Page 26 - Fiber Optic Communications Fund

Electromagnetics and Optics 7

Faraday’s law is stated as
d m
e.m.f. =− , (1.28)
dt
where e.m.f. is the electromotive force about a closed path L (that includes a conductor and connections to a
voltmeter), is the magnetic flux crossing the surface S whose perimeter is the closed path L, and d ∕dt is
m m
the time rate of change of this flux. Since e.m.f. is an integrated electric field intensity, it can be expressed as

e.m.f. = E ⋅ dl. (1.29)
∮
L
The magnetic flux crossing the surface S is equal to the sum of the normal component of the magnetic flux
density at the surface times the elemental surface area dS,

= ∫ B dS = ∫ B ⋅ dS, (1.30)
n
m
S S
where dS is a vector with magnitude dS and direction normal to the surface. Using Eqs. (1.29) and (1.30) in
Eq. (1.28), we obtain
d
E ⋅ dl =− B ⋅ dS
∮ dt ∫
L S
B
=− ⋅ dS. (1.31)
∫ t
S
In Eq. (1.31), we have assumed that the path is stationary and the magnetic flux density is changing with time;
therefore the elemental surface area is not time dependent, allowing us to take the partial derivative under the
integral sign. In Eq. (1.31), we have a line integral on the left-hand side and a surface integral on the right-hand
side. In vector calculus, a line integral could be replaced by a surface integral using Stokes’s theorem,

E ⋅ dl = (∇ × E) ⋅ dS (1.32)
∮ ∫
L S
to obtain
[ B ]
∇× E + ⋅ dS = 0. (1.33)
∫
S t
Eq. (1.33) is valid for any surface whose perimeter is a closed path. It holds true for any arbitrary surface
only if the integrand vanishes, i.e.,
B
∇× E =− . (1.34)
t
The above equation is Faraday’s law in the differential form and is one of Maxwell’s four equations.

1.4.1 Meaning of Curl
The curl of a vector A is defined as

curl A =∇ × A = F x + F y + F z (1.35)
x
y
z
where
A z A y
F = − , (1.36)
x
y z