Faraday’s Law and Inductance
• Historical Overview
– So far studied electric fields due to stationary charges and magentic
fields due to moving charges.
– Now study electric field due to a changing magnetic field.
– Faraday showed that an electric current can be induced in a circuit by
a changing magnetic field.
– These studies lead to Faraday’s law of induction.
• Faraday’s law of induction
– A straight metal wire lies in a uniform magnetic field directed into the
page. The wire contains free elctrons.
– Move the wire with a velocity v to the right.
– F = qv × B implies that a magnetic force acts on the wire.
– Can use the right hand rule to show that the force on the electrons is
downward, along the wire.
– The electrons move along the wire in response to this force: a current.
– Consider a loop of wire connected to a sensitive ammeter.
– Move a magnet toward the loop, the ammeter needle deflects in one
direction.
– If the magnet is stationary, no current is detected.
– If the magnet is stationary and the coil is moved either away or toward
it, a current is detected.
– http://web.mit.edu/jbelcher/anim.html#Faraday
– An electric force is set up in the coil as long as relative motion occurs
between the magnet and the coil.
– The current in the wire is called an induced current, produced by an
induced electromotive force or emf.
– Faraday concluded that an electric current can be produced by a timevarying magnetic field.
– In the example above, what causes the time-varying magnetic field?
– Current cannot be produced by a steady magnetic field.
– Discuss Figure 23.3
– To quantify this, define magnetic flux.
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– Consider an element of area dA on an arbitrarily shaped open surface.
If the magnetic field at the location of this element is B, the magnetic
flux through the element is B.dA, where doverlineA is a vector perpendicular to the surface whose magnitude equals area dA. Then the
total magnetic flux through the surface is
ΦB =
Z
B.A
– SI unit of magnetic flux is tesla-meter squared, where 1 weber or W b =
1T.m2 .
– an emf is induced in a circuit when the magnetic flux through the
surface bounded by the circuit changes with time.
– Faradays law of induction states that the emf induced in a circuit is
equal to the time rate of change of magnetic flux through the circuit.
=−
dΦB
,
dt
where ΦB is the magnetic flux through the surface bounded by the
circuit.
– If the circuit is a coil consisting of N identical and concentric loops
and if the field lines pass through all loops, the induced emf is
= −N
=−
dΦB
.
dt
d
(BAcosθ).
dt
– Why is cosθ needed here.
– Thus an emf can be induced in a circuit by changing the magnetic flux
in several ways:
∗ magnitude of B can vary with time
∗ the area A of the circuit can change with time
∗ the angle θ between B and the normal to the plane can change
with time
∗ any combination of these changes
∗ Try QuickQuiz 23.1, go over example 23.1.
• Motional emf
– An emf is induced in a conductor moving through a magnetic field motional emf.
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– Go over Figure 23.11.
• The Alternating Current generator
– The alternating-current (AC) generator is a device in which energy is
transferred in by work and out by electrical transmission:
– A coil of wire rotated in an external magnetic field by some external
agent - this is the work input.
– For example, in a hydroelectric power plant, falling water directed
against the blades of a turbine produces the rotary motion.
– As the loop rotates, the magnetic flux through it changes with time,
inducing an emf and a current in a circuit connected to the coil.
– Suppose the coil has N turns of the same area A and suppose the coil
rotates with a constant angular speed ω about an axis perpendicular
to the magnetic field. If θ is the angle between the magnetic field and
the direction perpendicular to the plane of the coil, the magnetic flux
through the loop at any time t is given by:
ΦB = BAcosθ = BAcosωt,
where θ = ωt.
– Thus the induced emf in the coil is = −N dΦdtB = −NAB dtd (cosωt) =
NABsinωt.
– Thus the emf varies sinusoidally with time - AC voltage.
• Lenz’s Law
– Negative sign in Faraday’s law.
– The polarity of the induced emf in a loop is such that it produces
a current whose magentic field opposes the change in magnetic flux
through the loop. That is the induced current is in a direction such
that the induced magnetic field attempts to maintain the original flux
through the loop.
– Go over Figure 23.16
• Induced emfs and Electric Fields
– A changing magnetic flux indices an emf and a current in a conducting
loop.
– The current is due to an electric field setup by a battery, for example.
– Can also think of this as the following: a changing magnetic field
creates an induced electric field. This electric field applies a force on
the charges to cause them to move.
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– An electric field is created in a conductor as a result of changing magnetic flux - or - An electric field is always generated by a changing
magnetic flux, even in free space where no charges are present.
– But the induced electric field has different properties to the electrostatic fields produced by stationary charges.
– Consider a conducting loop of radius r in a uniform magnetic field
that is perpendiuclar to the plane of the loop.
– If the magnetic field changes with time, Faraday’s law tells su that =
−dΦB /dt is induced in the loop. The induced current thus produced
implies an induced electric field, E, that must be tangent to the loop
so as to provide an electric force on the charges around the loop.
– The work done by the electric field on the loop in moving a test charge
q once around the loop is W = q. But the force on the charge is qE,
R
and W = F .dr = qe(2πr).
– Equating these two expressions, we get
E=
.
2πr
– Using this result, together with faraday’s law and ΦB = BA, we find
E=−
r dB
.
2 dt
– The negative sign indicates that the induced electric field E results in
a current that opposes the change in the magnetic field.
– This result is valid in the absence of a conductor or charges.
– Thus the general form of Faraday’s law of induction is
=
Z
E.ds = −
dΦB
.
dt
– The induced electric field E that appears above is a non-conservative
field that is generated by a changing magnetic field.
– Its non-conservative because the work done in moving a charge around
a closed path is not zero.
– Is this like an electrostatic filed?
• Self-Inductance
– Consider an isolated circuit consisting of a swicth, a resistor and a
source of emf.
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– When the switch is closed, the current doesnt jump from zero to /R
immediately.
– As the current increases with time, the magnetic flux through the loop
of the circuit itself due to the current also increases with time. This
increasing magnetic flux from the circuit induces an emf in the ciruit
(back emf) that opposes the change in the net magnetic flux through
the loop of the circuit. By Lenz’s law, the induced electric field in
the wires must therefore be opposite the direction of the current. the
opposing emf results in a gradual increase in the emf.
– The emf setup in this case is called self-induced emf.
– From Faraday’s law, the induced emf is the negative time rate of
change of the magnetic flux.
– The magnetic flux is proportional to the magnetic field (why?), which
in turn is proportional to the current in the circuit (why?).
– Thus the self induced emf is always proportional to the time rate of
change of the current.
– More quantitatively, for a closely spaced coil of N turns,
L = −N
dΦB
dI
= −L ,
dt
dt
where L is a proportionality constant called the inductance of the coil.
– The inductance of a coil containing N loops is L =
N ΦB
.
I
– The inductance is the ratio,
L=−
L
.
dI/dt
– The SI unit of inductance is the henry (H): 1H = 1V.s/A.
– Inductance of a coil depends on geometry.
– A circuit that contains a coil, such as a solenoid, has a self-inductance
that prevents the current from increasing or decreasing instantaneously.
• RL Circuits
– A circuit that contains a coil, such as a solenoid, has a self-inductance
that prevents the current from increasing or decreasing instantaneously.
– A circuit element whose main purpose is to provide inductance in a
circuit is called an inductor.
– Consider the circuit shown, consistin gof a resistor, an inductor, a
swicth and a battery.
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– Ignore the resistance of the battery and suppose the switch is closed
at t = 0.
– The current begins to increase, and due to the opposing current, the
inductor produces an emf that opposes the increasing current. The
back emf produced by the inductor is
L = −L
dI
.
dt
– Applying Kirchoof’s loop rule to the circuit:
− IR − L
dI
= 0,
dt
where IR is the voltage drop across the resistor. The potential difference across the inductor is given a negative sign because its emf is in
the opposite sense to that of the battery.
– This is a differential equation. Its solution will give I(t).
– Can show the solution is
I(t) =
(1 − e−t/τ ),
R
where τ is the time constant of the RL circuit and τ = L/R.
– Dimension of τ is time. It is the time interval required for the current
to reach (1 − e−1 ) = 0.632 of its final value /R.
– Note I(t) = 0 when t = 0. And at t = inf, I = /R.
– The final current, /R does not involve the inductance L because the
inductor has no effect on the circuit if the current is not changing.
• Energy Stored in a Magnetic Field
– Induced emf set up by an inductor prevents a battery from establishing
an instantaneous current. Part of the energy supplied by the batter
goes into the internal energy in the resistor. The remaining energy is
stored in the inductor.
– From Kirchoffls loop law , we have
I = I 2 R + LI
dI
.
dt
– This says that the rate, I, at which energy is supplied by the battery
equals the rate, I 2 R, at which energy is delivered to the resistor and
the rate, LI(dI/dt) at which energy is delivered to the inductor.
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– Thus, if UB is the energy stored in the inductor at any time,
dI
dUB
= LI .
dt
dt
SOlving this, we get
1
UB = LI 2 ,
2
where L is constant. This is the energy stored in the magnetic field of
the inductor when the current is I.
– Now consider a solenoid where L = µ0 n2 Al. The magnetic field of the
solenoid is B = mu0 nI. Then
B 2
B2
1 2 1
2
) =
(Al).
UB = LI = µ0 n Al(
2
2
µ0 n
2µ0
– Because Al is the volume of the solenoid, the energy stored per unit
volume in a magnetic field, or the megnetic energy density, is
uB =
B2
.
1µ0
– This formula is valid for any region of space in which a magnetic field
exists.
– Compare to eneergy per unit volume stored in an electric field, 21 0 E 2 .
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