Begin2.DVI

hand in the direction around C so that the thumb points in the direction of the

normal. Faraday’s law, obtained experimentally , states that the line integral of the

electric field around the closed curve Cequals the negative of the time rate of change

of the magnetic flux. This law can be written

∫

C

©E
·d
r =−∂

∂t

∫∫

S

B
·dS
 (9 .149)

Here E
is the electric field,
r is the position vector defining the closed curve C,B
is

the magnetic field and dS
=ˆendσ is a vector element of area on the surface S. The

Faraday law, given by equation (9.149), assumes the curve Cand surface Sare fixed

and do not change with time. The left-hand side of equation (9.149) is the work

done in moving around the curve Cwithin the electric field E
. The right-hand side

of equation (9.149) is the negative time rate of change of the magnetic flux across

the surface S. One can employ Stokes theorem and express equation (9.149) in the

form

∫∫

S

∇× E
·d
S=−

∫∫

S

∂B


∂t

·dS
or

∫∫

S

[

∇× E
+∂

B


∂t

]

·dS
= 0 (9 .150)

The equation (9.150) holds for all arbitrary surfaces S and consequently the inte-

grand must equal zero giving the Maxwell-Faraday equation

∇× E
=−∂

B


∂t

(9 .152)

which is the second Maxwell equation.

Example 9-12. The Biot-Savart law

Consider a volume V enclosed by a sur-

face S as illustrated and let J
=J
(x′, y′, z ′)

denote the current density within this vol-

ume. Let (x′, y′, z ′)denote a point inside V

where an element of volume dV =dx ′dy ′dz′

is constructed. In addition, construct the

vectors
r = xˆe 1 +yˆe 2 +zˆe 3 to a general

point (x, y, z) outside of the volume V and