Electromagnetism II (spring term 2020) E Goudzovski [email protected] http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2 Lecture 3 Maxwell’s equations in free space
Electromagnetism II (spring term 2020)
E Goudzovski [email protected]
http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2
Lecture 3
Maxwell’s equations in free space
Previous lecture (1)
1
Divergence (Gauss) theorem:
Integral of its divergence over the enclosed volume
KelvinStokes (curl) theorem:
Flux of a vector field through a closed surface
Line integral of a vector field around a closed curve
Flux of its curl through
any surface enclosed by the curve
Previous lecture (2)
2
Gauss’s law
Absence of magnetic poles
Faraday’s law
Ampere’s law
The principal EM laws in free space in the differential form:
This lecture
3
Lecture 3: Maxwell’s equations in free space
The continuity equation
Modification of Ampere’s law for non-steady currents
The displacement current
Maxwell’s equations of electrodynamics in free space
The continuity equation
4
Conservation of electric charge: any variation in the charge within
a closed surface is due to charges that flow across the surface
Differential form
(using the divergence theorem):
Divergence of both sides of Ampere’s law:
T1 from lecture 2 Continuity equation
Ampere’s law applies to steady currents only ( ),
and requires generalization
Modification to Ampere’s law
5
To generalize Ampere’s law for non-steady currents,
need to modify its RH side to make its divergence zero.
Gauss’ law: . Differentiation over t:
Displacement current density We found a quantity which has
zero divergence, and is suitable to modify Ampere’s law:
Not a “proof”: infinite number of ways to remove the contradiction.
Any solenoidal vector field (i.e. such that ) can be added to the RH side of the modified Ampere’s law.
(the AmpereMaxwell law)
The displacement current
6
Alternating current in a copper wire [conductivity =6×107 (·m)1]
Electric field: E = E0sin(t)
Conduction current density: JC = E = E0sin(t)
Displacement current density: JD = 0∂E/∂t = 0E0cos(t)
Ratio of maximum max conduction to displacement current:
JCmax / JD
max = E0 / 0E0 ~ 1019 s1 1
is very small for all frequencies used in practice.
A changing E-field produces B-field: Maxwell’s decisive step
towards electrodynamics (1865).
It took 30 years from Faraday’s discovery of induction to postulate the displacement current as the source of magnetic field.
Example 1
7
Q
S0
A charged sphere discharging
into external conductive medium.
By symmetry, is directed radially.
What about the field?
By symmetry, tangential component
BT=0, and radial component BR is
the same in each point of sphere S.
Assuming BR≠0 leads to
Conclude that B=0.
Ampere’s law leads to JC=0.
E-field at the surface of the sphere (lecture 1):
Conduction current at surface S0 is balanced by displacement current:
S0
Example 2
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Discharge of a capacitor:
S1
For the surface S1, only the
conduction current IC contributes.
For the surface S2, there is no conduction current; conclude that ID=IC.
S2
By definition, Lecture 1: E=/0
Check the equality ID=IC:
Example 3
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A thin parallel plate capacitor with
circular plates of radius R is being
charged. Find the magnetic field B(r).
Using axial symmetry, in the absence of conduction current (JC=0),
Inside the capacitor (r<R),
Outside the capacitor (r≥R),
Maxwell’s equations in free space
10
(M1)
(M2)
(M3)
(M4)
(remember them)
Displacement current density:
The laws of electrodynamics:
The continuity equation follows from (M1) and (M4).
Discussion and summary
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Maxwell’s equations in free space (2 scalar + 2 vector) are
equivalent to 8 scalar equations with 10 variables
(Ex, Ey, Ez, Bx, By, Bz, Jx, Jy, Jz, ).
They need to be complemented by the equations characterizing
the media. This will be discussed in lectures 59.
They are not symmetric wrt electric and magnetic fields:
no magnetic poles as sources of B field (M2);
no magnetic currents as sources of E field (M3).
For constant fields (∂E/∂t = ∂B/∂t = 0), two independent groups,
electrostatics: div E = /0; curl E = 0;
magnetostatics: div B = 0; curl B = 0J.