Introductory Physics Volume Two

116 Time Varying Fields 6.2
The induced current is caused by an electric field. This induced
electric field is not like the other electric fields we have seen. This is a
new type of electric field. The electric field is created by the changing
magnetic field.
The electric field wraps around the changing
magnetic field somewhat like a static magnetic field
wraps around a steady current. There are some additional
subtleties due to the fact that the induced
electric field is caushed by the change in the magnetic
field. The relationship between the induced electric
field and the changing magnetic field is stated in the
following law of physics.
Fact: Faraday’s Law
The line integral of the electric field around any closed curve is
equal to negative the rate of change of the magnetic flux through
any surface bounded by the curve.
∮
⃗E · ⃗dl = − d ∫
dt
⃗B · ⃗ dA
Notice that this is very similar to Ampere’s Law:
∮
∫
⃗B · ⃗dl = µ 0
⃗J · dA ⃗
Ampere’s Law and Faraday’s Law both relate the line integral around
closed curve to the flux of a vector field through the area enclosed by
the curve. The difference is that Faraday’s Law has the time derivative
of the flux.
Another important and potentially confusing thing to notice, is
that the line integral of the electric field ( ∫ ⃗ E · ⃗ dl) has so far been
referred to as the electric potential difference between the end points
of the curve. The confusing thing is that since it is a closed curve the
electric potential difference must be zero, ∆V = 0, while Faraday’s Law
tells us that the line integral of the induced electric field is not zero, but
equal to the rate of change of the magnetic flux. So we learn from this
that the force caused by the induced electric field is not a conservative
force field. We also see that there is no electric potential associated with
this electric field, ( ⃗ E ≠ − ⃗ ∇V ). To avoid confusion we will refer to the
line integral of the induced electric field by its historical name, the
electromotive force, or EMF. In equations the EMF is represented by
the symbol E, so that E = ∮ ⃗ E · ⃗ dl. Note that the unit of EMF is the
volt.