he current change) with a strength given by
dt
di
L
V
=
The faster the current changes, the larger the opposing voltage produced by the inductor. We
could also integrate this and get an expression for the current as a function of voltage
( )
( )
0
0
1
t
i
d
V
L
t
i
t
t
+
=
Because the formula for the power consumed by a circuit element is P = I V, we can calculate the
energy stored in an inductor as it goes from zero current to maximum current:
2
0
2
1
'
'
I
L
dI
I
L
U
dI
I
L
dt
P
I
=
= = Notice that we are not watching this inductor over a full cycle of sinusoidally oscillating voltage;
its just a current ramping up from 0 to some maximum value. The direction of the induced
voltage can be determined by realizing that it opposes the change in current.
Capacitors
Inductors produce a voltage in proportional to the derivative of current with respect to time, but
capacitors can be thought of as producing a voltage in response to the integral of current with
respect to time. We can write this as
dt
d
C
i
V
=
or
1 PHYS 3100
( )
( )
0
1
0
V
d
i
C
t
V
t
+
=
Finally, as we did with the inductor, we can write an expression for the power consumed as the
capacitor charges up to V volts:
2
0
2
1
'
'
V
C
dV
V
C
U
dV
V
C
dt
P
V
=
= = The purpose of a capacitor is to separate and store charge. The charging and discharging rates
may be the same, or they may be very different. Although current does not flow through a
capacitor (because it is just two conductors separated by some kind of dielectric), the buildup of
positive charge on one plate and negative charge on the other looks from the outside like a
current is flowing (positive charge is leaving the battery headed for the positive plate, and we can
think of the negative charge leaving the batterys negative terminal for the other plate as being
identical to positive charges coming from that plate to the batterys negative terminal, which
would look to the battery exactly like a regular current). This pseudo-current is called the
displacement current, and it is proportional to the change in voltage over time, as can be seen
by looking at the first equation in this section.
Series and Parallel Combinations
When multiple inductors are linked in series, they will all carry the same current, and the voltage
drop across the full chain will be equal to the individual voltage drops across each inductor. That
means that inductances in series work the same way as resistances in series: +
+
+
=
3
2
1
L
L
L
L
series
tot
If the inductors are in parallel, however, its the voltage across each that is the same. In this case,
the currents add to produce the total current, and the equivalent inductance to this group is +
+
+
=
3
2
1
1
1
1
1
L
L
L
L
par
tot
With capacitors, things are reversed. If you think about the capacitors as parallel-plate capacitors,
and remember from introductory physics that the capacitance of a parallel-plate capacitor is given
by
d
A
C
0
=
2 PHYS 3100
you can see that capacitors with the same separation and dielectric could be slightly redrawn into
a single capacitor. That equivalent capacitor would keep its plate separation and dielectric
constant, but its area would be equal to the sum of the areas of all of the other capacitors. +
+
+
=
3
2
1
C
C
C
C
par
tot
If the capacitors are in series they combine in a very different way. Notice that the right plate of
the left capacitor and the left plate of the right capacitor in the drawing below are connected by a
wire.
The plates must have charges which are equal in magnitude but opposite in sign since the
arrangement inside the dotted circle (plates of different capacitors connected by a wire) would be
neutral without a battery. That means that the plates facing them also have charges of the same
magnitude and opposite sign. Well call it Q and label them all
Since the voltage is split between the capacitors, the sum of the voltages across each should be
the batterys voltage.
Total
C
Q
C
Q
C
Q
V
V
V
=
+
=
+
=
2
1
2
1
The capacitance of several capacitors in series is then: +
+
=
2
1
1
1
1
C
C
C
series
tot
3
+
-
+
-
+Q
+Q
-Q
-Q PHYS 3100
Mutual Inductance
A current in a coil causes a magnetic field, and a changing magnetic field causes a current in a
coil. What happens if one coil is near another and the first (primary) coil is connected to a
generator? The primary coil will develop a magnetic field, which will of course change with
time since the voltage of the generator and therefore the current of the coil oscillates in time. As
this changing magnetic field encounters the other coil (not connected to a generator, and called
the secondary), it will cause an induced current to flow in it.
This effect is known as mutual induction. The emf induced in the secondary coil depends on
the change in magnetic flux through the secondary as time goes on. The magnetic flux in the
secondary depends on its area, its number of loops, and the magnetic field (produced by the
primary). The primarys magnetic field is proportional to the current through it, so we can say
the total flux through the secondary coil N
s s
is proportional to I
p
. If the two things are
proportional, we can include a constant and set them equal to each other. The constant will be
called M, for mutual inductance:
p
s
S
p
s
s
I
N
M
so
I
M
N =
= which we can rewrite as
dt
dI
M
V
p
s =
Keep in mind that M also depends critically on the arrangement of the coils, so it is a geometric
factor as well.
The direction of self-induced voltage has already been discussed, but since the direction of
mutually-induced voltage depends on the arrangement of the coils (which is not typically shown
in a circuit diagram), we have to have a way to indicate the direction of the voltage induced in
this case.
To do this, we use the dot convention described in your book. One end of each inductor is
marked with a dot. If you measure the voltage across each inductor from its un-dotted end to its
dotted end, youll see that the voltages oscillate in phase with one another. As your book states
it, When the reference direction for a current enters the dotted terminal of a coil, the reference
polarity of the voltage that it induces in the other coil is positive at its dotted terminal. In other
words, if the self-induced voltage in one coil is higher at its dotted end, the voltage produced by
mutual induction in the other coil will also be higher at its dotted end.
4 PHYS 3100
When you can see the coils, you can figure out the dotting if it is not present. Using the right-
hand rule, if the fingers in your right hand curl in the same direction that the current in one of the
coils is moving, your thumb will point in the direction of the magnetic flux. Put a dot on the
terminal of the first coil where the current enters. Now, pick one of the terminals of the other coil
and imagine a current entering it. Find the direction of the flux, again using the right-hand rule. If
the direction is the same as that from the flux due to the first coil, put the other dot by the
terminal through which the current entered the second coil. If not, the dot goes at the other
terminal of the second coil.
Mutual inductance combines the effect of the individual self-inductances of the coils involved
with a geometric factor which describes their degree of flux overlap. This factor is known as the
coefficient of coupling (represented by k) and it ranges from 0 for coils with no shared flux to 1
if two coils could completely share the same flux. This concept means that we can write the
mutual inductance M of two coils with self-inductances L
1
and L
2
and coupling coefficient k as
2
1
L
L
k
M
=
We will restrict ourselves to linear materials (meaning the flux is linearly related to the coil
current and the number of turns). This rules out iron and other ferromagnets, but we can still do a
significant amount within these restrictions. If we have only two coils, notice that the equation
above would be symmetric for the mutual inductance in either coil due to the other (i.e., M
21
=
M
12
= M).
If we want to examine the energy stored in two coils which are linearly coupled, we will find the
individual energies of the coils, as expected, but we will also get a term for the energy stored by
the mutual inductance:
2
1
2
2
2
2
1
1
2
1
2
1
I
I
M
I
L
I
L
U
+
+
=
As shown in your book, this extra term arises from the voltage induced in coil 1 (for example) as
current i
2
goes from 0 to I
2
. The voltage multiplied by the current and then integrated over time
gives the energy. This can be written as
2
1
0
2
1
2
I
I
M
di
M
I
dt
p
I
=
=
Natural Response of RL Circuits
Because a perfect inductor only resists changes in current, it will act like an ordinary wire (short
circuit) in a DC circuit after the current has reached its steady-state value. If the DC source is
disconnected, the inductor will begin to release its energy in the form of a potential across the
inductor which will drive a current. If we assume the part of the circuit containing the inductor
5 PHYS 3100
also contains a resistor, we can see that the energy stored in the inductor will eventually be
dissipated as heat in the resistor.
The current will move through the inductor in the same direction as the original current since the
inductor is resisting the change by trying to keep the current the same. The opposing voltage is
strongest where the rate of change is strongest, which is at the instant of disconnection. The
inductor cant completely replace the current, so its rate of change decreases with time, meaning