1. Chapter 23: Electromagnetic Waves
Chapter 31 opener. These circular disk antennas, each 25 m in diameter, are pointed to receive radio waves from out in space. Radio waves are electromagnetic (EM) waves that have frequencies from a few hundred Hz to about 100 MHz. These antennas are connected together electronically to achieve better detail; they are a part of the Very Large Array in New Mexico searching the heavens for information about the Cosmos. We will see in this Chapter that Maxwell predicted the existence of EM waves from his famous equations. Maxwell’s equations themselves are a magnificent summary of electromagnetism. We will also examine how EM waves carry energy and momentum.

2. 1. Changing Magnetic Fields Produce Electric Fields
Chapter Outline
1. Changing Magnetic Fields
Produce Electric Fields
Faraday’s
Law!
(Ch. 21)

3. Chapter Outline Maxwell’s Displacement Current
1. Changing Magnetic Fields
Produce Electric Fields
2. Modification of Ampère’s Law:
Maxwell’s Displacement Current
(Changing Electric Fields Produce Magnetic Fields)
Faraday’s
Law!
(Ch. 21)

4. Chapter Outline Maxwell’s Displacement Current
1. Changing Magnetic Fields
Produce Electric Fields
2. Modification of Ampère’s Law:
Maxwell’s Displacement Current
(Changing Electric Fields Produce Magnetic Fields)
3. Gauss’s Law for Magnetic Fields
Magnetic “Charge” Doesn’t Exist!
Faraday’s
Law!
(Ch. 21)

5. Chapter Outline Maxwell’s Displacement Current
1. Changing Magnetic Fields
Produce Electric Fields
2. Modification of Ampère’s Law:
Maxwell’s Displacement Current
(Changing Electric Fields Produce Magnetic Fields)
3. Gauss’s Law for Magnetic Fields
Magnetic “Charge” Doesn’t Exist!
4. Gauss’s Law for Electric Fields
Faraday’s
Law!
(Ch. 21)

6. Chapter Outline “Maxwell’s Equations”
1. Changing Magnetic Fields
Produce Electric Fields
2. Modification of Ampère’s Law:
Maxwell’s Displacement Current
(Changing Electric Fields Produce
Magnetic Fields)
3. Gauss’s Law for Magnetic Fields
Magnetic “Charge” Doesn’t Exist!
4. Gauss’s Law for Electric Fields
Faraday’s
Law!
(Ch. 21)
“Maxwell’s Equations”

7. “Maxwell’s Equations”
1. Changing Electric Fields Produce
Magnetic Fields
2. Modification of Ampère’s Law:
Maxwell’s Displacement Current
(Changing Electric Fields Produce
Magnetic Fields)
3. Gauss’s Law for Magnetic Fields
4. Gauss’s Law for Electric Fields
“Maxwell’s Equations”
From these come production of
Electromagnetic Waves & their Speed
And Light is an Electromagnetic Wave!

8. Derived from Maxwell’s Equations:

9. Derived from Maxwell’s Equations:
Light is an Electromagnetic Wave

10. Derived from Maxwell’s Equations:
Light is an Electromagnetic Wave
The Electromagnetic Spectrum

11. Derived from Maxwell’s Equations:
Light is an Electromagnetic Wave
The Electromagnetic Spectrum
Measuring The Speed of Light

12. Derived from Maxwell’s Equations: Electromagnetic Waves
Light is an Electromagnetic Wave
The Electromagnetic Spectrum
Measuring The Speed of Light
Energy & Momentum in
Electromagnetic Waves

13. Derived from Maxwell’s Equations: Electromagnetic Waves
Light is an Electromagnetic Wave
The Electromagnetic Spectrum
Measuring The Speed of Light
Energy & Momentum in
Electromagnetic Waves
The Poynting Vector

14. Derived from Maxwell’s Equations: Electromagnetic Waves
Light is an Electromagnetic Wave
The Electromagnetic Spectrum
Measuring The Speed of Light
Energy & Momentum in
Electromagnetic Waves
The Poynting Vector
Radiation Pressure

15. Derived from Maxwell’s Equations: Electromagnetic Waves
Light is an Electromagnetic Wave
The Electromagnetic Spectrum
Measuring The Speed of Light
Energy & Momentum in
Electromagnetic Waves
The Poynting Vector
Radiation Pressure
Radio & Television

16. Derived from Maxwell’s Equations: Electromagnetic Waves
Light is an Electromagnetic Wave
The Electromagnetic Spectrum
Measuring The Speed of Light
Energy & Momentum in
Electromagnetic Waves
The Poynting Vector
Radiation Pressure
Radio & Television
Wireless Communication

17. Electromagnetic Theory
The theoretical understanding of electricity & magnetism seemed complete by around 1850:
Coulomb’s Law & Gauss’ Law
explained electric fields & forces
Ampère’s Law & Faraday’s Law
explained magnetic fields & forces
These laws were verified in many experiments

18. Unanswered Questions (1850)
What is the nature of electric & magnetic fields?
What is the idea of action at a distance?
How fast do the field lines associated with a charge react to a movement in the charge?
James Clerk Maxwell studied some of these questions in the mid-1800’s
His work led to the discovery of electromagnetic waves

19. Discovery of EM Waves
A time-varying magnetic field causes the creation of an electric field

20. Discovery of EM Waves
A time-varying magnetic field causes the creation of an electric field
A magnetic field can produce an electric field

21. Discovery of EM Waves
A time-varying magnetic field causes the creation of an electric field
A magnetic field can produce an electric field
Maxwell proposed a modification to Ampère’s Law such that a time-varying electric field produces a magnetic field

22. Discovery of EM Waves
A time-varying magnetic field causes the creation of an electric field
A magnetic field can produce an electric field
Maxwell proposed a modification to Ampère’s Law such that a time-varying electric field produces a magnetic field
This gives a new way to create a magnetic field

23. Discovery of EM Waves
A time-varying magnetic field causes the creation of an electric field
A magnetic field can produce an electric field
Maxwell proposed a modification to Ampère’s Law such that a time-varying electric field produces a magnetic field
This gives a new way to create a magnetic field
It also gives the equations of electromagnetism some symmetry

24. A changing electric flux produces a magnetic field.
Symmetry of E and B
The correct form of Ampère’s Law (due to Maxwell):
A changing electric flux produces a magnetic field.
Since a changing electric flux can be
caused by a changing E, this was an
indication that a changing electric
field produces a magnetic field

25. A changing magnetic field produces an electric field
Symmetry of E and B
Faraday’s Law says that a changing magnetic flux produces an induced emf, & an emf is always associated with an electric field
Since a changing magnetic flux can be caused by a changing B, we can also say that
A changing magnetic field
produces an electric field

26. Symmetry of E and B

27. Maxwell’s Ideas & Reasoning
Consider Faraday’s Law:
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

28. Maxwell’s Ideas & Reasoning
Consider Faraday’s Law:
A time dependent Magnetic Field induces
an Electric Field.
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

29. Maxwell’s Ideas & Reasoning
Consider Faraday’s Law:
A time dependent Magnetic Field induces
an Electric Field.
Question: In an analogy with Faraday’s
Law, does a time dependent Electric Field
induce a Magnetic Field?
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

30. Maxwell’s Ideas & Reasoning
Consider Faraday’s Law:
A time dependent Magnetic Field induces
an Electric Field.
Question: In an analogy with Faraday’s
Law, does a time dependent Electric Field
induce a Magnetic Field?
The answer, based on experiment, is YES!!!
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

31. Maxwell’s Ideas & Reasoning
Consider Faraday’s Law:
A time dependent Magnetic Field induces
an Electric Field.
Question: In an analogy with Faraday’s
Law, does a time dependent Electric Field
induce a Magnetic Field?
The answer, based on experiment, is YES!!!
So, Ampère’s Law needs to be modified to
account for this!!!
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

32. Maxwell’s Ideas & Reasoning
Consider Faraday’s Law:
A time dependent Magnetic Field induces
an Electric Field.
Question: In an analogy with Faraday’s
Law, does a time dependent Electric Field
induce a Magnetic Field?
The answer, based on experiment, is YES!!!
So, Ampère’s Law needs to be modified to
account for this!!!
The modification is to add a time dependent
Electric Field to the right side of Ampère’s Law
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

33. Also: Consider Gauss’s Law:
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

34. Also: Consider Gauss’s Law:
Applies to Electric Fields & is about the E field
produced by electric charges.
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

35. Also: Consider Gauss’s Law:
Applies to Electric Fields & is about the E field
produced by electric charges.
Question: Is there an analogous Law for
Magnetic Fields?
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

36. Also: Consider Gauss’s Law:
Applies to Electric Fields & is about the E field
produced by electric charges.
Question: Is there an analogous Law for
Magnetic Fields?
The answer, based on experiment, is YES!!!
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

37. Also: Consider Gauss’s Law:
Applies to Electric Fields & is about the E field
produced by electric charges.
Question: Is there an analogous Law for
Magnetic Fields?
The answer, based on experiment, is YES!!!
However, experiments also show that there are
no “Magnetic Charges” (Magnetic Monopoles)
which are analogous to electric charges.
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

38. B = 0 For any CLOSED surface!
Also: Consider Gauss’s Law:
Applies to Electric Fields & is about the E field
produced by electric charges.
Question: Is there an analogous Law for
Magnetic Fields?
The answer, based on experiment, is YES!!!
However, experiments also show that there are
no “Magnetic Charges (Magnetic Monopoles)”
which are analogous to electric charges.
So, Gauss’s Law for Magnetic Fields is
B = 0 For any CLOSED surface!
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

39. B(2r) = 0Iencl for any closed loop This relates the magnetic field B
Changing Electric Fields Produce Magnetic Fields: Ampère’s Law & Displacement Current Maxwell’s Generalization of Ampère’s Law
A wire is carrying current I.
Recall Ampère’s Law:
B(2r) = 0Iencl
for any closed loop
This relates the magnetic field B
around a current to the current Iencl
through a surface
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

40. Faraday’s Law: So, changing Magnetic Fields produce currents
Changing Electric Fields Produce Magnetic Fields: Ampère’s Law & Displacement Current Maxwell’s Generalization of Ampère’s Law
Faraday’s Law:
“The emf induced in a
circuit is equal to the time
rate of change of magnetic
flux through the circuit.”
So, changing Magnetic
Fields produce currents
& thus Electric Fields
Figure Ampère’s law applied to two different surfaces bounded by the same closed path.

41. Maxwell’s reasoning about
Ampère’s Law:
For Ampere’s Law to hold, it can’t
matter which surface is chosen.
But look at a discharging capacitor;
there is current through surface 1
but there is none through surface 2:
Figure A capacitor discharging. A conduction current passes through surface 1, but no conduction current passes through surface 2. An extra term is needed in Ampère’s law.

42. Maxwell’s reasoning about
Ampère’s Law:
Therefore, Maxwell modified Ampère’s Law to include the creation of a magnetic field by a changing electric field.
This is analogous to Faraday’s Law which says that
electric fields can be produced by changing magnetic
fields. In the case shown, the electric field between
the plates of the capacitor is changing & so a magnetic
field must be produced between the plates.
Figure A capacitor discharging. A conduction current passes through surface 1, but no conduction current passes through surface 2. An extra term is needed in Ampère’s law.

43. Maxwell’s reasoning about
Ampère’s Law:
Results in a new version of Ampère’s Law:
B(2r) =
E
B(2r) =
Figure A capacitor discharging. A conduction current passes through surface 1, but no conduction current passes through surface 2. An extra term is needed in Ampère’s law.
B(2r) =
B(2r) =
B(2r) =

44. Example: Charging A Capacitor.
A C = 30-pF air-gap capacitor has circular plates of
area A = 100 cm2. It is charged by a V = 70-V battery
through a R = 2.0-Ω resistor. At the instant the battery is
connected, the electric field between the plates is
changing most rapidly. At this instant, calculate
(a) The current into the plates,
and
(b) The rate of change of the E field between the plates.
(c) The B field induced between the plates.
Assume that E is uniform between the plates at
any instant & is zero at all points beyond the
edges of the plates.
Solution: a. At t = 0, all the voltage is across the resistor, so the current is V/R = 35 A.
b. The field at any instant is (Q/A)/ε0. So dE/dt = (dQ/dt)/(Aε0) = (I/A)/ε0 = 4.0 x 1014 V/m·s.
c. Due to symmetry, the lines of B are circular and perpendicular to E. If E is constant over the area of the plates, Ampere’s law gives B = μ0ε0r0/2 dE/dt = 1.2 x 10-4 T.

45. Capacitor with Circular Plates

46. Solution: Circular Plate Capacitor
A C = 30-pF air-gap capacitor has circular plates of area A = 100
cm2. Charged by a V0 = 70-V battery through a R = 2.0-Ω resistor.
At the instant the battery is connected, the E field between the
plates is changing most rapidly. At this instant, calculate
(a) The current into the plates. We know, for charge Q
on capacitor plates, time dependence is:
I = (Q/t). At t = 0, this must
be given by Ohm’s Law: I = (V0/R) = 35A

47. Solution: Circular Plate Capacitor
A C = 30-pF air-gap capacitor has circular plates of area A = 100
cm2. Charged by a V0 = 70-V battery through a R = 2.0-Ω resistor.
At the instant the battery is connected, the E field between the
plates is changing most rapidly. At this instant, calculate
(b) The rate of change of the E field between the plates.
The E field between two closely spaced conductors is:
E = /0 = Q(t)/(0A). So (E/t) = (Q/t)/(0A) at t = 0
(E/t) = I/(0A) = 4  1014 V/(m s)

48. Solution: Circular Plate Capacitor
A C = 30-pF air-gap capacitor has circular plates of area A = 100
cm2. Charged by a V0 = 70-V battery through a R = 2.0-Ω resistor.
At the instant the battery is connected, the E field between the
plates is changing most rapidly. At this instant, calculate
(c) The B field induced between the plates.
Use Ampere’s Law with Displacement Current to solve this.
B(2πr) = 0μ0(E/t) = 0μ0(πr2)(E/t)
B = 0μ0(½)r (E/t) at outer radius r = 5.6 cm
B = 1.2  10-4 T

49. The second term in Ampere’s Law, first
written by Maxwell, has the dimensions of
current (after factoring out μ0), & is sometimes
called the Displacement Current:
B(2r) =
B(2r) =
B(2r) =
E 
where t

50. Gauss’s Law for Magnetism
Gauss’s law relates the electric field on a
closed surface to the net charge enclosed by
that surface. The analogous law for
magnetic fields is different, as there are no
single magnetic point charges (monopoles): BA BA

51. Maxwell’s Equations Maxwell’s Equations. In the absence of
We now have a complete set of equations that
describe electric and magnetic fields, called
Maxwell’s Equations. In the absence of
dielectric or magnetic materials, they are: EA BA
B
Eℓ = t
B(2r) =
E
B(2r) = t

52. Production of Electromagnetic Waves
Since a changing electric field produces
a magnetic field, and a changing
magnetic field produces an electric
field, once sinusoidal fields are created,
they can propagate on their own.
These propagating fields are called
electromagnetic waves.

53. Oscillating charges will produce electromagnetic waves:
Close to the antenna, the fields are complicated, and are called the near field:
Figure Fields produced by charge flowing into conductors. It takes time for the and fields to travel outward to distant points. The fields are shown to the right of the antenna, but they move out in all directions, symmetrically about the (vertical) antenna.

54. Far from the source, the waves are plane waves:
Figure (a) The radiation fields (far from the antenna) produced by a sinusoidal signal on the antenna. The red closed loops represent electric field lines. The magnetic field lines, perpendicular to the page, also form closed loops. (b) Very far from the antenna the wave fronts (field lines) are essentially flat over a fairly large area, and are referred to as plane waves.

55. The electric & magnetic waves are perpendicular
to each other, & to the propagation direction.
Figure Electric and magnetic field strengths in an electromagnetic wave. E and B are at right angles to each other. The entire pattern moves in a direction perpendicular to both E and B.