PES 1120 Spring 2014, Spendier
Lecture 38/Page 1
Today: Start last chapter 32
- Maxwell’s Equations
James Clerk Maxwell (1831-1879)
Scottish mathematical physicist.
He united all observations, experiments and
equations of electricity, magnetism, and optics into
a consistent theory.
Since light is an "electromagnetic" wave (more on
this in Physics 3), light also must satisfy Maxwell's
equations.
E&M Equations So Far
Gauss’s Law for E-Field :
Gauss’s law for electrostatics states that the electric flux through a closed surface is
proportional to the charge enclosed. The electric field lines originate from the positive charge
(source) and terminate at the negative charge (sink).
  q
 E   E  dA  enc
(ε0 = permittivity of free space)

0
surface
Gaussian surface
Gauss’s Law for B-Field
One would then be tempted to write down the magnetic equivalent as
  Qm
B
  dA  0
surface
where Qm is the magnetic charge (monopole) enclosed by the Gaussian surface. However,
despite intense search effort, no isolated magnetic monopole has ever been observed. Hence,
Qm = 0 and Gauss’s law for magnetism becomes

surface
 
B  dA  0
PES 1120 Spring 2014, Spendier
Lecture 38/Page 2
Gaussian surface
This implies that the number of magnetic field lines entering a closed surface is equal to the
number of field lines leaving the surface. That is, there is no source or sink. In addition, the
lines must be continuous with no starting or end points. In fact, as shown for a bar magnet,
the field lines that emanate from the north pole to the south pole outside the magnet return
within the magnet and form a closed loop.
Faraday’s Law of Induction
The most general form for Faraday’s Law is:
 
dB
dt
with
 
 B   B  dA
(induced emf)
(magnetic flux)
Now, let’s explore a very important and strange aspect of this.
Induced EMF
•Let’s take a very long (infinite) solenoid with a changing current.
•Remember that in this case, the magnetic field outside the solenoid is extremely weak.
•Now, outside of the conductor, let’s put a conducting wire loop, with a galvonometer to
measure current.
•It is safe to say that the magnetic field at the wire loop is zero.
•But there IS a changing flux through the loop.
•Will there be an induced current in the loop? YES!
PES 1120 Spring 2014, Spendier
Lecture 38/Page 3
But what force makes the charges move around the wire loop? It can't be a magnetic
force because the loop isn't even in a magnetic field. We are forced to conclude that there
has to be an induced electric field in the conducting ring caused by the changing
magnetic flux.
This is strange since we are accustomed to thinking about electric field as being caused
by electric charges, and now we are saying that a changing magnetic field somehow acts
as a source of electric field. This means that even if there is no conducting ring there,
there is an electric field!
What is the direction of the electric force on a positive point charge?
Let’s look at the equation for electric potential change going from some point a to point b
in the presence of a charge:
 
V  Vb  Va    E  ds
b
a
Now, what would we get if we turned around and returned to point a? In other words,
what is the integral of E dot ds around a closed path?

 
 
E  ds  
 E  ds  Va  Va  0
closed path
How would this answer change in the presence of a changing magnetic flux?
 
d B
E
  ds   dt
This is a new relation. It states that an electric field is induced along a closed loop by a
changing magnetic flux in the region encircled by that loop. This is a second way of
writing Faraday's Law of induction and is one of the 4 Maxwell's equations.
PES 1120 Spring 2014, Spendier
Lecture 38/Page 4
Correction to Ampere’s Law
Because symmetry is often so powerful in physics, we should be tempted to ask whether
induction can occur in the opposite sense; that is, can a changing electric flux induce a
magnetic field? The answer is that it can; furthermore, the equation governing the
induction of a magnetic field is almost symmetric with the above equation.


dE
(Maxwell's law of induction)
dt
(μ0 = permeability of free space)
 B  ds   
0 0
A magnetic field is induced along a closed loop by a changing electric flux in the
region encircled by that loop.
Example: A parallel-plate capacitor with circular plates of radius R is being charged. The
change of the electric field over time is dE/dt = 1.50 x 1012 V/(m*s). What is the field
magnitude B for r = R/5 = 11.0 mm and
PES 1120 Spring 2014, Spendier
Lecture 38/Page 5
Now recall that the left side Maxwell's law of induction, the integral of the dot product
around a closed loop, appears in another equation, namely, Ampere’s law:


 B  ds   I
0 enc
(Ampere's law)
where Ienc is the current encircled by the closed loop. Thus, our two equations that specify
the magnetic field produced by means other than a magnetic material (that is, by a current
and by a changing electric field) give the field in exactly the same form. We can combine
the two equations into the single equation


 B  ds   I
0 enc
 0  0
dE
(Ampere-Maxwell law)
dt
When there is a current but no change in electric flux (such as with a wire carrying a
constant current), the first term on the right side is zero, and so the equation reduces to
Ampere’s law. When there is a change in electric flux but no current (such as inside or
outside the gap of a charging capacitor), the second term on the right side is zero, and so
the equation reduces to Maxwell’s law of induction.
In 1865 (right after the American Civil War), James Clerk Maxwell was examining
Ampere’s Law and found that it needs this addition. Fixing the flaw led to a fundamental
shift in the way we understood nature. Because of that, all of the E&M equations were
renamed in his honor.
PES 1120 Spring 2014, Spendier
Lecture 38/Page 6
So now we are ready to write down all 4 Maxwell's equations:
Law

Gauss's law for E
Equation
  qenc
E
  dA   0
surface
Faraday's law
 E  ds  


Gauss's law for B

Physical Interpretation
Electric flux through a closed
surface is proportional to the
charged enclosed.
Changing
magnetic
flux
produces an electric field.
The total magnetic flux
through a closed surface is
zero.
(no
magnetic
monopoles)
Electric current and changing
electric flux produces a
magnetic field.
d B
dt
 
B  dA  0

surface
 
dE
B
  ds  0 I enc  0 0 dt
Ampere-Maxwell law
In the absence of sources where, qenc = Ienc = 0 (or in vacuum), the above equations become
Law
Equation
 
E
  dA  0

Gauss's law for E
surface
 
d B
E
  ds   dt
Faraday's law

Gauss's law for B

 
B  dA  0
surface


 B  ds   
Ampere-Maxwell law
0 0
dE
dt
An important consequence of Maxwell’s equations is the prediction of the existence of
electromagnetic waves that travel with speed of light
c
1

0 0
1
 4 10
7
T  m / A8.85 10
12
2
2
C / N m

 2.997 108 m / s .
The reason is due to the fact that a changing electric field produces a magnetic field and
vice versa, and the coupling between the two fields leads to the generation of
electromagnetic waves. The prediction was confirmed by H. Hertz in 1887.
(need to use Stokes theorem and wave equation to proof....)