The Geomagnetic Field - Quantitative Plate Tectonics - page 106

Z

By Stokes' theorem (see Appendix1 ) , this
relation implies, in turn, that:

ˆ.C/ D B dS (4.27)

S.C/

r E D 0 (4.24)

is called magnetic flux through the surface S .
Eq. ( 4.26 ) says that the emf associated with the
current through the circuit C is the opposite of
magnetic flux variation through S ( C ):

This equation implies that the electric field
generated by a system of charges is irrotational ,
so that it is not possible to maintain a stationary
current along a closed circuit. However, Faraday
discovered in 1840 that a variable magnetic field
could induce an electric current through a con-
ductor cable. In particular, he found that if he
moved a magnet near an electric circuit, it was
possible to induce a stationary current through
the circuit. This phenomenon also occurred if
he varied the current through a nearby wire.
Conversely, Faraday observed that the presence
of constant magnetic fields did not induce any
current, independently from the field strength. We
call electromotive force (emf)

@ˆ
@t

E .C/ D (4.28)

This equation has great practical importance,
because it is the fundamental operating principle
for the construction of a variety of electrical
devices (e.g., electrical motors and generators).
In fact, it implies that we can convert mechanical
energy , which is necessary to move the magnet
or the conductor, into electrical energy .Onemay
wonder: what happens if we keep the magnetic
field constant, so that @ B /@ t D 0, while moving
the wire? Of course, if the conductor C moves,
then the flux ˆ( S ) must change, because the inte-
gration surface in ( 4.27 ) also varies its orientation
in space. Furthermore, the wire will be subject to
a Lorentz force ( 4.20 ), which is non-conservative
and can maintain a stationary current.
Therefore, it is possible to show that also in
this case the induction law ( 4.28 ) holds, accord-
ing to Faraday's experiments, and we have:

E

the energy that
must be supplied by a source to move a unit of
charge along a closed walk through the circuit.
This quantity can be thought as the work done
by a special kind of electric field, which does not
originate from charge distributions.
In this representation, the emf

E

coincides
with the work done by a non - conservative electric
field E to move the charge along a closed loop :

I

E .C/ D E dr (4.25)

I Z

@
@t

@ˆ
@t

E .C/ D . v B/ dr D B dS D

C

S.C/

Faraday's experiments showed that the exis-
tence of this non-conservative electric field was
related to variations of magnetic field in a way
that will be clarified now. Let C beawireloop
and S ( C ) any open surface bounded by C .Also,
let B D B ( r , t )a variable magnetic field. Then,
Faraday ' slaw states that a non-conservative elec-
tric field E D E ( r , t ) and an emf through C are
generated, such that:

(4.29)

The induction law ( 4.28 ), which links the emf
to flux variations, does not have general validity,
because is some circumstances a true circuit does
not exist, and currents can flow through a vol-
ume. Faraday himself illustrated some of these
situations, one of which is shown in Fig. 4.3 .The
device in Fig. 4.3 is called Faraday ' s dynamo .It
is made by a disk that rotates about its axis in
presence of a constant magnetic field. The angu-
lar velocity of the disk, ¨, and the magnetic field,
B , are both aligned with the rotation axis. Any
point on the disk surface has velocity v D v ( r )
lying in the disk plane, with magnitude v D ¨ r , r

I Z

@ @t d S

E .C/ D E dr D (4.26)

C S.C/

By Stokes's theorem, the Maxwell-Faraday
Eq. ( 4.11 ) follows. The quantity: