Page 297 - Engineering Electromagnetics, 8th Edition

CHAPTER 9 Time-Varying Fields and Maxwell’s Equations 279

We need to define emf as used in (1) or (2). The emf is obviously a scalar, and
(perhaps not so obviously) a dimensional check shows that it is measured in volts.
We define the emf as

emf = E · dL (3)

and note that it is the voltage about a specific closed path.Ifany part of the path is
changed, generally the emf changes. The departure from static results is clearly shown
by(3),foranelectricfieldintensityresultingfromastaticchargedistributionmustlead
to zero potential difference about a closed path. In electrostatics, the line integral leads
to a potential difference; with time-varying fields, the result is an emf or a voltage.
Replacing in (1) with the surface integral of B,wehave

d

emf = E · dL =− B · dS (4)
dt S
where the fingers of our right hand indicate the direction of the closed path, and
our thumb indicates the direction of dS.A flux density B in the direction of dS and
increasing with time thus produces an average value of E which is opposite to the
positive direction about the closed path. The right-handed relationship between the
surface integral and the closed line integral in (4) should always be kept in mind
during flux integrations and emf determinations.
We will divide our investigation into two parts by first finding the contribution to
the total emf made by a changing field within a stationary path (transformer emf), and
then we will consider a moving path within a constant (motional, or generator, emf).
We first consider a stationary path. The magnetic flux is the only time-varying
quantity on the right side of (4), and a partial derivative may be taken under the integral
sign,
∂B

emf = E · dL =− · dS (5)
S ∂t
Before we apply this simple result to an example, let us obtain the point form of
this integral equation. Applying Stokes’ theorem to the closed line integral, we have
∂B

(∇× E) · dS =− · dS
S S ∂t
where the surface integrals may be taken over identical surfaces. The surfaces are
perfectly general and may be chosen as differentials,
∂B
(∇× E) · dS =− · dS
∂t
and
∂B
∇× E =− (6)
∂t