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MAGNETIC BRAKING
ABSTRACT
In this paper the basic of magnetic braking are introduced. Firstly, a simple theory is proposed
using Faraday's law and the Lorentz force. With this theory magnetic braking on copper
rectangular sheet moving linearly through the magnet is explained. Secondly, a magnetic drag
force and a magnetic lift force on a magnetic dipole moving over a nonmagnetic conducting
plane are explained with image method based on Maxwell’s equations. At the end of the
seminars the practical uses of forces on moving magnets are shown.
1. INTRODUCTION
The topic of magnetic braking has dramatically increased in popularity in recent years. Since
1987, numerous articles about magnetic braking were published. These articles describe both
experiments dealing with magnetic braking, as well as the theory behind the phenomenon.
Magnetic braking works because of induced currents and Lenz’s law. If you attach a metal
plate to the end of a pendulum and let it swing, its speed will greatly decrease when it passes
between the poles of a magnet. When the plate enters the magnetic field, an electric field is
induced in metal and circulating eddy currents are generated. These currents act to oppose the
change in flux through the plate, in accordance with Lenz’s Law. The currents in turn heat the
plate, thereby reducing its kinetic energy.
The practical uses for magnetic braking are numerous and commonly found in industry today.
This phenomenon can be used to damp unwanted nutations in satellites, to eliminate
vibrations in spacecrafts, and to separate nonmagnetic metals from solid waste [1].
2. THEORY
The subject of magnetic braking is rarely discussed in introductory physics texts. To calculate
the magnetic drag force on a moving metal object is generally difficult and implies solving
Maxwell's equations in time-dependent situation. This may be one of the reasons why the
phenomenon of magnetic braking, although conceptually simple to understand, has not
attracted the attention of textbooks authors. A simple approximate treatment is however
possible in some special cases. In our seminar we will try to explain magnetic braking with
the understandable (simple) theory. Reports in literature have made the theory behind this
phenomenon easily accessible. First we will be interested in the braking of a rectangular sheet
moving linearly through the magnet.
2. 1 Magnetic braking of a rectangular sheet moving linearly through the magnet
A good source for explaining why this braking happens we find in [2]. We assume that the
speed of the sheet is sufficiently small that the magnetic field generated by the induced
current is negligible in comparison with the applied magnetic filed. Under this condition just
stated, the magnetic drag force is seen to arise from mutual coupling between the induced
current and the applied magnetic field.
When the metal plate enters the magnetic field, a Lorentz force
F q(v B)
_ _ _
, (1)
3
is exerted on the conduction of electrons in the metal. Here, v
_
is the velocity vector of the
charge q, and B
_
is the magnetic field vector. The force on the electrons induces a current in
the metal (eddy current). An induced current moves along a closed path as if induced by an
electromotive force. Figure 1 shows these eddy currents in relation to the metal plate which
moves perpendicular to the magnetic field.
Figure 1: Induced currents in the metal plate [2].
We use Faraday’s law, which says that the magnitude of the induced emf is equal to the time
rate of change of the magnetic flux,
B dS v(B L).
dt
d
dt
d
Ui
_ _ _ _ _
_ (2)
A horizontal magnetic force is exerted on the portion of the eddy current that is within the
magnetic field. This force is transmitted to the metal sheet, and is the retarding force
associated with the braking:
F IL B,
_ _ _
(3)
where I is the current and L is the vertical height of the magnetic field.
Like we said when the metal sheet passes between the poles of the magnet, circulating
currents (eddy currents) are generated. As a result, a magnetic breaking force is induced on
the eddy currents which opposes the motion of the sheet. This is a simple theory of magnetic
braking, which assumes that the magnetic field generated by the induced current is negligible
in comparison with the applied magnetic filed. But we would like to have a theory, which
does not assume that the magnetic field generated by the induced current is negligible.
In next sections of our seminar equations for a magnetic drag force and a magnetic lift force
(a magnetic drag force acts together with a magnetic lift force) on a magnetic dipole moving
over a nonmagnetic conducting plane are shown. A magnet (a magnetic dipole) is moved
along the plane (in x direction), in which therefore the eddy currents are induced. Eddy
currents generate the magnetic field and in this magnetic field the magnet experiences the
magnetic force with two components: up (in z direction) and in opposite direction as the
magnet moves. As we already mentioned this are the magnetic lift force and the magnetic
drag force. To get equations for both we will use the image method based on Maxwell’s
equations [3].
The aim of this theory is also qualitatively to describe the magnetic field generated by the
induced eddy currents. These eddy currents are induced in the plane.
2. 2 Image method based on Maxwell’s equations (The Principle of Mirror Images)
We approximate movement of the magnet over the conducting plane with series of sudden
jumps. Firstly, we look example when at time t = 0 a magnetic dipole suddenly appears over
the conducting plane (Figure 2a). The eddy currents, which are generated in the plane, protect
the place on the other side of the plane (negative side of the plane) from changing the
magnetic field. In [3] it’s discussed:
Negative side: The magnetic field of eddy currents has together with the magnetic field of the
dipole in every point value 0. The magnetic field of the eddy currents on the negative side
equals to the magnetic field of the switched magnetic dipole on the positive side (Figure 2c).
Positive side (side, on which magnet is): Symmetry of the problem implies that the magnetic
field of the eddy currents is equal on both sides of the plane. The magnetic field of the eddy
currents on the positive side equals to the magnetic field, which is generated by mirror image
of the magnetic dipole on the negative side (Figure 2b).
Figure 2: The magnetic field of the magnetic dipole (a), magnetic field of induced eddy currents on
positive side of the plane (b) and magnetic field of induced eddy currents on negative side. The
complete magnetic field is shown in Figure 3 [3].
When the magnetic dipole suddenly appears on the positive side of the plane, there is no
magnetic field on the negative side of the plane, but on the positive side of the plane the
magnetic field of eddy currents has influence on the magnetic field of the magnetic dipole, it
fakes the magnetic field of the dipole (Figure 3).
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Figure 3: The magnetic field on positive side of the plane, when the magnetic dipole appears over the
conducting plane [3].
If we are interested in the force on the magnet, we are only interested in the magnetic field on
the positive side of the plane; therefore we will focus on mirror images of the magnetic dipole
on the negative side.
When the magnetic dipole suddenly disappears, two mirror images are created: one on the
positive side and the other on the negative side, magnetic fields are in opposite direction like
in a previous case.
2. 3 Velocity of mirror images
In superconductor eddy currents would last for ever, but in the conductor they disappear with
time (they are less and less stronger) and heat up the plane. How quickly eddy currents
disappear depends on the conductivity, the thickness c, and the permeability of the
metal [3]. Theory points out that the magnetic field of the eddy currents on the positive side is
weaken by time like mirror image on the negative side would move perpendicular away from
the metal plane with a velocity
c0
w (4)
2. 4 Force on magnet moving over conducting plane
2. 4. 1 Qualitative explanation with method of discrete steps
Imagine our movement of the magnetic dipole over conducting plane with small steps –
jumps. The magnetic dipole does not need any time for jumping on other place, on each place
the magnetic dipole waits short period of time dt. We are interested in the magnetic field on
the positive side of the plane, which is result of mirror images (of the magnetic dipole) on the
negative side (under the plane). When the dipole suddenly jumps on the next place, two
images are woken. One (negative) image appears under old location and other (positive)
under new location. Figure 4 shows 1. couple of images, made at last jump. At next jump the
story is the same, again couple of images is made and older couples propagate downward at
velocity w. The magnetic field, at point where dipole is, equals to sum of all magnetic fields
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of mirror images under the plane. If we want to know the force on a moving magnetic dipole,
we have to sum all magnetic fields of mirror images.
Figure 4: A magnet moves over conducting plane in right with the velocity v, under the plane there are mirror
images, which the magnet is leaving behind. Every step of the magnetic dipole (jump) takes no time and the
magnetic dipole stays on each place short period of time dt [4].
Figure 5: An example where the magnet moves over the conducting plane in left with the velocity v.
The velocity of magnet is less than the velocity of mirror images [4].
w
w > v
v
vdt
vdt
vdt
vdt
wdt
wdt
wdt
3. couple
4. couple
2. couple
vdt
z0
1. couple
Figure 6: An example where the magnet moves over the conducting plane in right with the velocity v.
The velocity of magnet is greater than the velocity of mirror images [4].
Two examples applying the image method are shown in Figure 5 and Figure 6. In the first
example (Figure 5) the velocity of the magnet is less than w. The positive image has moved
down the distance wdt when the negative image appears at the same location. Then, as the two
images move away head-to-tail, induced field falls to zero. In the second example (Figure 6),
the velocity is considerably greater than w. The positive image has moved only a small
distance when the negative image appears, and the two images nearly cancel each other
thereafter. In both figures (Figure 5 and Figure 6) slope of mirror images depends on both
velocities (
v
w
).