Download induced magnetic field

Electromagnetic
Induction
AP Physics C
Faraday’s Discovery
 Electromagnetic Induction is the process of using magnetic
fields to produce voltage, and in a complete circuit, a current.
 Michael Faraday first discovered it in 1831, using some of the
works of Hans Christian Oersted.
 He started by using different combinations of wires and
magnetic strengths and currents, but it wasn't until he tried
moving the wires that he got any success.
 It turns out that electromagnetic induction is created by just
that - the moving of a conductive substance through a
magnetic field.
Magnetic Flux
 The magnetic flux measures the amount of magnetic field passing through
a loop of area A if the loop is tilted at an angle θ from the field.
 We represent magnetic flux with a dot product, just like with electric flux.
 The unit for magnetic flux is a Webber.
Φ𝐵 = 𝐵 ∙ 𝐴
Magnetic Flux for Non-Uniform
Magnetic Fields
 If you have a non-uniform
magnetic field through a
loop, you have to integrate
to find the total flux.
 This often involves a change
of variable.
Faraday and Lenz’s Laws
 Faraday realized that changing the
magnetic flux in any way induced a voltage
(and therefor a current) in a circuit.
 Faradays law gives the magnitude of the
induced voltage.
 Lenz’s law (the negative sign) indicates the
direction of the induced current.
 Lenz’s law states that inductors resist
changes in magnetic flux.
𝑑Φ
𝜀=−
𝑑𝑡
Lenz’s Law
Example
The current in the straight wire is decreasing.
Which is true?
A. There is a clockwise induced current in the
loop.
B. There is a counterclockwise induced current
in the loop.
C. There is no induced current in the loop.
Example
The magnetic field is confined to the region
inside the dashed lines; it is zero outside. The
metal loop is being pulled out of the magnetic
field. Which is true?
A. There is a clockwise induced current in the
loop.
B. There is a counterclockwise induced current
in the loop.
C. There is no induced current in the loop.
Eddy Currents
Consider pulling a sheet of metal through
a magnetic field.
Two “whirlpools” of current begin to
circulate in the solid metal, called eddy
currents.
The magnetic force on the eddy currents
is a retarding force.
This is a form of magnetic braking.
Motional EMF
 Motional EMF is voltage generated by a conductive bar moving through a
magnetic field.
 The picture to the right shows a current being generated by motional EMF.
Example
An airplane with a wing span of 30.0 m flies parallel to the Earth’s surface
at a location where the downward component of the Earth’s magnetic
field is 0.60 x10-4 T. Find the difference in potential between the wing tips
is the speed of the plane is 250 m/s.
AC Generators
 AC generators convert mechanical energy into electrical energy.
 The change in magnetic flux through the loop induces a current, which then
goes to powering various devices.
 This is the basic principle behind wind power and hydroelectric power.
Transformers
A transformer sends an
alternating emf V1 through the
primary coil.
This causes an oscillating magnetic
flux through the secondary coil
and, hence, an induced emf V2.
The induced emf of the secondary
coil is delivered to the load:
Transformers
A step-up transformer, with N2 >> N1, can boost the
voltage of a generator up to several hundred thousand
volts.
Delivering power with smaller currents at higher
voltages reduces losses due to the resistance of the
wires.
High-voltage transmission lines carry electric power to
urban areas, where step-down transformers (N2 << N1)
lower the voltage to 120 V.
The Induced Electric Field
Faraday’s law and Lenz’s law may
be combined by noting that the emf
must oppose the change in Φm.
Mathematically, emf must have
the opposite sign of dB/dt.
Faraday’s law may be written as:
The Induced Magnetic Field
As we know, changing the magnetic
field induces a circular electric field.
Symmetrically, changing the electric
field induces a circular magnetic field.
The induced magnetic field was first
suggested as a possibility by James
Clerk Maxwell in 1855.
Maxwell’s Equations
 We have now been introduced to the four most important equations in all of
electromagnetic field theory. These formulas are known as Maxwell’s
equations, named after James Clerk Maxwell.
Electromagnetic Waves
 A changing electric field creates
a magnetic field, which then
changes in just the right way to
recreate the electric field,
which then changes in just the
right way to again recreate the
magnetic field, and so on.
 This is an electromagnetic wave
(a light wave).
Inductors
A coil of wire, or solenoid, can
be used in a circuit to store
energy in the magnetic field.
We define the inductance of a
solenoid having N turns, length l
and cross-section area A as:
The SI unit of inductance is the
henry, defined as:
1 henry = 1 H = 1 Wb/A = 1 T m2/A
A coil of wire used in a circuit for
the purpose of inductance is called
an inductor.
 If you divide both sides by time we get:
B
I
L
t
t
dI
  L
dt
 
d
dI
 L
dt
dt
Inductors in Circuits
 When an inductor is placed in a
circuit, it may experience either
a rise or a drop in voltage.
 If the current is increasing in the
inductor, the voltage decreases.
 If the current is decreasing in
the inductor, the voltage
increases.
 This is a consequence of energy
being stored in the inductor’s
magnetic field.
Energy in an Inductor
As current passes through an inductor, the electric power
is:
Pelec is negative because the current is losing energy.
That energy is being transferred to the inductor, which is storing energy UL at the
rate:
We can find the total energy stored in an inductor by integrating:
LC Circuits
The figure shows a capacitor with
initial charge Q0, an inductor, and
a switch.
The switch has been open for a
long time, so there is no current in
the circuit.
At t  0, the switch is closed.
How does the circuit respond?
The charge and current oscillate
in a way that is analogous to a
mass on a spring.
LC Circuits
LC Circuits
An LC circuit is an electric
oscillator.
The letters on the graph
correspond to the four
steps in the previous
slides.
The charge on the upper
plate is Q  Q0cosωt and
the current through the
inductor is I  Imaxsinωt,
where:
LR Circuits
 LR circuits involve inductors and resistors.
 When the switch is closed, the inductor resists changes in magnetic flux,
and therefor resists current.
 As time goes on, the current through the inductor reaches a steady state.
 The current through the inductor is modeled as logarithmic growth, while
the voltage across the inductor is modeled as exponential decay.
LR Circuits
 If the battery is removed from an LR circuit, the circuit no longer receives
emf from the battery, however there is still current for a short period of
time.
 The sudden change in the magnetic flux through the inductor induces an
emf whose current can be modeled with an exponentially decaying
function.
 L/R is often represented as τ, which is the time constant for an LR circuit.
Example
What is the battery current
immediately after the switch has
closed?
What is the battery current
immediately after the switch has
been closed for a very long time?