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Announcements
 Physics 2135 spreadsheets for all sections, with Exam 1 scores,
will be posted today on the Physics 2135 web site. You need your
PIN to find your grade.
 Preliminary exam average is about 72.1% (13 sections out of 13
reporting). Reasonable! Scores ranged from a low of 39 to a high
of 200 (6 students). I will fill in the ??’s during the “live” lecture and in its “.ppt” file.
Physics 2135 Exam 1 will be returned in recitation Thursday.
When you get the exam back, please check that points were
added correctly. Review the course handbook and be sure to
follow proper procedures before requesting a regrade. Get your
regrade requests in on time! (They are due by next Thursday’s
recitation.)
On a separate sheet of paper, briefly explain the reason for your regrade request. This should be based on the work actually
shown on paper, not what was in your head. Attach to the exam and turn it in by the end of your next Thursday’s recitation.
Today’s agenda:
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.
Energy Storage in Capacitors
Let’s 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 Q=CV ( V = q/C):
+
We start with zero charge on the capacitor,
and end up with Q, so
Q
Q
0
0
W   dW  
2 Q
q
q
Q2
dq 

.
C
2C 0 2C
-
dq
+
q is the amount of charge on
the capacitor at the time the
charge dq is being moved.
q
dW  V dq  dq
C
V
+q
-q
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.
Q2
U
.
2C
Because C, Q, and V are related through Q=CV, there are three
equivalent ways to write the potential energy.
Q2 CV 2 QV
U


.
2C
2
2
Q2 CV 2 QV
U


.
2C
2
2
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!
There are now four parameters you can determine for a
capacitor: C, Q, V, and U. If you know any two of them, you
can calculate the other two.
Example: a camera flash unit stores energy in a 150 F
capacitor at 200 V. How much electric energy can be stored?
CV 2
U
2
U
6
2
150

10
200

 
2
U3 J
If you keep everything in SI (mks) units, the result is “automatically” in SI units.
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
(we’ll learn how to calculate that later in the course).
A 100 F capacitor that operates at 12 V can deliver an
amount of energy U=CV2/2=7.2x10-3 joules.
If you want your capacitor
Truth in advertising: I set up the comparison in a way that makes the capacitor look bad.
to store lots of energy, store
it at a high voltage.
If a battery stores so much more energy, why use capacitors?
Application #1: short pulse magnets at the National Magnet
Laboratory, plus a little movie of a short pulse magnet at work.
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.
Application #2: quarter shrinker.
Application #3: can crusher.
Some links: shrinking, shrinking (can you spot
the physics mistake), can crusher,. Don’t do this
at home. Or this.
Energy Stored in Electric Fields
Energy is stored in the
capacitor:
A
C 0
and V  Ed
d
1
2
U  C  V 
2
V
+
E
1  0 A 
2
U 
  Ed 
2 d 
1
U    0 Ad  E 2
2
The “volume of the capacitor” is Volume=Ad
-
+Q
d
-Q
area A
Energy stored per unit volume (u):
V
+
1
 0 Ad  E 2 1 2
u 2
 0 E
Ad
2
The energy is “stored” in the electric
field!
-
E
+Q
d
-Q
area A
“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
1
u  0 E 2
2
This is on page 2 of your OSE
sheet. Do not use until later!
(Unless you really know what
you are doing.)
This is not a new “kind” of energy. It’s the
electric potential energy resulting from the
coulomb force between charged particles.
Or you can think of it as the electric energy
due to the field created by the charges.
Same thing.
V
+
-
E
+Q
f
-Q
area A
Today’s agenda:
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.
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.
In general, C = 0A / d.  is 1
for a vacuum, and  1 for air.
(You can also define  = 0
and write C =  A / d).
 A
C=
.
d
dielectric
The dielectric is the thin insulating sheet
in between the plates of a capacitor.
A lot of interesting physics happens in the dielectric, but we’ll skip that section.
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 >1.
Q = CV
  A
C=
d
Gives you a bigger kick when
Gives you a bigger kick when
Gives
you capacitor
a bigger kick when
you discharge
the
you discharge
the
you discharge capacitor
the capacitor
throughthrough
yourthrough
tongue!
your
tongue!
your tongue!
Homework hint: what if the
dielectric fills only half the
space between the plates?
dielectric
This is equivalent to two capacitors in parallel. Each of the
two has half the plate area. The two share the total
charge, and have the same potential difference
Q1 C 1
Q
C
Q2 C 2
Some things for you to ponder…
If you charge a capacitor and then remove the battery and
manipulate the capacitor, Q must stay the same but C, V,
and U may change. (What about E?)
If you charge a capacitor, keep the battery connected, and
manipulate the capacitor, V must stay the same but C, Q,
and U may change. (What about E?)
If exactly two capacitors are connected such that they have
the same voltage across them, they are probably in parallel
(but check the circuit diagram).
If you charge two capacitors, then remove the battery and
reconnect the capacitors with oppositely-charged plates
connected together…
draw a circuit diagram before and after, and use
conservation of charge to determine the total charge on
each plate before and after.
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.
A=10 cm2
  A
C=
d
C=
 6.7   8.85×10-12 10×10-4 
=6.7
5×10-3
C =1.19 10-11 F
V=300 V
d=5 mm
Q = CV




Q = 1.19 10-11  300   3.56 10-9 C = 3.56 nC
300 V
Example: how much charge would the capacitor on the
previous slide hold if the dielectric were air?
The calculation is the
same, except replace 6.7
by 1.
A=10 cm2
Or just divide the charge on the
previous page by 6.7 to get…
Q = 0.53 nC
=1
V=300 V
d=5 mm
300 V
Visit howstuffworks to read about capacitors and learn their
advantages/disadvantages compared to batteries!
Conceptual Example
V=0
A capacitor connected as shown
acquires a charge Q.
While the capacitor is still connected
to the battery, a dielectric material is
inserted.
Will Q increase, decrease, or stay the same?
Why?
V
V
Back to our Example: find the energy stored in the capacitor.
C =1.19 10-11 F
A=10 cm2
1
2
U = C  V 
2


1
2
U = 1.19 10-11  300 
2
-7
U = 5.36 10 J
=6.7
V=300 V
d=5 mm
300 V
Example: the battery is now disconnected. What are the
charge, capacitance, and energy stored in the capacitor?
The charge and capacitance are
unchanged, so the voltage drop
and energy stored are unchanged.
Q = 3.56 nC
A=10 cm2
=6.7
C =1.19 10-11 F
U = 5.36 10-7 J
V=300 V
d=5 mm
300 V
Example: the dielectric is removed without changing the plate
separation. What are the capacitance, charge, potential
difference, and energy stored in the capacitor?
A=10 cm2
 A
C=
d
8.85×10 10×10 

C=
-12
-4
=6.7
5×10-3
C =1.78 10-12 F
V=?
V=300 V
d=5 mm
Example: the dielectric is removed without changing the plate
separation. What are the capacitance, charge, potential
difference, and energy stored in the capacitor?
The charge remains unchanged,
because there is nowhere for it
to go.
A=10 cm2
Q = 3.56 nC
V=?
d=5 mm
Example: the dielectric is removed without changing the plate
separation. What are the capacitance, charge, potential
difference, and energy stored in the capacitor?
Knowing C and Q we can
calculate the new potential
difference.


3.56 10-9
Q
V = =
C
1.78 10-12
V = 2020 V
A=10 cm2


V=?
d=5 mm
Example: the dielectric is removed without changing the plate
separation. What are the capacitance, charge, potential
difference, and energy stored in the capacitor?
A=10 cm2
1
2
U = C  V 
2


1
2
-12
U = 1.78 10  2020 
2
U = 3.63 10-6 J
V=2020 V
d=5 mm
Ubefore = 5.36 10-7 J
Uafter = 3.63 10-6 J
Uafter
= 6.7
Ubefore
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.
U = Wexternal
A “toy” to play with…
http://phet.colorado.edu/en/simulation/capacitor-lab
(You might even learn something.)