Download 1 Induced current in a ring by a moving magnet

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Induced current in a ring by a moving magnet
Consider a metallic ring. When we move a magnet through the ring, the
change of the magnetic field will induce a current in the ring. However, if
we would observe this experiment moving along with the magnet, it seems
that the magnet does not move, but the ring moves instead. It would be
strange if the current in the ring would be different just by changing the
speed of the observer. The current through the ring should not depend
on the speed of the observer. Only the relative speed should be relevant.
With this observation started A. Einstein his paper on special relativity
“Zur Elektrodynamik bewegter Körper.
In this exercise we will check if the electrical and magnetic laws are indeed
invariant if we move either the ring or the magnet. For the calculation,
consider a metallic ring with a radius R in the xy-plane with its origin
located on the z-axis. Assume that the magnet generates the following
magnetic field


Br (r)
B(r, z) =  Bθ  ,
Bz (z)
(1)
So Bθ does not depend on the coordinate r, θ nor z.
a) The easiest way to calculate the change of the current is by moving
the ring with a speed v in the z-direction through the magnetic field
and calculating the force on the charge carriers in the ring. The timederivative of the current can be expressed as
q2v
J˙θ (t) =
Br (R),
m
(2)
where q is the charge of the charge carriers in the loop and m is their
mass. Derive this formula.
b) Now calculate J˙θ by moving the magnet through the loop. The magnet
moves in the opposite direction, so its speed is −v in the z-direction. The
magnetic field experienced by the ring is therefore B(r, z0 + vt). Show
that due to the change in the magnetic flux, the time-derivative of the
current in this frame of reference can be given as
2
q v R dBz
.
J˙θ (t) = −
m 2 dz
(3)
c) Both expressions look similar. However, they include different components of the magnetic field. However, these components are related. Use
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the fact that there are no magnetic monopoles to show that both expressions are equal, i.e.
Br (R) = −
R dBz
.
2 dz
(4)
Remember that the divergence in polar coordinates looks a bit funny
∇·F =
1 ∂
1 ∂
∂
rFr +
Fθ +
Fz .
r ∂r
r ∂θ
∂z
(5)
d) The fact that there are no magnetic monopoles restricts the shape of
our magnetic field. Since the Bz only depends on z and Br only on r
we can derive the shape by using separation of variables. Show that the
components of the magnetic field have the following form
c
cr
Br (r) = r + ,
2
r
Bz (z) = c z + cz ,
(6a)
(6b)
where c, cr and cz are constants determined by boundary conditions.
e) We showed in this specific example that the can either move the ring or
the magnet. Both give the same change in the current. However, this
should also hold for general loops and magnetic fields. Show for a general
constant velocity v, magnetic field, B(r) and form of the loop that the
force on a charged particle in the ring is the same, either moving the loop
and calculating the force on the particles or moving the magnetic field
and using the change in flux. You might need the following identity
∇ × (A × B) = A∇ · B − B∇ · A + (B · ∇)A − (A · ∇)B.
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(7)