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Let us consider, in a given system of reference, a flat
capacitor, moving with speed *v*. Let
*l _{0}* and

The two moving plates can be considered as electric currents and therefore they produce a magnetic field. If we suppose to divide each plate in many tight strips parallel to the direction of the movement, as shown in the picture here on the right, where the capacitor is viewed from above, we can imagine both the plates as a set of wires carrying rectilinear currents.

If the subdivisions, that is the number of wires, are in number of N, the negative plate will be filled by currents towards the negative direction of the x-axis, while the positive one will be filled by currents in the positive direction of the same axis.

The capacitor can be considered as a solenoid if we assume that every wire on the negative plate joins the correspondent on the positive plate, in order to lead to a closed circuit.

As in a normal solenoid, there will be a uniform magnetic field
inside, directed towards the positive z-axis, and given by the
following formulas (*n* is, as usual, the number of coils
per unit length):

.

To find the current *i* we can use the following
argument: the charge *Q*, on each plate, moves the length
*l* towards the positive direction of the x-axis, in a
time . The charge, *q*,
on each subdivision is , so the current *i*
is given by. For the field *B*
we have, considering also that
*w=w _{0}*: .

Now let us suppose we have a second frame of reference moving
towards the positive direction of the x-axis, with speed
*V*. Then, in this frame of reference, the
capacitor will have a different speed, *v'*, and, if
we put and , for the electric
and magnetic field measured by the new observer, the same
formulas we have obtained before will still be valid, with the
simple replacement of *u'* for *u*:

.

Now we can use the formula for the addition of speed; if we
divide both members by *c,* we obtain . Replacing this formula in
the one that gives *E'* and *B'*,
remembering that , we obtain: . In the same way,
considering *B' _{z}*, we obtain .

If we consider again the capacitor, but now moving as in the
following picture , we will find the same results as before, but
now concerning the field's components *E _{z}*
and

If the capacitor is placed as in the following figure, we find
that *E _{x}=E'_{x}*. We can
easily understand that the electric field is the same for the
two observers because the only thing that changes if viewed by
the new observer is the distance between the plates, but the
electric field of a capacitor doesn't depend on the distance
between the plates.

It is not easy to prove that, for the magnetic field, the link
*B _{x}=B'_{x}* is true when we change
the frame of reference, in spite of the apparent simplicity of
the formula, and we give up the calculations. Summarizing the
formulas we have found we can write:

.

It is necessary to underline that all what we have said in order to obtain these formulas is not a strict proof: these reasonings are rather an explanation of the role played by the Lorent transformations in the change of the electromagnetic field from an observer to another.

What is important in these formulas is the fact that they mix the fields: it makes no sense to speak about the electric or the magnetic field, even if we consider a case where the induction phenomenon or the shifting currents don't take part in, because the only important thing is the electromagnetic field. If, for example, we have a charge in quietness, in the first frame of reference we have the electric field of Coulomb's law and no magnetic field. In the second frame of reference we'll have an electric field , not much different from the previous one, at least for low speed, but we'll have also a magnetic field, , small with small speed when γ1 and β0, but different from zero. Naturally this was already known in classical theory, because a charge in motion produces a magnetic field, but its presence can now be considered as a simple proof that electric and magnetic fields are the expression of just one physic event: the electromagnetic field. Measuring one field or the other is only a matter of frame of reference. The deep unification of these two concepts is one of the most important concepts of the theory of relativity, that really is characterized by a lot of unifications (space and time, energy and momentum, electric and magnetic fields).

It is now easy to solve a famous paradox of the classical theory of magnetic interactions.

Let's consider a wire carrying a current *i*, as in
the following picture and let's place a negative charge
*q*, at a given distance from the wire, and moving, with
respect to the wire, with the same speed *v* of the
electrons of conduction in the wire (usually about
10^{-4} m/s). In the frame of the wire a force, the
force of Lorentz, will influence the charge, brignign it
nearer to the wire.In the frame of the charge, from the
calssical point of view, nothing changes for the magnetic field
(we have positive charges moving "to the left" instead
of negative ones moving "to the right", so the same
current as before), but now the charge is at rest and so ther
will be no magnetic effects. The charge is no more attracted
towards the wire (observe that the speed we are considering is
very low).

If we describe the experiment in terms of the formulas of the
electromganeic field transformations, we will find that, in the
reference of the wire, the charge feels the effect of a magnetic
field, which has only a *B _{z}* component
coming out from the known expression of the field of a straight
wire carrying a current. In the reference of the charge the
charge itself will feel (besides the magnetic field similar to
the previous one) also the effect of an electric field given by
the formula and directed towards the
positive y-axis. This electric fiield attracts the charge
towards the wire. (The fact that the charge comes closer to the
wire has to be an invariant property, independent from the frame
of reference).

copyright 2000 et seq. maddalena falanga & luciano battaia

first published on march 11 2002 - last updated on september 01
2003