We will now consider the consequences of the relative field transformations on a current loop
moving to the right with respect to a magnetic field generated in Frame A. In frame A, the Lorentz force on the
charges in the loop creates motional EMF which induces current in the loop.
In Frame B comoving with the loop, the loop appears to be at rest and
the region of magnetic field is seen to be moving at speed v to the left.
There is no Lorentz force on the charged particles in the loop in Frame B because the loop is not moving.
The electric field EB is generated by the motion of the B field.
We can determine the E and B fields in Frame B using the transform equations.
The B field is the same as that generated in Frame A. The E field in Frame A is zero, so the E field
in Frame B equals the cross product of v and B. The force that causes the current in the loop
arises from the E field generated by the relative motion of the frames.
Ampere's law and displacement current
Ampere's law states that the line integral of the B field around a closed loop equals the enclosed current.
To satisfy the general case to include the case of a charging or discharging capacitor, we extend the definition
of enclosed current to include the "displacement current." The displacement current is not a current in the
usual sense, there is no charge flowing, rather it is the temporally changing electric flux through a surface.
Induced magnetic field
An induced magnetic field is one that is caused by a changing electric field. For example, a charging capacitor
has an increasing electric field between its plates. This changing electric field induces a magnetic field
inside the capacitor plates.
Maxwell's equations
James Clerk Maxwell was responsible for the generalization of Ampere's law to include displacement current.
He combined the work of physicists who came before him with this discovery to formulate a set of equations
that, together with the Lorentz force law, describe the behavior of electric and magentic fields.
Here, we have included the magnetic equivalent of Gauss's law for electric fields. It is basically a
statement of the fact that there are no magnetic monopoles. A closed Gaussian volume in a region where
there is a magnetic field present has just as much magnetic flux entering as exiting. It is impossible
to have a nonzero net flux leaving or entering because that would imply a magnetic monopole must reside
within the volume.
