Introduction to Maxwell's Equations

Contributed by:

Jonathan James Thu, Jan 20, 2022 09:41 PM UTC

Faraday's law of induction, Ampere's circuital law, Origin of displacement current, Maxwell's equations

1. Chapter 24
Electromagnetic waves

2. So far you have learned
1. Coulomb’s Law – Ch. 19
2. There are no Magnetic Monopoles –
Ch. 22.1
3. Faraday’s Law of Induction – Ch. 23
4. Ampere’s Circuital Law – Ch. 22.9
Each law is empirical and Faraday’s is the most
remarkable of these.

3. Faraday’s Law of Induction is:
The EMF in a circuit is equal to the
1.Line integral of the magnetic field.
2.Surface integral of the curl of the magnetic
field.
3.Rate of change of the magnetic flux enclosed by
the circuit
4.Square of the hypotenuse of the current.
5.Inverse of the distance squared.

4. Faraday’s Law of Induction (Chapter 23)
  d  
 E dl  B da
C
dt S
implies an electric field is produced by a changing
magnetic field.
Imagine a highly resistive loop of wire in a changing magnetic field….
Long solenoid with changing B
.
E B

5. Ampere’s circuital law (Chapter 22.9)
   
B dl 0 I 0 J da
C S
is not correct for rapidly varying currents.

6. Something is rotten…
Remember my
“law”?
B ds  I
P
0
Apply it to this case…
For S1 : get I on the right
But for S2: get zero on the right
B ds  I
P boundary of S1
0
B ds 0
P boundary of S2
I is the current that crosses a surface bound by curve P

7. Something is rotten…
B ds  I
P
0
B ds  I
P boundary of S1
0  I 0
B ds 0
P boundary of S2

8. I can fix this!
For S1 : get I on the right and
E
dQ
I
dt
No current goes
through S2, but an Almost all
electric field does! electric flux goes
through S2
 E dA  E
 0 S1 S2
Gauss’s law

9. E
Take d/dt of this equation
S2
S1
Q
 E dA
 0 S1  S 2
dQ d E
I   0
Gives dQ/dt
dt dt
across S1
Gives dQ/dt
across S2

10. I propose to
replace I in
Ampere’s law
with: E
dQ d E
I 0
dt dt
The Ampere-Maxwell Law:
d E
 B d s   0 I   0 0
P
dt
The ordinary electrical current The displacement current

11. Origin of the term “Displacement Current”
Dielectric material
-Q +Q
I I
The dielectric material is polarized by the displacement of bound
charges, and there is a “displacement current” associated with the
movement of the bound charges.

12. Origin of the term “Displacement Current”
Vacuum
-Q +Q
I I
The amazing experimental fact is that there is a “displacement
current” associated with the [polarization of the] vacuum!
The conclusion is that
A magnetic field is produced by a changing
electric field.

13. Direct observation of the “Displacement
Current” is not so easy.
In fact only conduction currents contribute to the magnetic field at
low frequencies, so the first test by Hertz was really the best test
of the Ampere-Maxwell law: the existence of Hertzian waves.

14. And the Lorentz force from
E and B on A charges q:
My equations:
F q E  v B 
Q
E dA 
0
B dA 0
d B
E ds  dt
d E
B ds 0 I  0 0 dt

15. describe all electric and magnetic phenomena
F q E  v B 
Q
E dA 
0
and LIGHT
B dA 0
d B
E ds  dt
d E
B ds 0 I  0 0 dt

16. Maxwell’s equations
1.May be derived from pure thought.
2.Are mathematical descriptions of empirical fact.
3.Are irrelevant to modern physics.

17. Electromagnetic waves
If we look at Maxwell’s eqns where
there are no charges or currents -
after a bit of math we will get…
Units of
1/speed2
2 2
d E d E
 0 0 2  2
dt x constant
dx t constant

18. Suppose we start with a sine wave electric field in the
x direction traveling in the z direction:
E x E0 sin(kz  t )
The Ampere-Maxwell z
law says d
B( z )  0 0 E0 sin(kz ' t )dz '
dt 0
d 1
 0 0 E0   cos(kz ' t ) 0z
dt k
d 1
 0 0 E0 1  cos(kz ' t )
dt k

 0 0 E0 sin(kz ' t )
k

19. z
d
Then the Faraday law E x ( z )   B y ( z ')dz '
dt 0
of induction says
d  z
  0 0 E0  cos( kz '  t )  0
dt k2
d 1
 0 0 E0 2
 cos(kz  t )  1
dt k
2

 0 0 E0   sin( kz  t )
k
2

This can only be true if 0 0   1
k

20. 2 2
d E d E
 0 0 2  2
dt x constant
dx t constant
The speed of the
Units of waves is
1
c 2.99792 10 8 m/s
 0 0
• Light is an electromagentic wave
• It is described by Maxwell’s equations
• Electricity, magnetism and optics are
different aspects of the same theory

21. This is in the vacuum.
In a medium of dielectric constant
and magnetic permeability  the
speed of light is
1
c medium   2.99792 10 8 m/s


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