ELECTROMAGNETIC INDUCTION
10.1 Experimental Observations

- A conductor rod that slides at constant speed  on two conducting rails terminating in a galvanometer is
part of a closed passive circuit (without efm); generally, there is no current in the circuit. But, if the rod
moves inside a magnetic field, the galvanometer shows a current passing through the circuit (fig.1)
So, “a current appears in a conductor moving inside a magnetic field ”. Actually, this is not a “strange


effect”. The free electrons inside the conductor move with velocity  with respect to magnetic field B .


I


G
Fig.1

B

The magnetic force
(1)
FB = -e  x B
This force makes the electrons move “down”
and a current (in opposite direction) passes
through the circuit as long as the wire moves.
I2
(1.b)
- Consider now the rod at rest and a current I2 passing through a second wire located close by as
shown in figure 1.b. We know that this current builds a magnetic field around it. This field will penetrate
in the area limited by the circuit. The observations show that, starting from the moment t=0 (when the
current is set into the second wire) and during a very short time interval (Δt) the galvanometer needle moves;
so “a current appears in the circuit”. Also, for t > t+Δt, there is no more current in circuit. This effect was
observed for the first time by Joseph Henry.
-It is clear that, in the two cases, the appearance of an electric current in a circuit is due to “some kind of
action from the magnetic field”. Before going into a better understanding of these phenomena, keep in
mind that a current means charges in movement and this requires the presence of an electric field. So, in
these experiments the magnetic field action consists to the creation of an electric field in the circuit. The
electromagnetic induction is the induction of a current in a circuit due to a magnetic field action.
- From electrical point of view, one may say that, as a current passes through circuit, there is an emf in
action in the electrical circuit. It remains to define the magnitude and the direction of this emf.
a) Consistent observations showed that the magnitude of this inducted emf is related to the change of

magnetic flux ΔΦB inside the closed circuit. The flux ΦB of a magnetic field B through a loop,

n̂
B

(or a closed circuit) is defined as
 
 B  B * nˆA  B * A  BA cos
(2)

A  nˆA is the area vector of loop. A is the area of loop; n̂ is a unit vector with tail
θ
+ circulation
Fig.2

at loop center and directed perpendicular to loop plane such that angle to B at t=0
is θ<90o. Then, after aligning the thumb on n̂ direction, the right hand rule defines
the positive sense of circulation on the loop along the direction of curled fingers.

Note that for a uniform field B , the flux ΦB can be changed by changing the loop
area (fig 3a) or by changing the angle θ (fig3.b). The rotation of loop (change of θ)
is the situation generally met in practice (AC generators, electric motors,..).
1
Fig3.b
Fig3.a

Also, one may change the flux ΦB by changing the field B (magnitude or direction). In fig.4, the magnetic
field build by the primary circuit passes through the secondary circuit. One changes the magnitude of

B inside the area of secondary circuit by changing the current Ip in the primary circuit.
The unit of magnetic flux in SI system is the Weber (Wb). From the
expression (2), one may find that
(3)
1Wb  1T *1m 2
In this introduction for flux concept, we assumed a loop inside an
uniform magnetic field. In a more general situation, the magnetic field
may change from one location to another and the flux is defined as the
sum of all local contributions. This means that the general formulae for
the magnetic flux through a loop must be
Fig.4
B 


 B d A
(4)
Loop_ area
- Faraday defined the mathematical expression for the magnitude of inducted emf. He found that the
magnitude of inducted emf is proportional to the change rate of the magnetic flux through the loop

d B
dt
(5)
d B d ( BA cos  ) dB
dA
d


A cos   B cos   BA sin 
dt
dt
dt
dt
dt
As
(6)
the Faraday law shows that there are three ways to induct an emf (B change, A change, θ change).
Important: In practice, one deals mainly with the change of the first and the third factors “B ; θ”.
b) The direction of inducted emf is defined by Lenz law. The Lenz law is a particular expression of
action reaction behaviour in nature. It says that: While an exterior action is changing the flux, the
inducted emf in circuit has such a direction that the related current produces a magnetic field
tenting to keep the previous value of magnetic flux through the circuit.
+
Fig4.a
(b)
(c) t = 0
(d) t = dt
(e) t = dt
2
In fig4.a the external action is increasing the flux; the inducted current has such a direction that the related
field Bind (directed upside) works against the exterior action (by decreasing the net field). In fig4.b the
external action is decreasing the flux; the inducted current has such a direction that the related field Bind
(directed downside) works against the exterior action (by increasing the net field).
In fig4.c the circuit is in a stationary state; DC current constant magnetic field and there is no flux change
through the circuit (ε = 0). In fig4.d an external action is decreasing B magnitude (ΔΦB < 0). The inducted
current has a direction which tents to compensate for external action by creating an “up” directed Bind. In
fig4.e the external action is increasing B field (Δ ΦB > 0) and the inducted current direction is tenting to
compensate for external action, by creating a inducted magnetic field Bind directed “down” which tents to
decrease the net magnetic flux.

- Now, let’s use the figures 4.c,d,e to include the Lenz law into expression (5). At t = 0, there is a field B


through the circuit area. We chose the area vector A directed parallel to B so that at t = 0 ΦB is positive.
This way we fix the selection for the positive direction (fig4.c) of the current and emf around the circuit.
Then, at a moment t = dt, the figures 4.d,e show that the sign of inducted emf (following assigned definition)is
the opposite of sign at flux change ΔΦB. So, one gets the Faraday-Lenz law
 
In case of a coil with N turns the expression (7) becomes
 
d B
dt
(7)
d ( N B )
dt
(8)
Precisions.
a) Where is located the induced emf source? In the case of electromagnetic induction the emf
source is the magnetic field itself. It is distributed along the whole a circuit and its action is
distributed around the whole circuit.
b) Lines of a field are used to visualise the field direction and its magnitude (their density at a given location

B magnitude at this place) but they do not have any physical meaning. Meanwhile the
magnetic flux is a basic physical quantity at origin of a natural phenomenon.
is proportional to


c)The change of magnetic field B induces an electric field E which produces a current into the circuit.
This current induces an additional magnetic field in the surrounding space which brings a change of
magnetic flux and so on…. This short comment shows that there is a close relationship between the
changes of magnetic and electric fields. In fact, the electromagnetic induction acts in a much larger
frame. It is at the origin of all types of existing electromagnetic waves (light, TV, radio, X-rays,…).
10.2 GENERATORS OF ELECTRIC CURRENT
Essentially, an electric generator is a coil with N turns1(or loops) that rotates at constant angular velocity ω
(θ = ω*t) inside a uniform magnetic field (Fig.5) and its function is based on electromagnetic induction.



Assume that at t=0 the coil plan is perpendicular to B ; we assign the surface vector A parallel to B (θ =0 ).


So, at t = 0 the flux through the coil  B  N ( B* A)  NBA cos   NBA is positive and maximum.
1
The magnetic flux through the coil is Φcoil = N*Φone-loop
3

If the coil starts to rotate as shown, the angle between two vectors A and

B increases, the magnetic flux changes. At a moment t ≠ 0 the angle
between vectors is θ = ω*t and the flux is decreased to


(t ) B  N ( B* A)  NBA cos t
(9)
The equation (7) shows that in the coil is inducted the emf
 

B
Fig.5
d B
 NAB sin t
dt
(10)
This positive emf produces a current which flows along the positive
direction of circulation on the coil as shown in figure.
From expression (10) one can see that the inducted emf is maximum

 0  NAB

(11)

when θ = 900 i.e. when A perpendicular to B which means the coil plane is parallel to field B .
Now we can rewrite the expression (10) in form
(12)
   0 sin t
This expression shows that the inducted emf and consequently the current in the coil change their
direction in sinusoidal way. So, an alternating sinusoidal emf (fig.6) (and current) is induced in coil.
In a real AC generator the current goes out of coil into the external circuit by use of a set of two brush
contacts that slide on two rings which rotate together with the coil (see fig.7).
Fig.6
Fig.7
The commutator is a set of two half-rings welded to coil wires (from inside, Fig.8) and sliding over two
fixed brushes (from outside) while rotating with the coil. Let’s consider a CCW rotation at the moment
when θ = 900 (i.e. when ε =+εo ) and the current is sent outside by left side brush, as presented in fig.8.
I

B
Fig8
Fig.9
Fig.10
When the coil is rotated by 1800(i.e. θ becomes 2700) the direction of current in coil becomes "-" versus area
vector and this sends the current anew to the “left side brush”. So, the brush on the left gets from the coil
always the same positive part of current generated by the coil and sends it out versus the external circuit.
This device transforms the alternating current into a DC pulse current (Fig.9).
Wheatstone improved the quality of DC current (minimized the pulses) by using a system of multiple coils
and commutators oriented at different angles. This improvement brought to DC generators we use today.
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10.3 MOTIONAL EMF

-Let’s consider a conducting rod with length l moving at constant velocity  perpendicular to a uniform



magnetic field (Fig.11). Due to magnetic force ( FB  e x B ) the free electrons move to the lower end
and leave a net positive charge at the upper end of the rod. This situation builds up an electrostatic field



E 0 directed downward. The related electrostatic force acting upon the electron FE  e E0 is directed in




opposite sense of FB ( FE  FB ). The magnitude of electric force FE increases with the increase of charge
concentration at rod’s ends.
After a short interval of time it becomes equal to magnitude of

FB and electrons don’t move any more versus the rod’s ends
because the exerted net force becomes zero. This way a steady
state equilibrium is set up. Starting from this moment
(13)
eE0  eB _ i.e. _ E0  B
At steady state equilibrium there is a potential difference
Fig.11
Vb  Va  E0l  Bl
between rod’s ends
(14)
By referring to the terminology used for electric circuits, one should say that the potential difference
is due to the presence of an emf inside the rod. This kind of emf is known as motional emf. Since no
current is flowing through the rod one finds easily that potential difference between two rod extremes is
 motion Vb  Va  Bl
(15)
- What is the source of motional emf? To answer this question one must look closer the work spent
during the shift of electrons. There are two external factors that act on the rod; the magnetic field and the
external force that moves the rod right side. We will consider the action of each of them separately.

a) Work by Magnetic field Each electron has two components of velocity;  along the rod motion

and  d (drift velocity) directed downward (fig.12). The magnetic field interacts with both velocities.



The magnetic force on e- due to rod motion is F1 B  e  B . This force





provides to each e-, the power “work in 1sec” (P = F * net  F * (   d ) )







P1  F1 B * (   d )  ( e  B) * (   d )  ( eB) d

(16)


The magnetic force on e- due to its drift motion is F2 B  ed  B and it would

 



furnish the power P2  F2 B * (   d )  ( e d xB) *   ( e d B)  (eB) d
(17)
Fig.12
Opposite sense of vectors
5
So, the net power provided by magnetic field on each e- is P  P1  P2  (eB) d  (eB) d  0 (18)
This result does make sense because the magnetic force is always perpendicular to velocity which
means that it is perpendicular to displacement at any moment.
Hence, at any moment, their dot product is zero; so a magnetic force can never achieve work.

e
b) Work by force that shifts the rod. This force Frod
is exerted by the rod on each electron and it is

directed the same way as  . From Fig.12, one may see that the net force acting on each e- along the
direction of rod motion is constituted by two components




Fe  F e rod  ( e d x B)
(19)
Since the electron is moving at constant velocity right side (as a particle of the rod) it comes out that
the net force exerted on it is equal to zero. So, one gets:
F e rod  ed B  0  F e rod  ed B
(20)
If the conductor contains “n” (electron density) free electrons for m3, it has a section “A” and a length
“l”, it contains in total “n*A*l” free electrons. Since the same “part of rod force” is applied on each
of them it comes out that the net force applied on all free electrons in the rod has the magnitude
Frod  nAl * Fexte  nAl (ed B)  (nAed )lB  IlB
(21)
This result is not “unexpected” if we remember that this is the magnitude of force exerted by magnetic
field “B” on the wire with length “l ” carrying the current “I”. The external force applied on the rod
must balance the action of magnetic force so that the “wire can move at constant velocity”.
- From the energetic point of view, one may conclude that, the work spent by the external force goes to
shift the rod right side (motion at constant velocity) and to build the motional emf (15) in the rod.
Note that, although the magnetic force does not work by itself, it acts as a kind of intermediary agent
that conveys a part of external mechanical work versus “electrical energy ”.
-What happens if the rod is sliding over metal rails that close it in a circuit (fig.1)? One may assume that
the resistance of the rod is negligible versus the resistance R of the rails and this way a current I = ε / R
will flow through circuit. This current will tent to deplete the electric charge at rod ends. As a results, the
electric field in rod would decrease and the magnetic field would pump more electrons at rod terminals
to keep the same charge at terminals. The result would be a dynamic equilibrium with current I through
circuit (rails & rod) and a constant motional emf (ε =Bυl) at rod’s ends.
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