Electromagnetic Induction and Faraday’s Law
Induced EMF
Faraday looked for evidence
that a magnetic field would
induce an electric current
He found no evidence when the current was steady.
A current was induced when the switch was turned on or off.
A current is also
observed when a
magnet is moved
towards or away
from a coil
Therefore, a changing magnetic field induces an emf.
Faraday’s experiment used a magnetic field that was
changing because the current producing it was
changing; the previous graphic shows a magnetic
field that is changing because the magnet is moving.
The induced emf in a wire loop
is proportional to the rate of
change of magnetic flux
through the loop.
Magnetic flux:
Unit of magnetic flux: weber, Wb.
1 Wb = 1 T·m2
The magnetic flux is analogous to
the electric flux – it is proportional
to the total number of lines passing
through the loop.
Faraday’s Law
Lentz’s Law
There is an induced current in
a closed, conducting loop if and
only if the magnetic flux
through the loop is changing.
The direction of the induced
current is such that the
induced magnetic field opposes
the change in the flux.
An emf  is induced in a conducting
loop if the magnetic flux through the
loop changes. The magnitude of  is

 B

(1 loop)
t
 B
(N loops)
t
where  B is the change in magnetic
 N
flux in time t and the emf direction
is such as to drive an induced current
in the direction given by Lenz's law.
Magnetic flux will change if the area of the loop changes:
Magnetic flux
will also change
if the angle
between the
loop and the
field changes:
ConcepTest
In order to change the
magnetic flux through
the loop, what would
you have to do?
Magnetic Flux I
1) drop the magnet
2) move the magnet upwards
3) move the magnet sideways
4) only (1) and (2)
5) all of the above
ConcepTest
In order to change the
magnetic flux through
the loop, what would
you have to do?
Magnetic Flux I
1) drop the magnet
2) move the magnet upwards
3) move the magnet sideways
4) only (1) and (2)
5) all of the above
Moving the magnet in any direction would
change the magnetic field through the
loop and thus the magnetic flux.
ConcepTest
Magnetic Flux II
1) tilt the loop
In order to change the
magnetic flux through
the loop, what would
you have to do?
2) change the loop area
3) use thicker wires
4) only (1) and (2)
5) all of the above
ConcepTest
Magnetic Flux II
1) tilt the loop
In order to change the
magnetic flux through
the loop, what would
you have to do?
2) change the loop area
3) use thicker wires
4) only (1) and (2)
5) all of the above
Since  = B A cosq , changing the
area or tilting the loop (which varies
the projected area) would change
the magnetic flux through the loop.
Problem Solving: Lenz’s Law
1. Determine whether the magnetic flux is increasing,
decreasing, or unchanged.
2. The magnetic field due to the induced current points in
the opposite direction to the original field if the flux is
increasing; in the same direction if it is decreasing; and
is zero if the flux is not changing.
3. Use the right-hand rule to determine the direction of the
induced current.
4. Remember that the external field and the field due to the
induced current are different.
ConcepTest
Moving Bar Magnet I
If a North pole moves toward the
1) clockwise
loop from above the page, in what
2) counterclockwise
direction is the induced current?
3) no induced current
ConcepTest
Moving Bar Magnet I
If a North pole moves toward the
1) clockwise
loop from above the page, in what
2) counterclockwise
direction is the induced current?
3) no induced current
The magnetic field of the moving bar
magnet is pointing into the page and
getting larger as the magnet moves
closer to the loop. Thus the induced
magnetic field has to point out of the
page. A counterclockwise induced
current will give just such an induced
magnetic field.
Follow-up: What happens if the magnet is stationary but the loop moves?
ConcepTest
Moving Bar Magnet II
If a North pole moves toward
1) clockwise
the loop in the plane of the
2) counterclockwise
page, in what direction is the
3) no induced current
induced current?
ConcepTest
Moving Bar Magnet II
If a North pole moves toward
1) clockwise
the loop in the plane of the
2) counterclockwise
page, in what direction is the
3) no induced current
induced current?
Since the magnet is moving parallel
to the loop, there is no magnetic
flux through the loop. Thus the
induced current is zero.
ConcepTest
Falling Magnet I
A bar magnet is held above the floor
and dropped. In 1, there is nothing
between the magnet and the floor.
In 2, the magnet falls through a
copper loop. How will the magnet in
1) it will fall slower
2) it will fall faster
3) it will fall the same
case 2 fall in comparison to case 1?
S
S
N
N
copper
loop
ConcepTest
Falling Magnet I
A bar magnet is held above the floor
and dropped. In 1, there is nothing
between the magnet and the floor.
In 2, the magnet falls through a
copper loop. How will the magnet in
1) it will fall slower
2) it will fall faster
3) it will fall the same
case 2 fall in comparison to case 1?
When the magnet is falling from above
the loop in 2, the induced current will
produce a North pole on top of the loop,
which repels the magnet.
When the magnet is below the loop, the
induced current will produce a North
pole on the bottom of the loop, which
attracts the South pole of the magnet.
S
S
N
N
copper
loop
Follow-up: What happens in case 2 if you flip the magnet
so that the South pole is on the bottom as the magnet falls?
ConcepTest
If there is induced
current, doesn’t
that cost energy?
Where would that
energy come from
in case 2?
Falling Magnet II
1) induced current doesn’t need any energy
2) energy conservation is violated in this case
3) there is less KE in case 2
4) there is more gravitational PE in case 2
S
S
N
N
copper
loop
ConcepTest
If there is induced
current, doesn’t
that cost energy?
Where would that
energy come from
in case 2?
Falling Magnet II
1) induced current doesn’t need any energy
2) energy conservation is violated in this case
3) there is less KE in case 2
4) there is more gravitational PE in case 2
In both cases, the magnet starts with
the same initial gravitational PE.
In case 1, all the gravitational PE has
been converted into kinetic energy.
In case 2, we know the magnet falls
slower, thus there is less KE. The
difference in energy goes into making
the induced current.
S
S
N
N
copper
loop
ConcepTest
A wire loop is being pulled
through a uniform magnetic
field. What is the direction
Moving Wire Loop I
1) clockwise
2) counterclockwise
3) no induced current
of the induced current?
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
ConcepTest
Moving Wire Loop I
A wire loop is being pulled
through a uniform magnetic
field. What is the direction
1) clockwise
2) counterclockwise
3) no induced current
of the induced current?
x x x x x x x x x x x x
x x x x x x x x x x x x
Since the magnetic field is uniform, the
x x x x x x x x x x x x
magnetic flux through the loop is not
x x x x x x x x x x x x
changing. Thus no current is induced.
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
Follow-up: What happens if the loop moves out of the page?
ConcepTest
Moving Wire Loop II
A wire loop is being pulled
through a uniform magnetic
field that suddenly ends.
What is the direction of the
induced current?
x x x x x
x x x x x
x x x x x
x x x x x
x x x x x
x x x x x
x x x x x
1) clockwise
2) counterclockwise
3) no induced current
ConcepTest
Moving Wire Loop II
A wire loop is being pulled
through a uniform magnetic
field that suddenly ends.
What is the direction of the
1) clockwise
2) counterclockwise
3) no induced current
induced current?
x x x x x
The B field into the page is disappearing in
x x x x x
the loop, so it must be compensated by an
x x x x x
induced flux also into the page. This can
x x x x x
be accomplished by an induced current in
x x x x x
the clockwise direction in the wire loop.
x x x x x
x x x x x
Follow-up: What happens when the loop is completely out of the field?
ConcepTest
What is the direction of the
induced current if the B field
suddenly increases while the
loop is in the region?
Moving Wire Loop III
1) clockwise
2) counterclockwise
3) no induced current
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
ConcepTest
Moving Wire Loop III
What is the direction of the
induced current if the B field
suddenly increases while the
loop is in the region?
1) clockwise
2) counterclockwise
3) no induced current
The increasing B field into the page
x x x x x x x x x x x x
must be countered by an induced
x x x x x x x x x x x x
flux out of the page. This can be
x x x x x x x x x x x x
accomplished by induced current
in the counterclockwise direction in
the wire loop.
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
Follow-up: What if the loop stops moving while the field increases?
ConcepTest
If a coil is shrinking in a
magnetic field pointing into
the page, in what direction
is the induced current?
Shrinking Wire Loop
1) clockwise
2) counterclockwise
3) no induced current
ConcepTest
Shrinking Wire Loop
If a coil is shrinking in a
magnetic field pointing into
the page, in what direction
1) clockwise
2) counterclockwise
3) no induced current
is the induced current?
The magnetic flux through the loop is
decreasing, so the induced B field must
try to reinforce it and therefore points in
the same direction — into the page.
According to the right-hand rule, an
induced clockwise current will generate
a magnetic field into the page.
Follow-up: What if the B field is oriented at 90° to its present direction?
ConcepTest
If a coil is rotated as shown,
in a magnetic field pointing
to the left, in what direction
is the induced current?
Rotating Wire Loop
1) clockwise
2) counterclockwise
3) no induced current
ConcepTest
Rotating Wire Loop
If a coil is rotated as shown,
in a magnetic field pointing
to the left, in what direction
1) clockwise
2) counterclockwise
3) no induced current
is the induced current?
As the coil is rotated into the B field,
the magnetic flux through it increases.
According to Lenz’s Law, the induced
B field has to oppose this increase,
thus the new B field points to the right.
An induced counterclockwise current
produces just such a B field.
ConcepTest
Voltage and Current I
Wire #1 (length L) forms a one-turn loop,
and a bar magnet is dropped through.
Wire #2 (length 2L) forms a two-turn loop,
and the same magnet is dropped through.
Compare the magnitude of the induced
voltages in these two cases.
1) V1 > V2
2) V1 < V2
3) V1 = V2  0
4) V1 = V2 = 0
S
S
N
N
ConcepTest
Voltage and Current I
Wire #1 (length L) forms a one-turn loop,
and a bar magnet is dropped through.
Wire #2 (length 2L) forms a two-turn loop,
and the same magnet is dropped through.
Compare the magnitude of the induced
voltages in these two cases.
Faraday’s law:
  N 
t
1) V1 > V2
2) V1 < V2
3) V1 = V2  0
4) V1 = V2 = 0
B
depends on N (number of loops)
so the induced emf is twice as
large in the wire with 2 loops.
S
S
N
N
ConcepTest
Voltage and Current II
Wire #1 (length L) forms a one-turn loop,
and a bar magnet is dropped through.
Wire #2 (length 2L) forms a two-turn loop,
and the same magnet is dropped through.
Compare the magnitude of the induced
currents in these two cases.
1) I1 > I2
2) I1 < I2
3) I1 = I2  0
4) I1 = I2 = 0
S
S
N
N
ConcepTest
Voltage and Current II
Wire #1 (length L) forms a one-turn loop,
and a bar magnet is dropped through.
Wire #2 (length 2L) forms a two-turn loop,
and the same magnet is dropped through.
Compare the magnitude of the induced
currents in these two cases.
Faraday’s law:
  N 
t
1) I1 > I2
2) I1 < I2
3) I1 = I2  0
4) I1 = I2 = 0
B
says that the induced emf is twice
as large in the wire with 2 loops.
The current is given by Ohm’s law:
I = V/R. Since wire #2 is twice as
long as wire #1, it has twice the
resistance, so the current in both
wires is the same.
S
S
N
N
EMF Induced in a Rod Moving on a U-Shaped Rail
A conducting rod slides on a
conducting U-shaped rail at a
constant velocity perpendicular
both to the rod and to a uniform
magnetic field through the
conducting loop formed by the
rod and the rail
Since the area of he conducting
loop increases, the magnetic
flux increases.
According to Faraday’s law, an emf is induced
The induced current is in a direction that tends to slow the
sliding rod – it will take an external force to keep it moving.
EMF Induced in a Moving Conducting Rod
Magnetic force is exerted on
electrons in a conducting rod
moving in a magnetic field, in a
direction perpendicular to both
the rod and the magnetic field,
FB  qvB
The magnetic force does work separating charges across the length of
the rod, creating electric potential energy and a potential difference
across the length of the rod
U  qV  FBl  qvBl

V  Blv
The electric potential difference creates an electric field and an
electric force on the electrons that is of same magnitude as
magnetic force, but in the opposite direction
E
V
 vB
l
 FE  q E  qvB  FB
ConcepTest
Motional EMF
A conducting rod slides on a
conducting track in a constant
1) clockwise
B field directed into the page.
2) counterclockwise
What is the direction of the
3) no induced current
induced current?
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
v
ConcepTest
Motional EMF
A conducting rod slides on a
conducting track in a constant
1) clockwise
B field directed into the page.
2) counterclockwise
What is the direction of the
3) no induced current
induced current?
The B field points into the page.
The flux is increasing since the
area is increasing. The induced
B field opposes this change and
therefore points out of the page.
Thus, the induced current runs
counterclockwise according to
the right-hand rule.
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
v
x x x x x x x x x x x
Follow-up: What direction is the magnetic force on the rod as it moves?
ConcepTest
Magic Loop
A wire loop is in a uniform
(1) moves to the right
magnetic field. Current flows
(2) moves up
in the wire loop, as shown.
(3) remains motionless
What does the loop do?
(4) rotates
(5) moves out of the page
ConcepTest
Magic Loop
A wire loop is in a uniform
(1) moves to the right
magnetic field. Current flows
(2) moves up
in the wire loop, as shown.
(3) remains motionless
What does the loop do?
(4) rotates
(5) moves out of the page
There is no magnetic force on the top
and bottom legs, since they are parallel
to the B field. However, the magnetic
force on the right side is into the page,
and the magnetic force on the left side
is out of the page. Therefore, the entire
loop will tend to rotate.
This is how a motor works !!
Electric Generators
AC generator is shown to the right.
The axle is rotated by an external
force such as falling water or
steam. The brushes are in
constant electrical contact with the
slip rings. A sinusoidal emf is
induced in the rotating loop (N is
the number of turns, and A the
area of the loop):
A dc generator is similar, except that
it has a split-ring commutator
instead of slip rings.
EMF on an AC Generator
 ab  Blv

 Blv sin q  
cd
For a single loop,
   ab  cd  2Blv sin q
If loop rotates at constant angular velocity
2
  2 f 
T
h
 q   t and v   r  
2
Generator with N turns of loop area A  lh

  N  2 Bl h2 sin  t   NB A sin t


0  NB A
ConcepTest
Generators
A generator has a coil of wire
rotating in a magnetic field.
If the rotation rate increases,
1) increases
2) decreases
how is the maximum output
3) stays the same
voltage of the generator
4) varies sinusoidally
affected?
ConcepTest
Generators
A generator has a coil of wire
rotating in a magnetic field.
If the rotation rate increases,
1) increases
2) decreases
how is the maximum output
3) stays the same
voltage of the generator
4) varies sinusoidally
affected?
The maximum voltage is the leading
term that multiplies sin(t) and is
given by 0 = NBA. Therefore, if
 increases, then 0 must increase
as well.
  NBA sin( t )
Back EMF and Counter Torque
An electric motor turns
because there is a torque on
it due to the current. We
would expect the motor to
accelerate unless there is
some sort of drag torque.
That drag torque exists, and
is due to the induced emf,
called a back emf.
A similar effect occurs in a generator – if it is connected
to a circuit, current will flow in it, and will produce a
counter torque. This means the external applied torque
must increase to keep the generator turning.
Eddy Currents
Induced currents can flow in
bulk material as well as through
wires. These are called eddy
currents, and can dramatically
slow a conductor moving into
or out of a magnetic field.
Applications of this
phenomenon include
electromagnetic brakes and
metal detectors.
Transformers
A transformer consists of two
coils, either interwoven or
linked by an iron core. A
changing emf in one induces
an emf in the other.
The ratio of the emfs is equal
to the ratio of the number of
turns in each coil:
 B
VS  N S
t
 B
VP  N P
t
Energy must be conserved;
therefore, in the absence of
losses, the ratio of the currents
must be the inverse of the ratio of
turns:

P  I PVP  I SVS

This is a step-up transformer –
the emf in the secondary coil is
larger than the emf in the
primary coil.
ConcepTest
Transformers I
1) 30 V
What is the voltage
2) 60 V
across the lightbulb?
3) 120 V
4) 240 V
5) 480 V
120 V
ConcepTest
Transformers I
1) 30 V
What is the voltage
2) 60 V
across the lightbulb?
3) 120 V
4) 240 V
5) 480 V
The first transformer has a 2:1 ratio
of turns, so the voltage doubles.
But the second transformer has a
1:2 ratio, so the voltage is halved
again. Therefore, the end result is
the same as the original voltage.
120 V
240 V
120 V
ConcepTest
Transformers II
1) 1/4 A
Given that the intermediate
2) 1/2 A
current is 1 A, what is the
3) 1 A
current through the
4) 2 A
lightbulb?
5) 5 A
1 A
120 V
240 V
120 V
ConcepTest
Transformers II
1) 1/4 A
Given that the intermediate
current is 1 A, what is the
current through the
lightbulb?
2) 1/2 A
3) 1 A
4) 2 A
5) 5 A
Power in = Power out
240 V  1 A = 120 V  ???
1 A
The unknown current is 2 A.
120 V
240 V
120 V
ConcepTest
A 6 V battery is connected to
one side of a transformer.
Compared to the voltage drop
Transformers III
1) greater than 6 V
2) 6 V
across coil A, the voltage
3) less than 6 V
across coil B is:
4) zero
A
6V
B
ConcepTest
Transformers III
A 6 V battery is connected to
1) greater than 6 V
one side of a transformer.
2) 6 V
Compared to the voltage drop
across coil A, the voltage
3) less than 6 V
across coil B is:
4) zero
The voltage across B is zero.
Only a changing magnetic flux
induces an EMF. Batteries can
only provide DC current.
A
6V
B
Transmission of Power
Transformers work only if the current is changing;
this is one reason why electricity is transmitted as ac
Applications of Induction: Sound Systems
The microphone
The vibrating membrane
induces an emf in the coil
The seismograph
Has a fixed coil and a magnet
hung on a spring (or vice versa),
and records the current induced
when the earth shakes.
Applications of Induction
A ground fault circuit interrupter (GFCI)
Interrupts the current to a circuit that has shorted out in a very
short time, preventing electrocution.
Applications: Eddy Currents
• The magnetic force on the eddy current
acts as retarding force.
• Hence an external force is required to
pull a metal through a magnetic field
• The retarding force on the eddy currents
is used as a braking force in trains with
electromagnetic brakes.
Eddy currents acting as
electromagnetic brakes on a
rotating metal wheel
Eddy currents induced in a metal
objects by magnetic field due to a
current in a transmitter coil change the
magnetic flux through the receiver coil
and thus induce a current in it
Generalization of Faraday’s law
A magnetic field changing in time induces an electric
field that varies in space
The electric field will exist regardless of whether there
are any conductors around.
Exact mathematical equation stating this
generalization of Faraday’s law was derived by
Maxwell using calculus.
Maxwell’s Equations
Maxwell’s equations are the basic equations of
electromagnetism. They involve calculus; here is a
verbal summary:
1. Gauss’s law relates electric field to charge
(equivalent to Coulomb’s Law)
2. A law stating there are no magnetic “charges”
3. A changing magnetic field produces an electric field
4. A magnetic field is produced by an electric current,
and also by a changing electric field
Changing Electric Fields
Produce Magnetic Fields
Only one part of this is new –
that a changing electric field
produces a magnetic field.
In order for Ampère’s law to hold,
it can’t matter which surface we
choose. But look at a discharging
capacitor; there is a current
through surface 1 but none
through surface 2:
Ampère’s law relates the
magnetic field around a
current to the current through
a surface.
Therefore, Ampère’s law is
modified to include the creation
of a magnetic field by a
changing electric field:
 B l  0 Iencl  0 0
 E
t
Maxwell Equations
Fundamental equations of electromagnetism
Integral Form
q
Gauss's law:
E

dA


0
J. C. Maxwell
1831 - 1879
Differential Form
E 
Gauss's law for magnetism:
 B  dA  0
Faraday's law:
 E  ds  
Ampere-Maxwell law:
 B  ds  0 I through  0 0

0
B  0
d m
dt
B
t
J
1 E
 B 

0 c 2 c 2 t
 E  
d e
dt
Electric field can be created by charges and changing magnetic field
Magnetic fields can be created by currents and changing electric field
Transition to differential form uses:
 The differential operator
 Divergence theorem:
  iˆ
 ˆ  ˆ 
 j k ,
x
y
z
 E  dA     EdV ,
and Stoke's theorem:
 Expression of charges and currents through densities: q 
  dV ,
 E  ds     E  dA,
I
  J  dA,
through
 The zero limit of volumes and areas of closed surfaces, and of areas bounded by closed curves
 The relation  0 
1
0 c 2
Electromagnetic Waves
Maxwell’s Equations in free space (a region free of charges and currents)
Integral Form
Gauss's law:
Gauss's law for magnetism:
Faraday's law:
Ampere-Maxwell law:
Differential Form
 E  dA  0
 B  dA  0
d m
dt
d e
B

ds



0
0

dt
 E  ds  
E  0
B  0
B
t
1 E
 B  2
c t
 E  
• Induced electric field is created in a direction perpendicular to a changing
magnetic field, varying along a direction normal to both fields.
• Induced magnetic fields are created by in a direction perpendicular a changing
electric field, varying along a direction normal to both fields.
•Therefore perpendicular electric and magnetic fields can sustain each other as
long as they change in space and time.
What are the properties of self-sustaining
electromagnetic fields in free space?
Production of Electromagnetic Waves
Since a changing electric field produces a magnetic
field, and a changing magnetic field produces an electric
field, once sinusoidal fields are created they can
propagate on their own. These propagating fields are
called electromagnetic waves.
Oscillating charges will produce electromagnetic waves:
Far from the source, the waves are plane waves:
The electric and magnetic waves are perpendicular to each other,
and to the direction of propagation (i. e. they are transverse waves)
Light as an Electromagnetic Wave
When Maxwell calculated the speed of propagation of
electromagnetic waves, he found:
This is the speed of light in a vacuum.
For EM wave at same point in space: E  cB  B /  0  0
Light was known to be a wave; after producing
electromagnetic waves of other frequencies, it was
known to be an electromagnetic wave as well.
The frequency of an electromagnetic wave is related to
its wavelength:
Inductance
Mutual inductance: a changing
current in one coil will induce a
current in a second coil.
Example: a transformer.
Unit of inductance: the henry, H: 1 H = 1 V·s/A = 1 Ω·s.
Self-inductance: a changing current in a
coil will also induce an emf in itself:
Here, L is called the self-inductance.
Energy Stored in a Magnetic Field
 0 NI
1 2
U  energy  LI
2

  0 N   I 
For a solenoid: B 
;   N
=  NA 
  

l
t
 l   t 
2
1 2 1  0 N 2 A  2
1
1
  0 NI 
2
U B  LI  
I

Al

AlB
.



2
2 l
20  l 
20

U
1 2
Magnetic energy density
uB  B 
B
Al 2  0
1
Compare to electric energy density in a capacitor: uE  0 E
2
L
0 N 2 A
l
LR Circuit
(a) A circuit consisting of an inductor
and a resistor will begin with most of the
voltage drop across the inductor, as the
current is changing rapidly. With time,
the current will increase less and less,
until all the voltage is across the resistor.
where
(b) If the circuit is then shorted across the
battery, the current will gradually decay away:
AC Circuits
Resistors, capacitors, and inductors have different phase relationships
between current and voltage when placed in an ac circuit.
The current through
a resistor is in phase
with the voltage.
In a capacitor, the
current leads the
voltage by 90°.
The current through
an inductor lags the
voltage by 90°.
Reactance
Both the inductor and capacitor have an effective resistance
(ratio of voltage to current), called the reactance.
Inductor:
Capacitor:
Note that both depend on frequency.
LRC Series AC Circuit
Analyzing the LRC series AC circuit
is complicated, as the voltages are
not in phase – this means we
cannot simply add them.
Furthermore, the reactances
depend on the frequency.
Phasors
We calculate the voltage (and current)
using what are called phasors – these
are vectors representing the
individual voltages and currents
(a) Here, at t = 0, the current and
voltage are both at a maximum.
As time goes on, the phasors
will rotate counterclockwise.
(b) Some time t later, the phasors
have rotated.
The voltages across each device
are given by the x-component of
each, and the current by its xcomponent. The current is the same
throughout the circuit.
We find from the ratio of voltage to
current that the effective
resistance, called the impedance,
of the circuit is given by:
The rms current in an ac circuit is:
We see that Irms will be a maximum
when XC = XL; the frequency at which
this occurs is called the
resonance frequency