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Physics 213
General Physics
Lecture 13
Last Meeting: Self Inductance and
RL Circuits
Today: Finish RL Circuits, Energy
Stored, Electric Generators, and
Alternating Current
1
Inductor in a Circuit

Inductance can be interpreted as a measure of
opposition to the rate of change in the current
 Remember
resistance R is a measure of opposition to
the current

As a circuit is completed, the current begins to
increase, but the inductor produces an emf that
opposes the increasing current
 Therefore,
the current doesn’t change from 0 to its
maximum instantaneously
RL Circuit


When the current
reaches its maximum,
the rate of change and
the back emf are zero
The time constant, , for
an RL circuit is the time
required for the current
in the circuit to reach
63.2% of its final value
I 

1e


R
t / 
Energy Stored in an Inductor


The emf induced by an inductor prevents a
battery from establishing an instantaneous
current in a circuit
The battery has to do work to produce a current
 This
work can be thought of as energy stored by the
inductor in its magnetic field
 PEL = ½ L I2
Energy Stored in an Inductor,
Derived

A long, straight wire is in the same plane as a rectangular,
conducting loop.The wire carries a constant current I as shown in
the figure. Which one of the following statements is true if the wire
is suddenly moved toward the loop?
(a) There will be no induced emf and no induced current.
(b) There will be an induced emf, but no induced current.
(c) There will be an induced current that is clockwise around the
loop.
(d) There will be an induced current that is counterclockwise
around the loop. X
(e) There will be an induced electric field that is clockwise around
the loop.
AC Circuits


An AC circuit consists of a combination of circuit
elements and an AC generator or source
The output of an AC generator is sinusoidal and
varies with time according to the following
equation
 Δv



= ΔVmax sin 2ƒt
Δv is the instantaneous voltage
ΔVmax is the maximum voltage of the generator
ƒ is the frequency at which the voltage changes, in Hz
AC Generators

The emf generated by the
rotating loop can be found by
ε =2 B ℓ v=2 B ℓvsin θ

If the loop rotates with a
constant angular speed, ω,
and N turns
ε = N B A ω sin ω t



ε = εmax when loop is parallel
to the field
ε = 0 when when the loop is
perpendicular to the field
AC generator demo.
Resistor in an AC Circuit




Consider a circuit
consisting of an AC
source and a resistor
The graph shows the
current through and the
voltage across the resistor
The current and the
voltage reach their
maximum values at the
same time
The current and the
voltage are said to be in
phase
Dissipation Across Resistors in an
AC Circuit

The rate at which electrical energy is dissipated
in the circuit is given by P=i2R=v2/R


Where I and v are the instantaneous current and voltage
across resistor
The maximum current occurs for a small amount of time
Average current is zero.
Average power > zero.
rms Current and Voltage

The rms current is the direct current that would
dissipate the same amount of energy in a
resistor as is actually dissipated by the AC
current
Irms 

Imax
2
 0.707 Imax
Alternating voltages can also be discussed in
terms of rms values
Vrms 
Vmax
2
 0.707 Vmax
Power Revisited

The average power dissipated in resistor
in an AC circuit carrying a current I is

av  I
2
rms
R
Ohm’s Law in an AC Circuit

rms values will be used when discussing AC
currents and voltages
 Many
of the equations will be in the same form as
in DC circuits

Ohm’s Law for a resistor, R, in an AC circuit
 ΔVR,rms

= Irms R
Also applies to the maximum values of v and i
Capacitors in an AC Circuit



The current reverses
direction
The voltage across the
plates decreases as the
plates lose the charge
they had accumulated
The voltage across the
capacitor lags behind
the current by 90°
Capacitive Reactance and Ohm’s
Law

capacitive reactance
1
XC 
2 ƒC
 When

ƒ is in Hz and C is in F, XC will be in ohms
Ohm’s Law for a capacitor in an AC circuit
 ΔVC,rms
= Irms XC
Inductors in an AC Circuit




Consider an AC circuit
with a source and an
inductor
The current in the circuit
is impeded by the back
emf of the inductor
The voltage across the
inductor always leads
the current by 90°
v = L I/t
Inductive Reactance and Ohm’s
Law

inductive reactance
XL = 2ƒL


When ƒ is in Hz and L is in H, XL will be in ohms
Ohm’s Law for the inductor
 ΔVL,rms
= Irms XL
Combined Circuits: The RLC
Series Circuit


The resistor,
inductor, and
capacitor can be
combined in a circuit
The current in the
circuit is the same at
any time and varies
sinusoidally with
time
Current and Voltage Relationships
in an RLC Circuit



The instantaneous
voltage across the
resistor is in phase with
the current
The instantaneous
voltage across the
inductor leads the
current by 90°
The instantaneous
voltage across the
capacitor lags the
current by 90°
Resonance in an AC Circuit

Resonance occurs at
the frequency, ƒo,
where the current has
its maximum value



To achieve maximum
current, the impedance
must have a minimum
value
This occurs when XL =
XC
Then,
ƒo 
1
2 LC