21.9 Magnetic Materials
The intrinsic “spin” and orbital motion of electrons gives rise to the magnetic
properties of materials è electron spin and orbits act as tiny current loops.
In ferromagnetic materials groups of 1016 - 1019 neighboring atoms form
magnetic domains where the spins of electrons are naturally aligned with
each other; magnetic domain sizes are ~ 0.01 - 0.1 mm.
An external magnetic field can induce magnetism in ferromagnetic
materials by merging and aligning domains. Depending on the material,
the induced magnetism may or may not become permanent.
Putting iron in the center of a solenoid can create a strong electromagnet
with fields 100x - 1000x the applied fields (also, can turn fields on and off).
Chapter 22
Electromagnetic
Induction
22.1 Induced Emf and Induced Current
So far we have dealt only with currents and magnetic fields which are
constant in time, and we have shown that currents (i.e. moving charges)
produce magnetic fields.
Can magnetic fields produce currents? --> yes, if the magnetic field
going through a loop of wire is changing in time!
There are a number of ways a magnetic field can be used to
generate an electric current.
It is the changing field that produces the current.
22.1 Induced Emf and Induced Current
The current in the coil is called the induced current because it is brought
about by a changing magnetic field.
Since a source emf (electromotive force) is always needed to produce a
current, the coil behaves as if it were a source of emf. This emf is known
as the induced emf.
22.1 Induced Emf and Induced Current
An emf can be induced by changing the
area of a coil in a constant magnetic field
In each example, both an emf and a current are induced because
the coil is part of a complete circuit. If the circuit were open, there
would be no induced current, but there would be an induced emf.
The phenomena of producing an induced emf with the aid of a
magnetic field is called electromagnetic induction.
22.2 Motional Emf
THE EMF INDUCED IN A MOVING CONDUCTOR
downward
force on
electrons
Each charge within the conductor
is moving and experiences a
magnetic force
F = qvB
The separated charges on the
ends of the conductor give rise
to an induced emf, called a
motional emf.
22.2 Motional Emf
Derive the motional emf, E, when v, B,
and L are mutually perpendicular
Magnetic force on a charge in the rod
FM
FM = qvB
In equilibrium, the electric force of repulsion
from charge buildup at the ends, qE (where
E is the electric field), balances FM
qE = qvB è E = vB
Gradient equation è E = |ΔV/Δs| = E/L
E/L = vB
è E = vBL
22.2 Motional Emf
Example 1 Operating a Light Bulb with Motional Emf
Suppose the rod is moving with a speed of 5.0 m/s
perpendicular to a 0.80-T magnetic field. The rod
has a length of 1.6 m and a negligible electrical
resistance. The rails also have a negligible
electrical resistance. The light bulb has a
resistance of 96 ohms. Find (a) the emf produced
by the rod, (b) the current induced in the circuit,
(c ) the electric power delivered to the bulb, and
(d) the energy used by the bulb in 60.0 s.
22.2 Motional Emf
(a)
(b)
(c )
EE = vBL = (5.0 m s )(0.80 T )(1.6 m ) = 6.4 V
E 6.4 V
E
I= =
= 0.067 A
R 96Ω
P = IE = (0.067 A)(6.4 V) = 0.43 W
(d) Energy = Pt = (0.43 W)(60.0 s) = 26 J
22.2 Motional Emf
MOTIONAL EMF
AND
ELECTRICAL ENERGY
In order to keep the rod moving at constant velocity, the force
the hand exerts on the rod must balance the magnetic force on
the induced current which acts opposite the direction of Fhand
Fhand = F = ILB
Using the numbers from the last example,
Fhand = (0.067 A)(1.6 m)(0.80 T) = 0.086 N
and the work done by the hand in 60 s is
Whand = Fhandx = Fhandvt =(0.086 N)(5 m/s)(60 s) = 26 J = Energy
22.2 Motional Emf
F
The direction of the current in this
figure gives an induced force
consistent with the conservation of
energy è the direction of the induced
current tends to oppose the applied
motion -- it decelerates the rod once
the applied force is removed (i.e. it
takes energy to light the bulb).
The direction of the current in this
figure would produce a force which
violates the principle of conservation
of energy since it accelerates the rod
thus creating energy out of nothing.
22.2 Motional Emf
Conceptual Example 3 Conservation of Energy
A conducting rod is free to slide down between
two vertical copper tracks. There is no kinetic
friction between the rod and the tracks. Because
the only force on the rod is its weight, it falls with
an acceleration equal to the acceleration of
gravity.
Suppose that a resistance connected between the
tops of the tracks. (a) Does the rod now fall with
the acceleration of gravity? (b) How does the
principle of conservation of energy apply?
22.3 Magnetic Flux
RELATIONSHIP BETWEEN MOTIONAL EMF AND MAGNETIC FLUX
In time t0 an area A0 is swept out.
In time t an area A is swept out.
' x − xo $
' xL − xo L $
' A − Ao $
(
BA)− (BA)o
EE = vBL = %% t − t "" BL = %% t − t "" B = %% t − t "" B =
t − to
o #
o
o #
&
&
#
&
Φ = BA
è magnetic flux
Φ − Φ o ΔΦ
=
EE =
t − to
Δt
The motional emf equals the
change of the magnetic flux
per time
22.3 Magnetic Flux
GENERAL EXPRESSION FOR MAGNETIC FLUX
Φ = BA cos φ
è  depends on the angle at which the
B-field crosses the area
Units of magnetic flux: T m2 = Weber = Wb
22.3 Magnetic Flux
Example. A rectangular coil of wire is situated in a constant magnetic field
whose magnitude is 0.50 T. The coil has an area of 2.0 m2. Determine the
magnetic flux for the three orientations φ = 0o, 60o, and 90o, as shown.
Φ = BA cos φ
φ = 0o
Φ = (0.50)(2.0)cos 0o = 1.0 Wb
φ = 60o
Φ = (0.50)(2.0)cos 60o = 0.50 Wb
φ = 90o
Φ = (0.50)(2.0)cos 90o = 0 Wb
22.3 Magnetic Flux
GRAPHICAL INTERPRETATION OF MAGNETIC FLUX
The magnetic flux is proportional
to the number of field lines that pass
through a surface.
22.4 Faraday’s Law of Electromagnetic Induction
FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION
The average emf induced in a coil of N loops is
) Φ − Φo &
ΔΦ
$$ = − N
E = − N ''
E
Δt
( t − to %
SI Unit of Induced Emf: volt (V)
(The motional emf-Φ
relation we derived
is a special case of this.)
the minus sign
reminds us that
the induced emf
will oppose the
change in Φ
Faraday’s law states that an emf is generated if the magnetic flux
changes for any reason. Since Φ = BA cos φ, any change of B, A, or
φ will induce an emf.
22.4 Faraday’s Law of Electromagnetic Induction
Example 5 The Emf Induced by a Changing Magnetic Field
A coil of wire consists of 20 turns each of which has an area of 0.0015 m2.
A magnetic field is perpendicular to the surface. Initially, the magnitude of
the magnetic field is 0.050 T and 0.10 s later, it has increased to 0.060 T.
Find the average emf induced in the coil during this time.
BA cos φ − Bo A cos φ
ΔΦ
= −N
EE = − N
Δt
Δt
0.060 T − 0.050 T
( B − Bo %
2
= − NA cos φ &
# = −(20 ) 0.0015 m cos(0 )
0.10 s
' Δt $
= −3.0 ×10 −3 V
(
)