Today

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Todays agendum Electric potential energy. You
must be able to use electric potential energy in
work-energy calculations. Electric
potential. You must be able to calculate the
electric potential for a point charge, and use
the electric potential in work-energy
calculations. Electric potential and electric
potential energy of a system of charges. You must
be able to calculate both electric potential and
electric potential energy for a system of charged
particles (point charges today, charge
distributions next lecture). The electron
volt. You must be able to use the electron volt
as an alternative unit of energy.
3
This lecture introduces electric potential energy
and something called electric potential.
Electric potential energy is just like
gravitational potential energy.
Except that all matter exerts an attractive
gravitational force, but charged particles exert
either attractive or repulsive electrical
forcesso we need to be careful with our signs.
Electric potential is the electric potential
energy per unit of charge.
If you understand the symbols in the starting
equations, and avoid sign and direction mistakes,
homework and exams are not difficult.
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Electric Potential Electric Potential Energy
Electric Potential Energy
Work done by Coulomb force when q1 moves from a
to b
r
b
dr
q1 ()
ds
FE
rb
a
q1 ()
ra
q2 (-)
You dont need to worry about the details of the
math. They are provided for anybody who wants to
study them later.
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r
b
dr
q1 ()
ds
I did the calculation for a charge moving away
from a charge you could do a similar
calculation for , -, and .
FE
rb
a
ra
q2 (-)
The important point is that the work depends only
on the initial and final positions of q1.
In other words, the work done by the electric
force is independent of path taken. The electric
force is a conservative force.
Disclaimer this is a demonstration rather than
a rigorous proof.
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The next two slides are intended to draw a
parallel between the electric and gravitational
forces.
Instead of saying ugh, this is confusing new
stuff, you are supposed to say, oh, this is
easy, because I already learned the concepts in
Physics 103.
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A bit of review
y
If released, it gains kinetic energy and loses
potential energy, but mechanical energy is
conserved EfEi. The change in potential energy
is Uf - Ui -(Wc)i?f.
Ui mgyi
yi
Uf 0
x
What force does Wc? Force due to gravity.
graphic borrowed from http//csep10.phys.utk.edu
/astr161/lect/history/newtongrav.html
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A charged particle in an electric field has
electric potential energy. It feels a force
(as given by Coulombs law).

F
E
It gains kinetic energy and loses potential
energy if released. The Coulomb force does
positive work, and mechanical energy is
conserved.
- - - - - - - - - - - - - - - - - - -
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The next two slides define electrical potential
energy.
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Now that we realize the electric force is
conservative, we can define a potential energy
associated with it.
The subscript E is to remind you this is
electric potential energy. After this slide, I
will drop the subscript E.
The change in potential energy when a charge q0
moves from point a to point b in the electric
field of another charge q is
r
b
dr
q0
ds
FE?
rab
a
The minus sign in this equation comes from the
definition of change in potential energy. The
sign from the dot product is automatically
correct if you include the signs of q and q0.
ra
? on FE means the direction depends on the signs
of the charges.
q
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(from the previous slide)
is equivalent to your starting equation
i and f refer to the two points for which we
are calculating the potential energy difference.
You could also use a and b like your text
does, or 0 and 1 or anything else convenient.
I use i and f because I always remember that
?(anything) (anything)f (anything)i.
The next two slides use this definition of
electrical potential energy to derive an equation
for the electrical potential energy of two
charged particles.
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starting with an equation from two slides back
r
b
dr
q0
ds
FE?
rab
a
ra
By convention, we choose electric potential
energy to be zero at infinite separation of the
charges.
this diagram shows q0 after it has moved from a
to b
q
If there are any math majors in the room, please
close your eyes for a few seconds. We should be
talking about limits.
0
0
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This provides us with the electric potential
energy for a system of two point charges q and
q0, separated by a distance r
You can call the charges q and q0, or q1 and q2,
or whatever you want. If you have more than two
charged particles, simply add the potential
energies for each unique pair of particles.
Example calculate the electric potential energy
for two protons separated by a typical
proton-proton intranuclear distance of 2x10-15 m.
What is the meaning of the sign in the result?
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Really Important fact to keep straight.
The change in potential energy is the negative of
the work done by the conservative force which is
associated with the potential energy (the
electric force).
If an external force moves an object against
the conservative force, then
Always ask yourself which work you are
calculating.
for example, if you push two negatively charged
balloons together
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Todays agendum Electric potential energy. You
must be able to use electric potential energy in
work-energy calculations. Electric
potential. You must be able to calculate the
electric potential for a point charge, and use
the electric potential in work-energy
calculations. Electric potential and electric
potential energy of a system of charges. You must
be able to calculate both electric potential and
electric potential energy for a system of charged
particles (point charges today, charge
distributions next lecture). The electron
volt. You must be able to use the electron volt
as an alternative unit of energy.
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Electric Potential
Previously, we defined the electric field by the
force it exerts on a test charge q0
Similarly, it is useful to define the potential
of a charge in terms of the potential energy of a
test charge q0
The electric potential V is independent of the
test charge q0.
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From
we see that the electric potential of a point
charge q is
The electric potential difference between points
a and b is
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Things to remember about electric potential
? Electric potential and electric potential
energy are related, but are not the same.
Electric potential difference is the work per
unit of charge that must be done to move a charge
from one point to another without changing its
kinetic energy.
? The terms electric potential and potential
are used interchangeably.
? The units of potential are joules/coulomb
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Things to remember about electric potential
? Only differences in electric potential and
electric potential energy are meaningful.
It is always necessary to define where U and V
are zero. Here we defined V to be zero at an
infinite distance from the sources of the
electric field.
Sometimes it is convenient to define V to be zero
at the earth (ground).
It should be clear from the context where V is
defined to be zero, and I do not foresee you
experiencing any confusion about where V is zero.
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Two more starting equations
and
so
(potential is equal to potential energy per unit
of charge)
Potential energy and electric potential are
defined relative to some reference point, so it
is better to use
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Two conceptual examples.
Example a proton is released in a region in
space where there is an electric potential.
Describe the subsequent motion of the proton.
The proton will move towards the region of lower
potential. As it moves, its potential energy will
decrease, and its kinetic energy and speed will
increase.
Example an electron is released in a region in
space where there is an electric potential.
Describe the subsequent motion of the electron.
The electron will move towards the region of
higher potential. As it moves, its potential
energy will decrease, and its kinetic energy and
speed will increase.
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Todays agendum Electric potential energy. You
must be able to use electric potential energy in
work-energy calculations. Electric
potential. You must be able to calculate the
electric potential for a point charge, and use
the electric potential in work-energy
calculations. Electric potential and electric
potential energy of a system of charges. You must
be able to calculate both electric potential and
electric potential energy for a system of charged
particles (point charges today, charge
distributions next lecture). The electron
volt. You must be able to use the electron volt
as an alternative unit of energy.
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Electric Potential Energy of a System of Charges
To find the electric potential energy for a
system of two charges, we bring a second charge
in from an infinite distance away
r
q1
q1
q2
before
after
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To find the electric potential energy for a
system of three charges, we bring a third charge
in from an infinite distance away
r12
r12
q1
q2
q1
q2
r13
r23
q3
before
after
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Electric Potential and Potential Energy of a
Charge Distribution
Collection of charges
P is the point at which V is to be calculated,
and ri is the distance of the ith charge from P.
Charge distribution
dq
r
P
Potential at point P.
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Example a 1 ?C point charge is located at the
origin and a -4 ?C point charge 4 meters along
the x axis. Calculate the electric potential at
a point P, 3 meters along the y axis.
y
P
3 m
x
q2
q1
4 m
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Example how much work is required to bring a 3
?C point charge from infinity to point P? An
external force moves q3 slowly from an infinite
distance to the point P.
0
y
q3
P
0
3 m
x
q2
q1
4 m
The work done by the external force was negative,
so the work done by the electric field was
positive. The electric field pulled q3 in (keep
in mind q2 is 4 times as big as q1).
Positive work would have to be done by an
external force to remove q3 from P.
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Example find the total potential energy of the
system of three charges.
y
q3
P
3 m
x
q2
q1
4 m
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Todays agendum Electric potential energy. You
must be able to use electric potential energy in
work-energy calculations. Electric
potential. You must be able to calculate the
electric potential for a point charge, and use
the electric potential in work-energy
calculations. Electric potential and electric
potential energy of a system of charges. You must
be able to calculate both electric potential and
electric potential energy for a system of charged
particles (point charges today, charge
distributions next lecture). The electron
volt. You must be able to use the electron volt
as an alternative unit of energy.
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The Electron Volt
An electron volt (eV) is the energy acquired by a
particle of charge e when it moves through a
potential difference of 1 volt.
This is a very small amount of energy on a
macroscopic scale, but electrons in atoms
typically have a few eV (10s to 1000s) of
energy.
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Todays agendum Electric potential of a charge
distribution. You must be able to calculate the
electric potential for a charge
distribution. Equipotentials. You must be able
to sketch and interpret equipotential
plots. Potential gradient. You must be able to
calculate the electric field if you are given the
electric potential. Potentials and fields near
conductors. You must be able to use what you have
learned about electric fields, Gauss Law, and
electric potential to understand and apply
several useful facts about conductors in
electrostatic equilibrium.
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Electric Potential of a Charge Distribution
Example potential and electric field between two
parallel conducting plates.
Assume V0ltV1 (so we can determine the direction
of the electric field). Also assume the plates
are large compared to their separation, so the
electric field is constant and perpendicular to
the plates.
Also, let the plates be separated by a distance d.
E
V0
V1
d
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y
x
E
z
dl
V0
V1
d
Ill discuss in lecture why the absolute value
signs are needed.
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Important note the derivation of
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Example A rod of length L located along the
x-axis has a total charge Q uniformly distributed
along the rod. Find the electric potential at a
point P along the y-axis a distance d from the
origin.
y
?Q/L
P
r
dq?dx
d
dq
x
dx
x
L
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y
P
A good set of math tables will have the integral
r
d
dq
x
dx
x
L
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Example Find the electric potential due to a
uniformly charged ring of radius R and total
charge Q at a point P on the axis of the ring.
dq
Every dq of charge on the ring is the same
distance from the point P.
r
R
P
x
x
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dq
r
R
P
x
x
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Example A disc of radius R has a uniform charge
per unit area ? and total charge Q. Calculate V
at a point P along the central axis of the disc
at a distance x from its center.
dq
The disc is made of concentric rings. The area of
a ring at a radius r is 2?rdr, and the charge on
each ring is ?(2?rdr).
r
P
x
x
R
We can use the equation for the potential due to
a ring, replace R by r, and integrate from r0 to
rR.
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dq
r
P
x
x
R
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dq
r
P
x
x
R
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See your text for other examples of potentials
calculated from charge distributions, as well as
an alternate discussion of the electric field
between charged parallel plates.
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PRACTICAL APPLICATION
For some reason you think practical applications
are important.
Well, I found one!
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Todays agendum Electric potential of a charge
distribution. You must be able to calculate the
electric potential for a charge
distribution. Equipotentials. You must be able
to sketch and interpret equipotential
plots. Potential gradient. You must be able to
calculate the electric field if you are given the
electric potential. Potentials and fields near
conductors. You must be able to use what you have
learned about electric fields, Gauss Law, and
electric potential to understand and apply
several useful facts about conductors in
electrostatic equilibrium.
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Equipotentials
Equipotentials are contour maps of the electric
potential.
http//www.omnimap.com/catalog/digital/topo.htm
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Equipotential lines are another visualization
tool. They illustrate where the potential is
constant. Equipotential lines are actually
projections on a 2-dimensional page of a
3-dimensional equipotential surface. (Just
like the contour map.)
The electric field must be perpendicular to
equipotential lines. Why?
Otherwise work would be required to move a charge
along an equipotential surface, and it would not
be equipotential.
In the static case (charges not moving) the
surface of a conductor is an equipotential
surface. Why?
Otherwise charge would flow and it wouldnt be a
static case.
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Here are some electric field and equipotential
lines I generated using an electromagnetic field
program.
Equipotential lines are shown in red.
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Todays agendum Electric potential of a charge
distribution. You must be able to calculate the
electric potential for a charge
distribution. Equipotentials. You must be able
to sketch and interpret equipotential
plots. Potential gradient. You must be able to
calculate the electric field if you are given the
electric potential. Potentials and fields near
conductors. You must be able to use what you have
learned about electric fields, Gauss Law, and
electric potential to understand and apply
several useful facts about conductors in
electrostatic equilibrium.
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Potential Gradient (Determining Electric Field
from Potential)
The electric field vector points from higher to
lower potentials.
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For spherically symmetric charge distribution
In one dimension
In three dimensions
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Calculate -dV/d(whatever) including all signs. If
the result is , E vector points along the
(whatever) direction. If the result is -, E
vector points along the (whatever) direction.
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Example In a region of space, the electric
potential is V(x,y,z) Axy2 Bx2 Cx, where A
50 V/m3, B 100 V/m2, and C -400 V/m are
constants. Find the electric field at the origin.
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Todays agendum Electric potential of a charge
distribution. You must be able to calculate the
electric potential for a charge
distribution. Equipotentials. You must be able
to sketch and interpret equipotential
plots. Potential gradient. You must be able to
calculate the electric field if you are given the
electric potential. Potentials and fields near
conductors. You must be able to use what you have
learned about electric fields, Gauss Law, and
electric potential to understand and apply
several useful facts about conductors in
electrostatic equilibrium.
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Potentials and Fields Near Conductors
When there is a net flow of charge inside a
conductor, the physics is generally complex.
When there is no net flow of charge, or no flow
at all (the electrostatic case), then a number
of conclusions can be reached using Gauss Law
and the concepts of electric fields and
potentials
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Summary of key points (electrostatic case)
The electric field inside a conductor is zero.
Any net charge on the conductor lies on the outer
surface.
The potential on the surface of a conductor, and
everywhere inside, is the same.
The electric field just outside a conductor must
be perpendicular to the surface.
Equipotential surfaces just outside the conductor
must be parallel to the conductors surface.
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Another key point the charge density on a
conductor surface will vary if the surface is
irregular, and surface charge collects at sharp
points.
Therefore the electric field is large (and can be
huge) near sharp points.
Another Practical Application
To best shock somebody, dont touch them with
your hand touch them with your fingertip.
Better yet, hold a small piece of bare wire in
your hand and gently touch them with that.
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Todays agendum Capacitance. You must be able
to apply the equation CQ/V. Capacitors
parallel plate, cylindrical, spherical. You must
be able to calculate the capacitance of
capacitors having these geometries, and you must
be able to use the equation CQ/V to calculate
parameters of capacitors. Circuits containing
capacitors in series and parallel. You must be
understand the differences between, and be able
to calculate the equivalent capacitance of,
capacitors connected in series and parallel.
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Capacitors and Dielectrics
Capacitance
A capacitor is basically two parallel conducting
plates with air or insulating material in between.
E
V0
V1
L
A capacitor doesnt have to look like metal
plates.
Capacitor for use in high-performance audio
systems.
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The symbol representing a capacitor in an
electric circuit looks like parallel plates.
Heres the symbol for a battery, or an external
potential.

-
When a capacitor is connected to an external
potential, charges flow onto the plates and
create a potential difference between the plates.
Capacitor plates build up charge.
V
The battery in this circuit has some voltage V.
We havent discussed what that means yet.
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In a battery, chemical reactions release energy.
The energy is expended by transporting charged
particles from one terminal of the battery to
the other. A charge separation is maintained.
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-
If the external potential is disconnected,
charges remain on the plates, so capacitors are
good for storing charge (and energy).
conducting wires
-

V
Capacitors are also very good at releasing their
stored charge all at once. The capacitors in
your tube-type TV are so good at storing energy
that touching the two terminals at the same time
can be fatal, even though the TV may not have
been used for months.
High-voltage TV capacitors are supposed to have
bleeder resistors that drain the charge away
after the circuit is turned off. I wouldnt bet
my life on it.
Graphic from http//www.feebleminds-gifs.com/.
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assortment of capacitors
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Q
-Q
Heres this V again. It is the potential
difference provided by the external potential.
For example, the voltage of a battery.
C
-

V
The magnitude of charge acquired by each plate of
a capacitor is QCV where C is the capacitance of
the capacitor.
C is always positive.
The unit of C is the farad but most capacitors
have values of C ranging from picofarads to
microfarads (pF to ?F).
micro ?10-6, nano ?10-9, pico ?10-12 (Know
for exam!)
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Todays agendum Capacitance. You must be able
to apply the equation CQ/V. Capacitors
parallel plate, cylindrical, spherical. You must
be able to calculate the capacitance of
capacitors having these geometries, and you must
be able to use the equation CQ/V to calculate
parameters of capacitors. Circuits containing
capacitors in series and parallel. You must be
understand the differences between, and be able
to calculate the equivalent capacitance of,
capacitors connected in series and parallel.
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Parallel Plate Capacitance
Q
-Q
We previously calculated the electric field
between two parallel charged plates
E
V0
V1
d
This is valid when the separation is small
compared with the plate dimensions.
A
We also showed that E and ?V are related
This lets us calculate C for a parallel plate
capacitor.
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Reminders
Q is the magnitude of the charge on either plate.
V is actually the magnitude of the potential
difference between the plates. V is really ?V.
Your book calls it Vab.
C is always positive.
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Parallel plate capacitance depends only on
geometry.
Q
-Q
E
This expression is approximate, and must be
modified if the plates are small, or separated by
a medium other than a vacuum.
V0
V1
d
A
Greek letter Kappa. For todays lecture use
Kappa1.
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Isolated Sphere Capacitance
An isolated sphere can be thought of as
concentric spheres with the outer sphere at an
infinite distance and zero potential.
We already know the potential outside a
conducting sphere
The potential at the surface of a charged sphere
of radius R is
so the capacitance at the surface of an isolated
sphere is
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Capacitance of Concentric Spheres
Lets calculate the capacitance of a concentric
spherical capacitor of charge Q.
In between the spheres
b
a
Q
-Q
You need to do this derivation if you have a
problem on spherical capacitors!
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alternative calculation of capacitance of
isolated sphere
b
a
Q
-Q
Let a?R and b?? to get the capacitance of an
isolated sphere.
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Coaxial Cylinder Capacitance
We can also calculate the capacitance of a
cylindrical capacitor (made of coaxial cylinders).
?
L
The next slide shows a cross-section view of the
cylinders.
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Gaussian surface
b
r
a
Q
E
-Q
dl
This derivation is sometimes needed for homework
problems!
Lowercase c is capacitance per unit length
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Example calculate the capacitance of a capacitor
whose plates are 20 cm x 3 cm and are separated
by a 1.0 mm air gap.
d 0.001
area 0.2 x 0.03
If you keep everything in SI (mks) units, the
result is automatically in SI units.
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Example what is the charge on each plate if the
capacitor is connected to a 12 volt battery?
0 V
?V 12V
12 V
Remember, its the potential difference that
matters.
If you keep everything in SI (mks) units, the
result is automatically in SI units.
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Example what is the electric field between the
plates?
0 V
?V 12V
E
d 0.001
12 V
If you keep everything in SI (mks) units, the
result is automatically in SI units.
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Todays agendum Capacitance. You must be able
to apply the equation CQ/V. Capacitors
parallel plate, cylindrical, spherical. You must
be able to calculate the capacitance of
capacitors having these geometries, and you must
be able to use the equation CQ/V to calculate
parameters of capacitors. Circuits containing
capacitors in series and parallel. You must be
understand the differences between, and be able
to calculate the equivalent capacitance of,
capacitors connected in series and parallel.
79
Capacitors in Circuits
Recall this is the symbol representing a
capacitor in an electric circuit.
And this is the symbol for a battery
-

or this
or this.
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Circuits Containing Capacitors in Parallel
Vab
Capacitors connected in parallel
C1
C2
a
b
C3

-
V
The potential difference (voltage drop) from a to
b must equal V.
Vab V voltage drop across each individual
capacitor.
Note how I have introduced the idea that when
circuit components are connected in parallel,
then the voltage drops across the components are
all the same. You may use this fact in homework
solutions.
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C1
Q C V
Q1
C2
-

a
? Q1 C1 V
Q2
Q2 C2 V
C3
Q3
Q3 C3 V

-
V
Ceq
Now imagine replacing the parallel combination of
capacitors by a single equivalent capacitor.
a
Q

By equivalent, we mean stores the same total
charge if the voltage is the same.
-
V
Q1 Q2 Q3 Ceq V Q
Important!
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Summarizing the equations on the last slide
C1
Q1 C1 V Q2 C2 V Q3 C3 V
C2
a
b
Q1 Q2 Q3 Ceq V
C3

-
Using Q1 C1V, etc., gives
V
C1V C2V C3V Ceq V
C1 C2 C3 Ceq (after dividing both
sides by V)
Generalizing
Ceq ?Ci (capacitors in parallel)
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Circuits Containing Capacitors in Series
Capacitors connected in series
C1
C2
C3

-
Q
-Q
V
An amount of charge Q flows from the battery to
the left plate of C1. (Of course, the charge
doesnt all flow at once).
An amount of charge -Q flows from the battery to
the right plate of C3. Note that Q and Q must
be the same in magnitude but of opposite sign.
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The charges Q and Q attract equal and opposite
charges to the other plates of their respective
capacitors
C1
C2
C3
A
B
Q
-Q
-Q
Q
-Q
Q

-
V
These equal and opposite charges came from the
originally neutral circuit regions A and B.
Because region A must be neutral, there must be a
charge Q on the left plate of C2.
Because region B must be neutral, there must be a
charge --Q on the right plate of C2.
85
Vab
C1
C2
C3
A
B
a
b
Q
-Q
-Q
Q
-Q
Q
V3
V2
V1

-
V
The charges on C1, C2, and C3 are the same, and
are
Q C1 V1 Q C2 V2 Q C3 V3
But we dont know V1, V2, and V3 yet.
We do know that Vab V and also Vab V1 V2
V3.
Note how I have introduced the idea that when
circuit components are connected in series, then
the voltage drop across all the components is the
sum of the voltage drops across the individual
components. This is actually a consequence of the
conservation of energy.
86
Lets replace the three capacitors by a single
equivalent capacitor.
Ceq
Q
-Q
V

-
V
By equivalent we mean V is the same as the
total voltage drop across the three capacitors,
and the amount of charge Q that flowed out of the
battery is the same as when there were three
capacitors.
Q Ceq V
87
Collecting equations
Q C1 V1 Q C2 V2 Q C3 V3
Important!
Vab V V1 V2 V3.
Q Ceq V
Substituting for V1, V2, and V3
Substituting for V
Dividing both sides by Q
88
Generalizing
(capacitors in series)
89
Example determine the capacitance of a single
capacitor that will have the same effect as the
combination shown. Use C1 C2 C3 C.
C2
C1
C3
I dont see a series combination of capacitors,
but I do see a parallel combination.
C23 C2 C3 C C 2C
90
Now I see a series combination.
C1 C
C23 2C
91
Example for the capacitor circuit shown, C1
3?F, C2 6?F, C3 2?F, and C4 4?F. (a) Find
the equivalent capacitance. (b) if ?V12 V, find
the potential difference across C4.
C2
C1
Ill work this at the blackboard.
C4



C3

?V
92
Overview
Electric charge and electric force Coulombs
Law Electric field calculating electric
field motion of a charged particle in an
electric field Gauss Law electric
flux calculating electric field using Gaussian
surfaces properties of conductors
93
Overview
Electric potential and electric potential
energy calculating potentials and potential
energy calculating fields from
potentials equipotentials potentials and fields
near conductors Capacitors capacitance of
parallel plates, concentric cylinders,
(concentric spheres not for this
exam) equivalent capacitance of capacitor
network Dont forget the concepts from Physics
103 that were frequently used!
94
Three charges Q, Q, and Q, are located at the
corners of an equilateral triangle with sides of
length a. What is the force on the charge located
at point P (see diagram)?
y
F1
P
?
Q
?
F2
a
Q
-Q
x
95
y
F1
P
?
Q
?
F2
a
Q
-Q
x
96
What is the electric field at P due to the two
charges at the base of the triangle?
You can repeat the above calculation, replacing F
by E.
y
F1
P
?
Or
Q
?
F2
a
Q
-Q
x
97
An insulating spherical shell has an inner radius
a and outer radius b. The shell has a total
charge Q and a uniform charge density. Find the
magnitude of the electric field for rlta.
98
An insulating spherical shell has an inner radius
a and outer radius b. The shell has a total
charge Q and a uniform charge density. Find the
magnitude of the electric field for altrltb.
99
An electron has a speed v. Calculate the
magnitude and direction of an electric field that
will stop this electron in a distance D.
Know where this comes from!
Do you understand that these Es have different
meanings?
Use magnitudes if you have determined direction
of E by other means.
100
Two equal positive charges Q are located at the
base of an equilateral triangle with sides of
length a. What is the potential at point P (see
diagram)?
P
a
Q
Q
101
Three equal positive charges Q are located at the
corners of an equilateral triangle with sides of
length a. What is the potential energy of the
charge located at point P (see diagram)?
P
Q
a
Q
Q
102
For the capacitor system shown, C16.0 ?F, C22.0
?F, and C310.0 ?F. (a) Find the equivalent
capacitance.
103
For the capacitor system shown, C16.0 ?F, C22.0
?F, and C310.0 ?F. The charge on capacitor C3 is
found to be 30.0 ?C. Find V0.
104
Todays agendum Energy Storage in
Capacitors. You must be able to calculate the
energy stored in a capacitor, and apply the
energy storage equations to situations where
capacitor configurations are altered. Dielectrics
. You must understand why dielectrics are used,
and be able include dielectric constants in
capacitor calculations.
105
Energy Storage in Capacitors
Lets calculate how much work it takes to charge
a capacitor.
The work required for an external force to move a
charge dq through a potential difference ?V is dW
dq ?V.
From QC?V (? ?V q/C)
?V

-
dq
q is the amount of charge on the capacitor during
the time the charge dq is being moved.

We start with zero charge on the capacitor, and
end up with Q, so
q
-q
106
The work required to charge the capacitor is the
amount of energy you get back when you discharge
the capacitor (because the electric force is
conservative).
Thus, the work required to charge the capacitor
is equal to the potential energy stored in the
capacitor.
Because C, Q, and V are related through QCV,
there are three equivalent ways to write the
potential energy.
107
All three equations are valid use the one most
convenient for the problem at hand.
It is no accident that we use the symbol U for
the energy stored in a capacitor. It is just
another version of electrical potential energy.
You can use it in your energy conservation
equations just like any other form of potential
energy!
108
Example a camera flash unit stores energy in a
150 ?F capacitor at 200 V. How much electric
energy can be stored?
If you keep everything in SI (mks) units, the
result is automatically in SI units.
109
Example compare the amount of energy stored in a
capacitor with the amount of energy stored in a
battery.
A 12 V car battery rated at 100 ampere-hours
stores 3.6x105 C of charge and can deliver at
least 4.3x106 joules of energy (well learn how
to calculate that later in the course).
A 100 ?F capacitor that stores 3.6x105 C at 12 V
stores an amount of energy UCV2/27.2x10-3
joules.
If you want your capacitor to store lots of
energy, store it at a high voltage.
If a battery stores so much more energy, why use
capacitors?
106 joules of energy are stored at high voltage
in capacitor banks, and released during a period
of a few milliseconds. The enormous current
produces incredibly high magnetic fields.
110
Energy Stored in Electric Fields
Energy is stored in the capacitor
?V

-
E
d
Q
-Q
area A
The volume of the capacitor is VolumeAd
111
Energy stored per unit volume (u)
?V

-
The energy is stored in the electric field!
E
Weve gone from the concrete (electric charges
experience forces)
d
Q
-Q
to the abstract (electric charges create
electric fields)
area A
to an application of the abstraction (electric
field contains energy).
112
The energy in electromagnetic phenomena is the
same as mechanical energy. The only question is,
Where does it reside? In the old theories, it
resides in electrified bodies. In our theory, it
resides in the electromagnetic field, in the
space surrounding the electrified bodies.James
Maxwell
?V

-
E
This is not a new kind of energy. Its the
electric potential energy resulting from the
coulomb force between charged particles.
f
Q
-Q
Or you can think of it as the electric energy due
to the field created by the charges. Same thing.
area A
113
Todays agendum Energy Storage in
Capacitors. You must be able to calculate the
energy stored in a capacitor, and apply the
energy storage equations to situations where
capacitor configurations are altered. Dielectrics
. You must understand why dielectrics are used,
and be able include dielectric constants in
capacitor calculations.
114
Dielectrics
If an insulating sheet (dielectric) is placed
between the plates of a capacitor, the
capacitance increases by a factor ?, which
depends on the material in the sheet. ? is the
dielectric constant of the material.
dielectric
In general, C ??0A / d. ? is 1 for a vacuum,
and ? 1 for air. (You can also define ? ??0
and write C ? A / d).
115
The dielectric is the thin insulating sheet in
between the plates of a capacitor.
dielectric
Any reasons to use a dielectric in a capacitor?
?Makes your life as a physics student more
complicated.
?Lets you apply higher voltages (so more charge).
?Lets you place the plates closer together (make
d smaller).
?Increases the value of C because ?gt1.
116
Example a parallel plate capacitor has an area
of 10 cm2 and plate separation 5 mm. 300 V is
applied between its plates. If neoprene is
inserted between its plates, how much charge does
the capacitor hold.
A10 cm2
?6.7
?V300 V d5 mm
117
Example how much charge would the capacitor on
the previous slide hold if the dielectric were
air?
A10 cm2
The calculation is the same, except replace 6.7
by 1.
Or just divide the charge on the previous page by
6.7 to get.
?1
?V300 V d5 mm
118
V0
Conceptual Example
A capacitor connected as shown acquires a charge
Q.
V
While the capacitor is still connected to the
battery, a dielectric material is inserted.
V
Will Q increase, decrease, or stay the same?
Why?
119
Example find the energy stored in the capacitor
in slide 13.
A10 cm2
?6.7
?V300 V d5 mm
120
Example the battery is now disconnected. What
are the charge, capacitance, and energy stored in
the capacitor?
A10 cm2
The charge and capacitance are unchanged, so the
voltage drop and energy stored are unchanged.
?6.7
?V300 V d5 mm
121
Example the dielectric is removed without
changing the plate separation. What are the
capacitance, charge, potential difference, and
energy stored in the capacitor?
A10 cm2
?6.7
?V? d5 mm
?V300 V d5 mm
122
Example the dielectric is removed without
changing the plate separation. What are the
capacitance, charge, potential difference, and
energy stored in the capacitor?
A10 cm2
The charge remains unchanged, because there is
nowhere for it to go.
?V? d5 mm
123
Example the dielectric is removed without
changing the plate separation. What are the
capacitance, charge, potential difference, and
energy stored in the capacitor?
A10 cm2
Knowing C and Q we can calculate the new
potential difference.
?V? d5 mm
124
Example the dielectric is removed without
changing the plate separation. What are the
capacitance, charge, potential difference, and
energy stored in the capacitor?
A10 cm2
?V2020 V d5 mm
125
Huh?? The energy stored increases by a factor of
???
Sure. It took work to remove the dielectric. The
stored energy increased by the amount of work
done.