9.2 Induced Electric Fields
9.2.1 Motional Electromotive Force
We consider conducting material to have a number
(essentially infinite) number of electric charge
carriers that are free to move (albeit with
resistance unless the conductor is a perfect
conductor) throughout the material in any way
demanded by external electric and magnetic fields.
That means we can consider the source of an induced
EMF by looking at the direct effects of an external
magnetic field on charges in a moving conductor.
For example, if we consider a conducting bar of
length L moving in a plane perpendicular to a constant
magnetic field, we see that the positive and negative
electric charges within the bar will separate as shown
in figure 9.8.
Figure 9.8: A conducting bar moving in a magnetic field
experiences separation of positive and negative
charges within the bar. This leads to an
induced EMF across the bar.
The positive charges are pushed to the bottom by the
velocity cross magnetic field force and the negative
charges are pushed in the opposite direction. The
charge separation creates an electric field in the
downward direction. This electric field builds as the
charge separation grows until the force on the charges
due to the resulting electric field cancels the force
due to motion in the magnetic field. That means that
we have, if we assume the charge q to mean positive
charge
where the motional EMF,
e, is just the
average electric field divided by the length of the
bar. It is induced by the motion of the bar in the magnetic
field. The magnitude of this EMF is equal in magnitude
to the product of the velocity, the magnetic field
magnitude, and the length of the bar. If we attach
this bar to frictionless rails which provide a resistance
R as shown in figure 9.9
and provide an external force on the bar to maintain its
constant velocity, then the bar is a source of
EMF and just like any other EMF source, can drive a
current through the circuit.
Figure 9.9: A conducting bar moving on frictionless
rails acts as an EMF source for pushing
current through the circuit formed by
the bar and the rails.
Note that the motional EMF is identical to the
induced EMF we calculated through Faraday's Law. In fact,
motional EMF is one example of Faraday's Law. It has a
number of practical applications.
- Problem 1:
- An electromagnetic flowmeter can be
used to measure flowrate of a conductive
fluid when it is important not to interrupt
the flow itself (e.g. blood through an
artery during heart surgery). Suppose a
voltmeter is used to determine the potential
difference across a tube of diameter d as
shown in figure 9.10
in the presence of a magnetic field. What is the
velocity magnitude, v, if the voltmeter
registers 6.12 mV for B = 0.120 T and
d = 1.2 cm?
Figure 9.10: Measuring the velocity of a conductive fluid
through a tube of diameter d in the presence
of a magnetic field.
- Solution:
- The motional EMF across the fluid to
opposite sides of the tube wall should be
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6.12×10-3 volts
(0.120 T)(0.012 m)
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| (9.2.1.15) |
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- Problem 2:
- Reconsider
problem 3 of the previous lecture
in terms of motional EMF. What is the
terminal velocity of the loop?
- Solution:
- Since all parts of the loop move at the
same terminal velocity, vy, at equilibrium,
we reconsider figure
9.7
in terms of motion EMF on each part of the loop.
First, note that the left and right sides are
completely symmetric so any forces produced by
currents induced by motional EMF must cancel.
For the top and bottom parts of the loop, we
consider each as an independent EMF source to
find
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®
v
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× |
®
B
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|·l = vy(B0ytop)l |
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| | |
®
v
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× |
®
B
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|·l = vy(B0ybot)l |
| (9.2.1.16) |
|
Note that the direction of the EMF in both the
top and bottom pieces of the loop point to the
right. Hence the bottom EMF pushes current
counter-clockwise about the loop and the top
EMF pushes current clockwise as shown in
figure 9.11.
Figure 9.11: The motional EMF on each of the four sides of the
conducting loop can be calculated. The net EMF
pushes current counter-clockwise around the loop.
The net EMF for the circuit is therefore
|
enet = ebot - etop = B0lvy(ybot - ytop) = B0lwvy |
| (9.2.1.17) |
if we arbitrarily assign counter-clockwise
as the positive direction. The induced
current in the loop is
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Iinduced = |
enet
R
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= |
B0lwvy
R
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| (9.2.1.18) |
We see that this is the same as the
current we derived
in the solution to problem 3 so the conclusion
as to the net force acting on the loop and the
terminal velocity of the loop is the same.
9.2.2 Induced Electric Fields
In general, since the force on charge carriers in a conductor
is given by
we can calculate the motional EMF for a conductor moving in
a magnetic field via the following formulation.
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| ó
(ç)
õ
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®
E
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·d |
®
l
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= | ó
(ç)
õ
| ( |
®
v
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× |
®
B
|
)·d |
®
l
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Þ |
| |
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| (9.2.2.20) |
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Hence we see that, in general, we can calculate
motional EMF's, i.e. the EMF generated in
moving conductors through the interaction of the
motion and external magnetic field. The second
equation shows that Faraday's Law predicts the
creation of an
induced electric field
in the presence of a time-varying magnetic flux.
Note that this second equation and the original
expression of Faraday's Law are equivalent and
always valid. The calculation of motional
EMF's is consistent with Faraday's Law only
for the case of motion of a conductor in a magnetic
field.
- Problem 3:
- A square conducting plate of
width and length d and mass m
falls in a constant gravitational field
with acceleration g. The plate falls
in the presence of a magnetic field B
directed into the page as shown in
figure 9.12.
Find the induced EMF between points
a and b as a function of time
assuming the plate starts falling from
rest. What electric field exists between
these points as a function of time?
Figure 9.12: A conducting square plate falls in the
presence of a magnetic field. An induced
EMF across the horizontal plate can be
calculated as a function of the vertical
velocity of the plate.
- Solution:
- The motion of the plate causes a
motional EMF which we calculate as
follows.
The electric field is strictly speaking
only established for equilibrium charge
separation, but if we consider charges
as instantaneously at rest at time t,
then the electric field is
Note that since there is no circuit,
no steady current flows through the plate.
If the magnetic field is not uniform,
however, there will be
eddy currents
which circulate throughout the plate.
9.2.3 Ampere-Maxwell Law
James Clerk Maxwell was responsible for setting down our
current understanding of the fundamental importance of
the equations that make up Gauss's Law, Faraday's Law,
and Ampere's Law. In deciding on a suitable description
of electrodynamics, he sought to place the mathematical
description on the same footing as Newton's Laws of Motions.
That he succeeded is testimony to his mathematical skill
and great physical insight.
To start, we should review here the form of the laws as
we currently describe them.
| Gauss's Law for Static Electric Charges | (ò)E·dA = q/e0 |
| Gauss's Law for Magnetism | (ò)B·dA = 0 |
| Ampere's Law | (ò)B·dl = m0Ienc. |
| Faraday's Law | (ò)E·dl = -dFB/dt |
where the second equation represents the fact that there
are no magnetic charges observed (so far) in the universe,
hence there are no point-like sources or sinks of the
magnetic field.
We first note the degree of mathematical symmetry present in
the equations despite the differences in behavior of electric
and magnetic fields. It's also clear that this symmetry between
electric and magnetic fields would be perfect except for two
obvious asymmetries within the equations themselves. The first is
that there are no magnetic charges as there are electric charges
(something that has motivated physicists to seek out such charges for
decades) and the second is that, electric fields can be created in
two ways: either from static electric charges (Gauss's Law)
or induced by time-varying magnetic fields (magnetic fluxes to be exact),
but there is no corresponding possibility of creating magnetic fields
from either static magnetic charges (they have not been found
to exist) nor from time-varying electric fields. If we simply assumed
that the latter symmetry had to exist, why would it not be apparent in
experiments that were done up to the 1800's? In this, we can make a
small digression that points out one of the prime means of gaining
insights into physics research.
Let us infer that nature is respectful of symmetry mathematically. What
would Maxwell's Equations look like completely symmetrically? First,
the existence of magnetic charges would change the form of Gauss's Law
applied to static charges. We would have
if units are to be kept consistent. What about currents involving
electric and magnetic charges? Certainly there would be contributions
to both Ampere's Law and Faraday's Law. These contributions, again
looking at consistent units, would look as follows:
where the subscripts E and B depict electric or
magnetic charges. The k in the revamped Ampere's Law
is a placeholder for the correct proportionality constant
for this term. We can find the value of this constant
in terms of the fundamental constants m0 and
e0 purely from dimensional analysis.
Since magnetic monopoles do not exist, we deal simply with
the additions necessary to consider induced magnetic fields
from time-varying electric field fluxes. This would affect
only Ampere's Law and the term added would assume that
a time-varying electric flux would induce a B field. To make
sure that the units are correct, we would want to have
Ampere's Law extended as follows:
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(ç)
õ
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®
B
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·d |
®
l
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= m0(Ienc. + Idisplacement) |
| (9.2.3.25) |
where we want
That is, this new term Idisplacement is proportional to
the time-rate of change of electric flux. The simplest form
of converting electric flux (SI units of N/(C·m2·s)) to current (C/s) is, interestingly enough, multiplication by
a constant with the units C2/(N·m2). Note that these
are precisely the units of the permittivity constant, e0!
Hence, an ansatz, or educated guess would be
This comes directly from dimensional analysis and
a "belief" in the need for symmetry among the equations
for electricity and magnetism. Note that this ansatz
also gives rise to an assumption as to why the
effect might be hard to observe. Note that,
if the above form is correct, then Ampere's Law
is changed to have the form
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(ç)
õ
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®
B
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·d |
®
l
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= m0Ienc. + m0e0 |
dFE
dt
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| (9.2.3.28) |
This means that the time-varying electric flux term is
multiplied by two small terms rather than just
one. This might make the effect weak enough to be
masked by other experimental interferences unless one
is very careful.
In fact, the form we have derived for Idisplacement,
referred to as the displacement
current, does
exist and can be verified experimentally. It was motivated
by Maxwell theoretically not from the symmetry and dimensional
analysis argument above, but from the following consideration
of a weakness in Ampere's mathematical description of his
law. Consider the case of a capacitor connected to conducting
lines. If one of the lines carries a current
I, then the
other line connected to the other plate must also carry such
a current due to the electric field of the previous plate.
Figure 9.13: A charging capacitor and the corresponding
time-varying electric field between its plates.
If we apply Ampere's Law to a circular disk around the
wire, then the magnetic field is expected to be constant
in magnitude and to have a direction which follows the
boundary. The magnitude of the B field is proportional
to the current penetrating the area bound by the path.
Figure 9.14: Magnetic field around a current as it approaches
the capacitor.
Unfortunately, the mathematical description of the area
bounded by a curve is rather loose. The area shown in
figure 9.15
is bounded by the same curve.
Figure 9.15: Magnetic field around the same perimeter which
encloses a different area.
Ampere's Law, as stated, is a line integral
which can be shown to be equivalent for the two areas and
in fact for any area bounded by the same circular curve
around the wire in the region outside the capacitor.
The unfortunate part, though, is that no current penetrates
the second surface and hence the prediction is that the
magnetic field should be zero around the boundary. To resolve
the obvious inconsistency, Maxwell generalized Ampere's Law to
include the displacement current matching just the characteristics
we gave on the basis of the dimensional argument above.
To verify the form of the displacement current, note that
a parallel-plate capacitor builds charge according to the
current impinging on the left plate for our figure.
This current is related to the change in electric flux
as follows:
If we multiply this by e0, we see that we
get the displacement current as being equal to the current
in the wires attached to the plates of the capacitors, hence
we expect to get the same value for the magnetic field no
matter whether we draw the surface so as to have the current
in the wire penetrate the surface or to have the time-varying
electric field between the capacitor plates penetrating the surface.
Let's now look at an example with numbers included.
- Problem 4:
- Find the magnetic field expected for a
capacitor with circular plates of diameter
10 cm, distance between the plates of 1 cm,
and a current of 1 A going to one of the
plates. Evaluate the magnetic field at
both 0.5 cm from the central axis of the
capacitor and at distance 20 cm from the
axis.
Figure 9.16: Magnetic field in the region between the plates
of a charging capacitor.
- Solution:
- Let's assume that the induced magnetic field
follows the symmetry of the plate, namely
that it is circular and therefore of equal
magnitude at every point around a circle of
radius r. Then, we can evaluate the electric
flux through a ring of radius r1 < R with
R = 5 cm.
For r2 > R, we note that the electric field
extends only to the edge of the capacitor, so
The time rate of change of the electric flux for these two
cases is
For either case, the integral of B over
an Amperian path gives B(2pr). Therefore,
we have, from the Ampere-Maxwell formula
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m0I
2p
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r1
R2
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= |
(4p×10-7 T·m/A)(1 A)
2p
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0.005 m
(0.05 m)2
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= 4×10-7 T |
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m0I
2pr2
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= |
(4p×10-7 T·m/A)(1 A)
2p(0.2 m)
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= 10-6 T |
| (9.2.3.33) |
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Detecting the magnetic field in the region of the
capacitor and separating out any influence
due to the magnetic field from the wires is difficult.
Fortunately, Maxwell was able to verify the correctness
of his ansatz through the theoretical discovery of even
more remarkable consequences of his revision of
Ampere's Law coupled with the other Maxwell equations.
Send comments to larryg@upenn5.hep.upenn.edu.
This page was last modified on 03/21/2003 at 10:23:59 (EST).
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