Lecture 17 Maxwell’s equations and the displacement current

Introduction

By studying the physical properties of electric (E-field) and magnetic (B-field) fields we have been able to describe these properties by four, relatively simple, equations (Maxwell’s equations) plus some additional  subsidiary information.

  We will see in the next lecture how these equations correctly predict the existence of electromagnetic waves. However, first, we show that one of the equations is still incomplete.

Maxwell’s equations

These four equations can be expressed in both an integral and differential form 

a)                

this result arises from Coulomb’s and Gauss’s laws and states that free charges act as sources or sinks of D. 

b)                

this result arises from the application of Gauss’s law to magnetic fields and the non-existence of magnetic monopoles. There are no sources or sinks of B. 

c)                 

this equation describes Faraday’s law of electromagnetic induction. An electromotance is produced in a circuit when the magnetic flux through the circuit changes. 

d)

this result describes Ampère’s circuital law (which is derived from the Biot-Savart law). A conduction current produces an H-field.

Subsidiary information is provided by

(a) the relationships between E and D and B and H

 D=e₀E+P or D=e₀e_rE

 H=B/m₀-M or H=B/(m₀m_r)

 (b) the total electric and magnetic force on a charged particle

 F=Q(E+v´B)

 From this equation the magnetic force on a current carrying element may be derived.

(c) the relationship between current and charge which can be expressed by the following equation

       =

The total current outwards through a closed surface equals the rate at which the total charge enclosed within that surface decreases. The final term simply states that the total charge contained within the surface is given by the integral of the free charge density r_c over the appropriate volume t.

Displacement current

Are the Maxwell’s equations as stated in the previous section and the subsidiary information mutually consistent in all situations?

Writing the equations again in their differential form

As they stand the equations appear to be some what asymmetric. In particular we have a term ¶B/¶t in the third equation for curl E but no corresponding ¶E/¶t in the fourth equation for curl H.

Although this lack of symmetry does not prove that the above equations are inconsistent, by considering the following situation we will see that an additional term must be added to the equation for curl H.

Consider the above circuit element which consists of two wires connected to the plates of a parallel plate capacitor. If a current I_c flows in the wires then the charge on the plates of the capacitor must change I_c=dQ/dt.

We now apply the circuital law for H to the above circuit.

In words this equation states that the line integral of H around a closed path L equals the conduction currents encircled by L.

By encircled we mean passing through a surface S bounded by L.

However in the above circuit for the path L there are two choices of surface S.

·        Surface S₁ passes through one of the wires and hence gives a contribution I_c to the circuital law.

·        Surface S₂ passes through the space between the capacitor plates. It does not pass through any conductor and hence does not give a contribution to the circuital law as it presently stands.

The circuital law for the path L and the two surfaces S₁ and S₂ is hence

(a)         for S₁

(b)          for S₂

However this result should be independent of the surface chosen as they are both bounded by the same path L.

(a) must be correct as we can move a long distance away from the capacitor where its effects can be neglected. Hence we need to modify the circuital law so that (b), even though S₂ does not pass through any conduction currents, gives the same result as (a).

Although S2 does not have a conduction current passing through it, it does have a time varying flux of D due to the charge on the plates of the capacitor.

For a parallel plate capacitor of plate area A

 E=Q/(e₀A) Þ D=Q/A as D=E/e₀ Þ Q=DA

 Now I_c=dQ/dt so I_c=d(DA)/dt

 We hence define a new quantity, known as the displacement current I_d, which is given in the present case by

 I_d=d(DA)/dt

 If the circuital law is now written in a modified form to include both conduction and displacement currents we obtain

 With this modification both surfaces S₁ and S₂ in the capacitor problem give the same result.

Modification of the circuital law

For the special case considered above the displacement current is given by I_d=d(DA)/dt from which we can define a more general result

J_d=¶D/¶t is the displacement current density.

Modification of Maxwell’s equation

The circuital law in the presence of both conduction and displacement currents becomes

                  Integral form

    Differential form        

Although the modification to these equations has been derived from consideration of a special system it can be shown that it is valid in all situations.

Final form of Maxwell’s equations

In differential form

                   Coulomb-Gauss’s laws - free charges are sinks/sources of E-fields

                     No sinks/sources of B-fields – no magnetic monopoles

         Faraday’s law of electromagnetic induction

   Ampère’s law with the addition of displacement currents.

In terms of B and E (using D=e₀e_rE and H=B/(m₀m_r) to eliminate D and E.

Finally in vacuo J=r=0 and m_r=e_r=1

Conclusions

·        Maxwell’s equations and subsidiary information derived from studies of electrostatics and magnetism

·        Inconsistency of Maxwell’s equation for Ñ´H in its initial form

·        Requirement for the inclusion of the displacement current I_d (or in terms of displacement current density J_d=¶D/¶t).

·        Final form of Maxwell’s equations for E, D, B and H (integral and differential forms)

·        Maxwell’s equations in terms of E and B

·        Maxwell’s equations in vacuo.

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