Transcript AC Circuits - San Jose State University

Electromagnetic Waves Ch. 32
Maxwell’s equations
Plane EM waves & speed of light
The EM spectrum
C 2009 J. Becker
(sec. 32.1)
(sec. 32.2)
(sec. 32.6)
MAXWELL’S EQUATIONS
The relationships between electric
and magnetic fields and their
sources can be stated compactly in
four equations, called
Maxwell’s equations.
Together they form a complete
basis for the relation of E and B
fields to their sources.
C 2004 Pearson Educational / Addison Wesley
A capacitor being charged by a current ic has a
displacement current equal to iC between the plates,
with displacement current iD = e A dE/dt. This changing
E field can be regarded as the source of the magnetic
field between the plates.
A capacitor being charged by a current iC has a
displacement current equal to iC in magnitude between
the plates, with
DISPLACEMENT CURRENT
iD = e A dE/dt
From C = e A / d and DV = E d we can use
q = C V to get
q = (e A / d ) (E d ) = e E A = e F E and
from iC = dq / dt = e A dE / dt = e dF E / dt = iD
We have now seen that
a changing E field can produce a B field
and from Faraday’s Law
a changing B field can produce an E field (or emf)
C 2009 J. Becker
MAXWELL’S EQUATIONS
The relationships between electric
and magnetic fields and their
sources can be stated compactly in
four equations, called
Maxwell’s equations.
Together they form a complete
basis for the relation of E and B
fields to their sources.
C 2004 Pearson Educational / Addison Wesley
An electromagnetic wave front. The plane representing
the wave front (yellow) moves to the right with speed c.
The E and B fields are uniform over the region behind
the wave front but are zero everywhere in front of it.
Gaussian surface for a
plane electromagnetic wave.
The total electric flux and
total magnetic flux through
the surface are both zero.
Applying Faraday’s law to a plane wave.
E
dl = - d FB /dt
1.  E o dl = -Ea
(cos 90o = 0)
2. In time dt the wave front
moves to the right a distance
c dt. The magnetic flux
through the rectangle in the
xy-plane increases by an
amount d FB equal to the flux
through the shaded rectangle
in the xy-plane with area ac
dt, that is,
d FB = Bac dt. So
-d FB / dt = -Bac and
E = Bc
Applying Ampere’s law to a plane wave.
 B dl = mo eo d FE /dt
1.  B dl = Ba
o
(cos
o
90
= 0)
2. In time dt the wave front moves
to the right a distance c dt. The
electric flux through the
rectangle in the xz-plane
increases by an amount d FE equal
to E times the area ac dt of the
shaded rectangle, that is,
d FE = Eac dt. Thus
d FE / dt = Eac.
Ba = mo eo Eac  B = mo eo E c
and from E = Bc and B = mo eo E c
1/2
We must have c = 1 / (mo eo)
c = 3.00
8
(10)
m/sec
Faraday’s law applied
to a rectangle with
height a and width Dx
parallel to the xy-plane.
Ampere’s law applied to
a rectangle with height a
and width Dx parallel to
the xz-plane.
Representation of the electric and magnetic fields in a
propagating wave. One wavelength is shown at
time t = 0. Propagation direction is E x B.
Wave front at time dt after it passes through a stationary
plane with area A. The volume between the plane and the wave
front contains an amount of electromagnetic energy uAc dt.
A standing electromagnetic wave does not propagate
along the x-axis; instead, at every point on the x-axis
the E and B fields simply oscillate.
Examples of standing electromagnetic waves
Microwave ovens have a standing wave with l =
12.2 cm, a wavelength that is strongly absorbed
by water in foods. Because the wave has nodes
(zeros) every 6.1 cm the food must be rotated
with cooking to avoid cold spots!
Lasers are made of cavities of length L with
highly reflecting mirrors at each end to reflect
waves with wavelengths that satisfy L = m l / 2,
where m = 1, 2, 3, …
C 2009 J. Becker
The electromagnetic spectrum. The frequencies and
wavelengths found in nature extend over a wide range.
The visible wavelengths extend from approximately
400 nm (blue) to 700 nm (red).
One cycle in the production of an electro-magnetic wave
by an oscillating electric dipole antenna. The red arrows
represent the E field. (B not shown.)
Review
See www.physics.edu/becker/physics51
C 2009 J. Becker