Download 1830 Joseph Henry in the US and Michael Faraday in England had

Chapter 21
LECTURE NOTES
By 1830 Joseph Henry in the US and Michael Faraday in England had found that a changing
magnetic field can induce an electric current to flow. This set the stage for large-scale
generation of electric power.
A steady magnetic field does not produce electricity in a wire held at rest in the field, but a
change in magnitude or direction of the magnetic field (or motion of the wire across the field
line) causes current to flow as if there was a battery or source of emf connected to the wire. This
induced emf is said to be produced by electromagnetic induction.
Faraday’s Law of Induction
Consider a loop or coil of wire with area A. Magnetic flux ΦB = B ۰ A = BAcos θ
A
B
θ
where B is the magnetic field strength in tesla (T)
and A is a vector representing a surface area (in
m2). A is perpendicular to the surface and θ is the
angle between B and the normal to the surface.
Flux is a measure of the number of magnetic field
lines passing through the coil. Flux has units of
T ۰ m2 .
 B
 B
where
is the rate of change of flux
t
t
and N is the number of loops or turns of wire. The negative sign is a manifestation of Lenz’s
Law: the induced emf gives rise to a current that creates a magnetic field that opposes the
original change in flux.
Faraday found the induced emf (in volts) E = -N
Example 1
Consider a coil or loop of wire at rest in the plane of the paper. Stab at it from above with a bar
magnet whose N pole is closest to the wire. There is motion of wire across field lines; this drives
charges in the wire by v x B, creating an induced current in the loop that produces its own field.
The current flows counterclockwise as seen from above because by a right hand rule this creates
a magnetic field with N pole pointing out toward the approaching N pole of the bar magnet. The
current is induced in just the right direction as to oppose the change creating it. This really is
conservation of energy.
Motional emf
A metal bar of length  slides on two rails to the right at velocity v, completing a circuit that
includes resistor R. There is an external magnetic field B out of the plane of the paper. The
induced emf causes a conventional current to flow clockwise. Consider v x B and see the +
charges are driven to the bottom of the bar leaving a net – charge at the top. The + charges flow
clockwise with current I = E/R.
Chapter 21
LECTURE NOTES
⊙
⊙
R
⊙
Bout
⊙
v
+
Faraday’s law of induction with N = 1 gives |ℰ| =
ℓ
⊙
⊙
  B ( BA ) BA
=
=
since
t
t
t
Bvt
where vΔt is the change in x as the bar moves right or |ℰ| = BℓV.
t

 E
BV
and E = = vB is the electric field created along the bar. We can also determine
Thus I =
R

the direction of the induced current using Lenz’s Law. To fight the increase in flux caused by
ΔA, the induced current must create a magnetic field to oppose the external B. To create Bin
requires a clockwise current around the loop by a right hand rule.
B is constant =
A generator works by rotating coils of wire (the armature) through a stationary magnetic field.
This produces alternating current with slip rings and brushes used to remove the current without
tangling wires. Power plants and alternators on cars employ rotating electromagnets (rotors)
moving past stationary banks of wire (stators) for better efficiency. Every generator run in
reverse is a motor (and vice-versa). If we do mechanical work spinning the armature of a
generator we produce electricity. If we feed electricity into the device we can turn the armature
by v x B forces (a motor).
Now when we turn a motor it works like a generator, but the voltage induced opposes the
changes creating it. This back emf obeys the formula ℰ – ℰinduced = IRmotor. Here ℰ is the input
emf (120v if plugged into wall), ℰinduced the back emf, I the current flowing in the armature, and
R the resistance of the motor. The greater the speed of the motor, the greater is the ℰinduced. If
there is no load, it is easy to spin the armature. But when current flows the moving current loop
in the magnetic field feels a counter torque and it is harder to spin the motor. We can burn out a
motor by putting too high a load on it for too long. When just barely turning (heavy loud or
turned on) ℰinduced = 0 so I = ℰ/Rmotor is higher. This can produce enough I2R heating to burn up
the motor. When running at speed, ℰinduced is nearly ℰ so I is very small. When a large appliance
like an air conditioner or refrigerator starts, the lights may dim as the high current draw of the
appliance causes some voltage drop across the household wiring before the outlet.
Transformers
These devices, invented at the end of the 19th century by Tesla, can step up or step down the
current or voltage of AC electricity. They made long distance transmission of electricity
possible. The device is simply two coils of wire (primary and secondary) insulated and wrapped
together or wound on a common iron core.
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Chapter 21
LECTURE NOTES
vp
vs
Apply AC at the primary coil at VP. The changing B
produced by the changing current induces an AC voltage
of the same frequency in the secondary.
VP = NP ΔΦB/Δt and Vs = NsΔΦB/Δt not worrying about signs and noting the same flux lines
V
N
both coils through the common core. Thus s = s . If the number of turns in the secondary is
VP N P
less than in the primary, Vs is less than VP and we have a step-down transformer.
Is N P
(so if the voltage goes down, current goes up) because the power out is equal

IP NS
(almost) to the power in and P = IV.
Note:
Now how do these devices help in power transmissions? The wires between power plant and
city suffer I2R losses. To minimize loss of energy power should be transmitted at low current
(and high voltage to keep power up). The voltage must be stepped down for use in homes.
Since transformers only work with AC, this is the reason our power network uses AC.
Inductance
As in a transformer, when two coils are adjacent, a changing current in one induces an emf in the
I
other. ℰ 2 = -M 1 when M, the mutual inductance, has units of  s where we define 1 henry
t
I
(H) = 1  ۰s. Also ℰ1= -M 2 .
t
Even a single coil induces a back-emf in itself. A charging current causes B to expand or
collapse across the wire of the coil. The back-emf ℰ= -L ΔI/Δt where L is the self-inductance in
henry. An inductor
is a coil in a circuit possessing inductance. It presents an impedance to
changes in current smoothing out sudden increases or decreases of current.
Self-inductance of a solenoid is given by L = μ0N2A/ℓ where N is the number of coil, A the cross
sectional area, and ℓ the length of the solenoid.
Energy
Energy is stored in the magnetic field of an inductor (U = 1/2LI2) as in the electric field of a
(1 / 2)B 2
capacitor (U = (1/2)CV2). The energy density is u =
or (1/2)ε0E2.
0
3
Chapter 21
LECTURE NOTES
R
+
L
-
LR Series Circuit
When the switch closes at t = 0, the current is changing at a great rate so the counter emf induced
in the inductor opposes the emf of the battery almost completely, limiting the current I. We find
I = Imax (1 – et/τ) where τ = L/R is another time constant in seconds and Imax = V/R is the current
reached after several time constants have passed.
When the switch is opened with the battery absent, the energy stored in the magnetic field of the
inductor is released to continue driving the current preventing it from suddenly going to zero.
Lenz’s law tells us the current induced opposes the changes in flux creating it so it tries to
maintain current flow.
AC Circuits
An AC power source produces current I = I0cos(2πft) at Vrms =
V0
I
, Irms = 0 .
2
2
Resistor
 Ohm’s Law applies for both DC and AC circuits.
 I and VR increases and decreases in phase.
 Energy consumed appears as heat at the average rate P = IV
P = I2rms R = V2rms/R
Inductor
 VL= -V0sin(2πft), leading current by 90°
 VL, I are out of phase by ¼ cycle.
 P = 0 over 1 cycle
 L impedes flow of AC: Vapplied = Iinduced XL where XL = inductive reactance in ohms (  )
Capacitor
VC = V0 sin (2πft), lags current by 90°
 P = 0 over 1 cycle

Vapplied = I XC where XC = capacitive reactance =

Note capacitors block DC but pass AC.
1
in ohms
2fC
LRC Series Circuit
Since VR = IR, VL = IXL, VC = IXC we can find a combined impedance Z = R 2  X L  X C 
which is reasonable when we draw a phasor diagram showing how the voltages lead or lag the
current and considering these values as vectors.
2
4
Chapter 21
LECTURE NOTES
VL
VR
I
VC
We define a power factor cos Φ and write P = IrmsVrmscosΦ where tan Φ =
XL  XC
.
R
1
1
. Here the impedance Zmin = R is at its lowest, the current in the
2 LC
circuit a maximum. We have a tuner that maximizes current at a specific frequency. Variable
capacitors or other elements let us tune a radio. The frequency f0 is the resonant frequency at
which the system likes to oscillate.
Note XL = XC at f0 =
Impedance matching
To maximize power transfer between two circuits or devices we require the input and output
impedances match (be equal). We can use a transformer to make this so with
ZP
N 2
= NPs .
Zs
( )
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