If the magnetic field is uniform over the entire surface and the entire surface lies
in the same plane, then the expression further simplifies to
1i^i d (" i r -^dBnorm
I E i s = -Aloop d ( = -AodBnorm (3-77)
Sloop dt dt
The quantity Aloop is the total area of the surface and n is the unit vector normal
to the surface. Hence, the line integral of the electric field about the closed path C is
equal to the negative of the area of the surface bound by C times the time-derivative
of the magnitude of the normal component of magnetic field through the surface.
The above expression is only valid if the magnetic field is uniform over the entire
surface of interest and the surface lies in a single plane. If B is a component of an
electromagnetic wave, then the expression is only valid if the longest dimension of the
surface is much smaller than a quarter of a wavelength.
The term on the left-hand side of Equation 3-77 is defined as the electromotive
force, or EMF, and is expressed in units of V. This can be interpreted by saying that
if a perfectly conducting wire placed along the path C is broken, the voltage measured
between the two open ends of the wire is equal the time-derivative of the normal
component of magnetic field through the surface bound by the wire times the area
bounded by the wire. Hence, a loop of wire can be used to sense the component of
the magnetic field which is normal to the plane of the loop. The open circuit voltage,
vioop(t), of a broken loop of wire in the presence of a time-varying magnetic field is
given by
vods = Aloop dBr(t) (3-78)
cVloopW = dt
A wire-loop antenna in the presence of a uniform time-varying magnetic field can
be viewed as a voltage source whose magnitude is proportional to the time derivative
of the component of the magnetic field which is normal to the plane of the loop. This
is the Thevenin equivalent voltage of a loop antenna and the basis of the equivalent