Historical burdens on physics
91 Two phenomena of electromagnetic induction
Subject:
When introducing electromagnetic induction often two realizations of induction experiments are distinguished.
In one of them an electric conductor is moved through a uniform magnetic
field that is constant in time, Fig. 1a. The interpretation of the experiment is
as follows: A Lorentz force acts on the charge carriers and displaces them
until the electrostatic force which results from the displacement equilibrates
the Lorentz force. Between the ends of the conductor an electric potential
difference has built up that can be measured with a voltmeter.
In the second induction experiment the electric conductor remains at rest,
and the magnetic field strength is changed, by moving the magnet, Fig. 1b.
Again the voltmeter deviates. This experiment cannot be interpreted with a
Lorentz force. It seems to be based on another physical effect.
However, both results can be summarized in one equation, the law of electromagnetic induction:
Uind = – dφ/dt
This procedure is commented in the following way: “Surprisingly, two different physical causes of the electromagnetic induction can be summarized in
a single equation.“
a
b
Abb. 1. Electromagnetic induction in two different reference frames. (a) Magnet at rest, wire
loop moving; (b) mangnet moving, wire loop at rest.
Deficiencies:
The same experiment is described in two different reference frames.
We begin by considering the second experiment, Fig. 1b. The wire loop is
at rest and the magnet is moving. For the interpretation the second Maxwell
equation is needed:


rot E = –B .
The magnetic flux density inside of the wire loop is changing. Thereby a

non-conservative electric field ( rot E ≠ 0 ) is created. By means of the integral form of Maxwell’s second equation
a
b
Abb. 2. A small body Q that is electrically charged moves with respect to a magnetic pole P.
The process is represented in two different reference frames. (a) The magnet is at rest, the
charged body moves in the direction perpendicular to the drawing plane. (b) Body Q is at
rest, the magnetic pole moves out of the drawing plane. The lines in the upper figures (a) are
magnetic, in the lower figure electric field lines. The two drawings at the left represent the
contributions of P and Q separately. The figures at the right show the resulting field.
∫
 
Edr = –Φ
this fact can be expressed as follows: The magnetic flux that traverses the
wire loop is changing. Thereby within the conductor a emf is created.
Now the experiment of Fig. 1a: A Lorentz force acts on the charge carriers.
This is equilibrated by an electrostatic force. There is a conservative electric


field ( rot E = 0 ). The magnetic flux density B does not depend on time. In
order to conciliate the version of the experiment with Maxwell’s second
equation one often used a somewhat inelegant mathematical trick. When
calculating the magnetic flux as the surface integral of the flux density one
admits that the integration surface changes with time. In a strict sense this
corresponds to a hidden change of the reference frame.
We see that one and the same experiment was described in two different
reference frames. When passing from one frame to the other field strengths
transform, i. e. change their values. Only in this way is it possible that in
one case we have a conservative electric field and in the other a nonconservative.
To better understand the consequences of a change of the frame of reference, let us consider an experiment that is even more simple:
A single magnetic north pole P (the end of a long permanent magnet) is
moving relative to a small body Q that is positively charged, Fig. 2. The
movement is perpendicular to the straight line connecting P and Q. We describe what happens in two reference frames: the frame in which P is at
rest (upper figures) and that in which Q is at rest (lower figures).
1. Reference frame of P
The electric charge that moves with body Q (Fig. 2a, movement into the
plane of drawing) represents an electric current. This current is surrounded
by a magnetic field. (Maxwell’s first equation is responsible for this field.) P
“feels” this field and experiences a force in the upwards direction. Meanwhile a Lorentz force that is oriented downwards acts on Q. Thus, the interaction between P and Q is mediated by a magnetic field. In the left part of
Figure 2a the contributions or P and Q to the magnetic field are represented
separately. In this way the force on P in the field of Q and the force of Q in
the field of P can be read. These forces can also be read from the right part
of the figure, in which the resulting field strengths are represented. The field
lines are denser above Q than below. Since there is compressional stress
in the direction perpendicular to the field lines, Q is pushed by the field
downwards. Moreover, the field lines are denser above P than below. Since
there is tensional stress in the direction of the field lines, P is pulled upwards by the field.
2. Reference frame of Q
The moving magnetic pole P represents a magnetic „displacement current“
(Fig. 2b, movement out of the drawing plane). This is surrounded by an
electric field. (Maxwell’s second equation is responsible for this field.) Q
feels this electric field and experiences a force in the downward direction.
Meanwhile a force that is oriented upwards acts on P: The electric analogue to the magnetic Lorentz force (magnetic current within an electric
field). Thus, the interaction between P and Q is mediated by an electric
field. Here too, the forces can be read from left as well as from the right figure.
The example shows, that in electrodynamics a change of the reference
frame can require that a phenomenon has to be described sometimes by
means of Maxwell’s first equation and sometimes by Maxwell’s second
equation, and that one and the same interaction is mediated sometimes by
an electric and sometimes by a magnetic field.
Origin:
We usually discuss reference frame changes only in mechanics, and are
not trained to at reference frame effects in other fields of physics, such as
electrodynamics or thermodynamics.
Disposal:
We know from mechanics that a change of the reference frame brings
complications, and that an inappropriately chosen of the reference frame
can make the description of a phenomenon cumbersome. If the change of
reference frames is not the actual teaching objective, we recommend to
elude the subject.
Friedrich Herrmann, Karlsruhe Institute of Technology