and mathematically inconsistent if we are no longer considering static situations, To show this, take the divergence of Ampere's law:
∇⋅(∇×B) = µ0∇⋅Jtotal Mathematically speaking, the divergence of the curl (shown on the left) is always zero, leading to:
0 = ∇⋅Jtotal This equation was fine for static situations, but for non-static situations, the continuity equation states: ∂ ∂ - ρtotal / t = ∇⋅Jtotal
15 Chiranjibi Ghimire Maxwell’s Equation
- It took the genius of Maxwell to realize this problem and figure out how to fix it. For this accomplishment he is now honored with the distinction of having the final four equations named after him. - Maxwell realized that to remove the contradiction, he could add an extra term to Ampere's law that would automatically make the continuity equation hold true. - Let us start with the continuity equation and work backwards to see what the more complete form of Ampere's law should look like
16 Chiranjibi Ghimire How Maxwell fix Ampere's law We can rewrite Ampere's law,
(We know B = µ0H) ∇×B = µ0 Jtotal ∇× µ0 H = µ0 Jtotal ∇×H = Jtotal Taking Divergence on both sides,
∇⋅ Jtotal = 0 (only valid for static (steady-state) problem) But for non-static situations, the continuity equation states: ∂ ∂ ∇⋅ Jtotal = - ρtotal / t What Maxwell saw was that the continuity equation could be converted into a vanishing divergence by using coulomb’s law.
17 Chiranjibi Ghimire Cont’d... In term of partial field instead of total field, and in terms of free current/charge instead of total current/charge, ∇⋅J = - ∂ ρ /∂t ∇⋅(J + ∂ D/ ∂t )= 0 ( using coulomb’s law ∇⋅D = ρ ) Then Maxwell replaced J in Ampere’s law by its generalization J→J + ∂ D/ ∂t For time-dependent field. Thus Ampere’s law becomes, ∇×H = J + ∂ D/ ∂t Still the same, experimentally verified, law for steady state phenomena, but now mathematically consistent with the continuity equation for time dependent field. 18 Chiranjibi Ghimire Cont’d…
We now have four equations which form the foundation of electromagnetic phenomena: ∇⋅D = ρ ∇×H = J + ∂ D/ ∂t
∇⋅B = 0 ∇×E + ∂ B/ ∂ t =0 An important consequence of Maxwell’s equations, as we shall see, is the prediction of the existence of electromagnetic waves that travel with 2 speed of light c =1/ μ0 ε0. The reason is due to the fact that a changing electric field produces a magnetic field and vice versa, and the coupling between the two fields leads to the generation of electromagnetic waves. 19 Chiranjibi Ghimire Thanks!!Thanks!! For your patience……
20 Chiranjibi Ghimire Backup slides…
−12 2 2 ε0 is the electric ε0 =8.854 × 10 C /(N ⋅ m ) constant µ is the magnetic 0 πµ =4 × 10−7 N/A 2 constant 0 −7 2 µ π ε 0 0 =4 × 10 [N/A ] ×8.854 × 10−12 [C2 /(N⋅ m 2 ) ] =1.1 13 × 10−17 [s2/m 2 ]
21 Chiranjibi Ghimire ……
From −17 2 2 µ ε0 0 =1.113 × 1 0 [s /m ] we can write 1 =2.998 × 10 8 m/s µ ε0 0
which is the speed of light in vacuum !
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