Week 8
Faraday’s Law
Inductance
Energy

The law states that the induced electromotive
force or EMF in any closed circuit is equal to the
time rate of change of the magnetic flux through
the circuit:
where
◦ Ɛ is the electromotive force (EMF) in Volts
◦ ΦB is the magnetic flux through the circuit in Webers

The magnetic flux ΦB through a surface S, is
defined by an integral over a surface:


If the loop enters the field at a constant
speed, the flux will increase linearly and EMF
will be constant.
What happens when the entire loop is in the
field?



A conducting rod of length L moves in the xdirection at a speed vax as show below.
The wire cuts into a magnetic field B az.
Calculate the induced EMF.

Compute the total flux ΦB
ΦB = B A = B L (xo + vt)

Calculate the rate of change
Ɛ = - dΦB/dt = - vBL

Since the magnetic flux through the loop
increases linearly the EMF induced in the loop
must be constant.



An EMF is
induced on the
rectangular
loop of the
generator.
The loop of
area A is
rotating at an
angular rate ω
(rads).
The magnetic
field B is held
constant.



Assume B = Bo ax
the square loop rotates so its normal unit
vector is an = ax cos ωt + ay sin ωt
Determine the flux as a function of time and
the induced EMF

Inductance is the property in an electrical
circuit where a change in the current flowing
through that circuit induces an EMF that
opposes the change in current.


The term 'inductance' was coined by Oliver
Heaviside. It is customary to use the symbol
L for inductance, possibly in honor of
Heinrich Lenz.
Other terms coined by Heaviside
◦
◦
◦
◦
Conductance G
Impedance Z
Permeability μ
Reluctance R



The electric current produces a
magnetic field and generates a
total magnetic flux Φ acting on
the circuit.
This magnetic flux tends to act to
oppose changes in the flux by
generating an EMF that counters
or tends to reduce the rate of
change in the current.
The ratio of the magnetic flux to
the current is called the selfinductance which is usually simply
referred to as the inductance of
the circuit.

Derive an expression for the inductance of a
solenoid
Since B = μNI/ℓ
Φ = BA = μNIA/ℓ
thus L = NΦ/I = μN2A/ℓ

Derive an expression for the inductance of a
toroid. Recall that
B = μNI
2πr


The pickup from an electric guitar captures
mechanical vibrations from the strings and
converts them to an electrical signal.
The vibration from a string modulates the
magnetic flux, inducing an alternating electric
current.

Taking the time derivative of the flux linkage
λ = NΦ = Li
yields
=v
This means that, for a steady applied voltage
v, the current changes in a linear.

Let the current through an inductor be
i(t) = Io cos(ωt).
Determine v(t) across the inductor.



In practice, small
inductors for
electronics use may
be made with air
cores.
For larger values of
inductance and for
transformers, iron is
used as a core
material.
The relative
permeability of
magnetic iron is
around 200.

Here, L11 and L22 are the
self-inductances of circuit
one and circuit two,
respectively. It can be
shown that the other two
coefficients are equal: L12 =
L21 = M, where M is called
the mutual inductance of
the pair of circuits.


Consider for example two circuits carrying
the currents i1, i2.
The flux linkages of circuits 1 and 2 are given
by
L12 = L21 = M

Describe what will happen when the switch is
on.

The energy density of a magnetic field is
u = B2
2μ

The energy stored in a magnetic field is
U = ½ LI2


In a certain region of space, the magnetic
field has a value of 10-2 T, and the electric
field has a value of 2x106 V/m.
What is the combined energy density of the
electric and magnetic fields?


Read Sections 6-1, 6-2, 6-4, and 6-5.
Solve end-of-chapter problems 6.2, 6.5,
6.10, and 6.12.