Chapter 32
Inductance L and the stored
magnetic energy
RL and LC circuits
RLC circuit
Resistance, Capacitance and
Inductance
∆V
Ohm’s Law defines resistance: R ≡
I
Resistors do not store energy, instead they transform
electrical energy into thermo energy at a rate of:
P = ∆V ⋅ I =
∆V 2
R
= I 2R
Q
Capacitance, the ability to hold charge: C ≡
∆V
Capacitors store electric energy once charged:
1 Q2 1
2
∆U E =
= C ( ∆V )
2 C 2
Inductance, the ability to “hold” current (moving charge).
Inductors store magnetic energy once “charged” with current,
i.e., current flows through it.
Inductance, the definition
When a current flows through a coil,
there is magnetic field established.
If we take the solenoid assumption E
for the coil: B = µ0 nI
When this magnetic field flux
changes, it induces an emf, EL,
called self-induction:
d ( NAB )
d ( NAµ0 nI )
dΦ B
dI
E =−
=−
=−
= − µ n 2V
L
dt
dt
dI
or: E L ≡ − L
dt
dt
0
dt
I
+
EL
–
≡ −L
For a solenoid: L = µ0 n 2V
dI
dt
Where
n: # of turns per unit length.
N: # of turns in length l.
A: cross section area
V: Volume for length l.
This defines the inductance L, which is constant related only to the coil. The selfinduced emf is generated by current flowing though a coil. According to Lenz
Law, the emf generated inside this coil is always opposing the change of the
current which is delivered by the original emf.
Inductor
We used a coil and the solenoid assumption to
introduce the inductance. But the definition
L≡−
EL
dI
dt
holds for all types of inductance, including a straight
wire. Any conductor has capacitance and inductance.
But as in the capacitor case, an inductor is a device
made to have a sizable inductance.
An inductor is made of a coil. The symbol is
Once the coil is made, its inductance L is defined. The
self-induced emf over this inductor under a changing
current I is given by:
dI
EL = − L
dt
Unit for Inductance
The SI unit for inductance is the henry (H)
V ⋅s
1H = 1
A
Named for Joseph Henry:
1797 – 1878
American physicist
First director of the Smithsonian
Improved design of electromagnet
Constructed one of the first motors
Discovered self-inductance
Discussion about Some Terminology
Use emf and current when they are caused
by batteries or other sources
Use induced emf and induced current when
they are caused by changing magnetic fields
When dealing with problems in
electromagnetism, it is important to
distinguish between the two situations
Example: Inductance of a coaxial
cable
Start from the definition
EL = −
We have dΦ B = LdI , or Φ B = LI
µo I
dr
Φ B = ∫ B dA = ∫
a 2πr
µ I b
= o ln  
2π
a
b
So the inductance is
Φ B µo  b 
L=
=
ln  
I
2π  a 
dΦ B
dI
= −L
dt
dt
Put inductor L to use: the RL Circuit
An RL circuit contains a
resistor R and an inductor L.
There are two cases as in
the RC circuit: charging and
discharging. The difference
is that here one charges
with current, not charge.
Charging:
When S2 is connected to
position a and when switch
S1 is closed (at time t = 0),
the current begins to
increase
Discharging:
When S2 is connected to
position b.
PLAY
ACTIVE FIGURE
RL Circuit, charging
Applying Kirchhoff’s loop rule to the
circuit in the clockwise direction gives
ε −IR −L
dI
=0
dt
Here because the current is increasing,
the induced emf has a direction that
should oppose this increase.
Solve for the current I, with initial
condition that I(t=0) = 0, we find
ε
ε
−Rt L
I=
1− e
≡
1 − e −t τ
R
R
(
)
(
)
Where the time constant is defined as:
L
τ≡
R
RL Circuit, discharging
When switch S2 is moved to position
b, the original current disappears.
The self-induced emf will try to
prevent that change, and this
determines the emf direction (Lenz
Law).
Applying Kirchhoff’s loop rule to the
previous circuit in the clockwise
direction gives
dI
−I R + L
=0
dt
Solve for the current I, with initial
condition that I ( t = 0 ) = E R we find
ε −Rt L ε −t τ
I= e
≡ e
R
R
Energy stored in an inductor
In the charging case, the current I from
the battery supplies power not only to
the resistor, but also to the inductor.
From Kirchhoff’s loop rule, we have
dI
ε =IR +L
dt
Multiply both sides with I:
εI = I 2 R + LI
dI
dt
This equation reads: powerbattery=powerR+powerL
So we have the energy increase in the inductor as:
dUL
dI
= LI
dt
dt
I
1 2
Solve for UL: UL = ∫ LId I = LI
2
0
Stored energy type and
the Energy Density of a Magnetic Field
Given UL = ½ L I2 and assume (for simplicity) a solenoid with L =
µo n2 V
2
2


1
B
B
UL = µo n 2V 
V
 =
2
2 µo
 µo n 
Since V is the volume of the solenoid, the magnetic energy
density, uB is
UL B 2
uB ≡
=
V
2 µo
So the energy stored in the
solenoid volume V is
magnetic (B) energy.
And the energy density is
proportional to B2.
This applies to any region in which a magnetic field exists (not
just the solenoid)
RL and RC circuits comparison
RL
Charging
ε
I=
1 − e −Rt L
R
Discharging
ε
I = e −Rt L
R
Energy
(
1
UL = LI 2
2
RC
)
−t
ε
I ( t ) = e RC
R
−t
Q RC
I (t ) =
e
RC
Q2 1
UC =
= C (∆V ) 2
2C 2
Magnetic field Electric field
Energy density
B2
uB =
2 µo
uE =
1
εo E 2
2
Energy Storage Summary
Inductor and capacitor store energy through
different mechanisms
Charged capacitor
When current flows through an inductor
Stores energy as electric potential energy
Stores energy as magnetic potential energy
A resistor does not store energy
Energy delivered is transformed into thermo energy
LC Circuits
LC: circuit with an inductor and a
capacitor.
Initial condition: either the C or
the L has energy stored in it.
The “show” starts: when the
switch S closes, t = 0 and the
time starts.
Your physics intuition: neither C
nor L consumes energy, the
initially stored energy will oscillate
between the C and the L.
LC Circuits, the calculation
Initial condition: Assume that the
capacitor was initially charged to Qmax.
when the switch S closes, t = 0 and the
time starts.
Apply Kirchhoff’s loop rule:
q
dI
∆VC + E L = − L = 0
C
dt
Here q is the charge in the capacitor at
time t. Because charges flow out of the
capacitor to form the current I, we have:
−dq
=I
dt
d 2I
1
Combine these two equations:
I =0
+
2
dt
LC
1
2
, and I max = ωQmax
Solve for the current I: I = I max sin (ωt ) with ω ≡
LC
q = Qmax cos (ωt )
Here we also have
LC Circuits, the oscillation of charge and
current
Oscillations: simply plot the results, we find
out that the charge stored in the capacitor
and the current “stored” in the inductor
oscillate. The phase difference is T/2.
q = Qmax cos (ωt )
I = I max sin (ωt )
This means that when the capacitor is fully
charged, the current is zero. When the
capacitor has no charges in it, the current
reaches its maximum in magnitude through
the inductor.
From the formulas for the energies stored
in a capacitor and an inductor, we know
that this oscillation happens between
electric energy and magnetic energy.
Q2
UC =
2C
UL =
1 2
LI
2
q
LC Circuits, the oscillation of energy
From the following four formulas
q = Qmax cos (ωt ) I = I max sin (ωt )
q2
1
UC =
UL = LI 2
2C
2
We have the oscillation of the energies in
the capacitor and the inductor:
2
Qmax
C
UC =
cos 2 (ωt ) = Emax
cos 2 (ωt )
2C
UL =
1 2
L
LImax sin 2 ( ωt ) = Emax
sin 2 ( ωt )
2
From energy conservation:
C
L
Emax
= Emax
, or Imax = ωQmax
Move from the ideal LC circuit to the
real-life RLC circuit
In actual circuits, there is always some resistance,
therefore, there is some energy transformed to thermo
energy by the resistance in the system and dissipates to
the environment.
Radiation is also inevitable in this type of circuit, and
energy will be radiated out of the LC system as
electromagnetic wave through space.
The total energy in the circuit continuously decreases as
a result of these processes
Here we will only discuss about the energy dissipated
through the resistance.
The RLC Circuit and the analysis
Concentrating the resistance in the
system into a resistor, with the inductor
and the capacitor, we model the circuit
with an RLC Circuit.
The capacitor is charged with the
switch at position a. At time t = 0, the
switch is thrown to position b to form
the RLC circuit.
Apply Kirchhoff’s loop rule:
q
dI
−dq
− L − IR = 0, and
=I
C
dt
dt
We have:
d 2 q R dq
1
2
2
+
+
ω
q
=
0
,
and
ω
=
dt 2 L dt
LC
Solve for q:
q = Qmax e
−
R
t
2L
cos (ωd t ) , with ωd2 =
And: I = Qmax e
−
R
t
2L
1
R
−
LC 2 L
R


ω
sin
ω
t
+
cos
ω
t
(
)
(
)
d
d 
 d
2L


PLAY
ACTIVE FIGURE
Mutual Inductance
The magnetic flux through the area enclosed
by a circuit often varies with time because of
time-varying currents in nearby circuits
This process is known as mutual induction
because it depends on the interaction of two
circuits
Mutual Inductance and transformers
The current in coil 1 sets up a magnetic
field that varies as I1.
When magnetic field lines pass through
coil 2, cause the magnetic flux in coil 2 to
change and induce current I2 in coil 2.
This process is called mutual inductance.
If coil 1 has a current I1 and N1 turns, and
coil 2 has N2 turns. When the field lines
that go through coil 1 completely go
through coil 2, we have a transformer.
Coil 1 and 2 are called prime and second
coils. The terminal voltages at these two
coils are
∆V1 N1
=
∆V2 N 2
From energy conservation: ∆V1 I1 = ∆V2 I 2
I 2 N1
We have:
=
I1 N 2
If coil 2 connects to a resistor R2, the
resistance coil 1 “sees” is
2
N 
R1 =  1  R2
 N2 