Videos
SP212
Ch. 25 – Capacitance
Maj Jeremy Best USMC
Physics Department, U.S. Naval Academy
February 23, 2016
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
1 / 22
Capacitors
What is a Capacitor: https://youtu.be/pnBRFXgaTMo
Capacitors:
http://www.youtube.com/watch?v=PAPGTuvHSRo
How Caps work:
http://www.youtube.com/watch?v=t9Qwx75eg8w
Capacitor Charging http://youtu.be/IvFVu7Jxa2I
Capacitor Size demo
http://demonstrations.wolfram.com/
ParallelPlateCapacitorsAndRCCircuits/
http://demonstrations.wolfram.com/
ParallelPlateCapacitors/
PHET Capacitor Lab http://phet.colorado.edu/en/
simulation/capacitor-lab
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
2 / 22
Capacitors
We’ve seen capacitors before. In essence, a capacitor
consists of two isolated conductors of area A, separated
by a distance d. These conductors are called plates,
regardless of their shape and we say the capacitor is
charged when one plate has charge +q and the other −q.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
3 / 22
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
4 / 22
Capacitors in circuits
The equal but opposite charges on a conductor create a
potential difference V between the two plates. This
related to the charge q :
As you know, capacitors are mainly used in circuits:
Capacitance
−
+
C
B
V
q = CV
The SI unit of capacitance is the farad (F)
(1 F = 1 C/V).
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
5 / 22
Calculating the Capacitance
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
6 / 22
The Parallel Plate Capacitor
We need to be able to calculate the capacitance of
capacitors with various geometries. We will follow three
basic steps:
Use Gauss’s Law to relate the charge q on the
positive plate and the electric field ~E
Integrate along the field (from negative to positive)
to find the potential
Plug both of these into q = CV
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
7 / 22
We draw our Gaussian surface to just enclose the charge
on the positive plate: qenc = q
Our clever choice of Gaussian surface also makes the flux
easy:
I
~
Φ = ~E · d A
= EA
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
8 / 22
Other Capacitors
q = 0 EA
We need to find V by integrating the field, which we do
from the negative to positive plate:
Z +
Z d
V =
E ds = E
ds = Ed
−
The Geometry of other capacitors is indicated in the
book.
0
Plugging q and V into q = CV , we solve for C for a
parallel plate capacitor:
Parallel Plate Capacitance
Cylindrical Capacitor
C=
0 A
d
C = 2π0
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
9 / 22
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
L
ln(b/a)
February 23, 2016
10 / 22
Capacitors in Circuits
When electrical components (such as capacitors) are
connected into circuits, we often want to simplify the
circuit by finding an equivalent capacitance for all (or
at least most) of them.
Spherical Capacitor
C = 4π0
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
ab
b−a
February 23, 2016
11 / 22
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
12 / 22
Series vs. Parallel
Finding the Equivalent Capacitance
Capacitors in parallel add plain (like you’re used to):
Ceq =
n
X
Ci
i=1
Capacitors in series add “strange” (that’s a technical
term):
Capacitors in Parallel
n
Capacitors in Series
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
13 / 22
Book Method: Equiv Capacitance
X 1
1
=
Ceq
Ci
i=1
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
14 / 22
Energy Storage in Capacitors
Everything in physics comes down to conservation of
energy. (that’s actually not much of an overstatement).
Capacitors store energy in the form of their electric field.
The energy stored is:
q2
U=
= (1/2)CV 2
2C
We can turn this into an energy density by dividing by
AD, the volume between the plates:
u = (1/2)0 E 2
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
15 / 22
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
16 / 22
Dielectrics
E field in a Dielectric
Since capacitors are charged plates if they touch
then they are just like wires, and no longer a
capacitor.
What goes in between these charged plates can
dramatically change the E field and capacitance.
Typically we deal with air , which happens to have
a dielectric constant κ of 1.
Real Capacitors always have a dielectric, and
usually the higher the κ the better. Why?
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
17 / 22
Dielectrics
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
18 / 22
Book Method: Gauss Law Dielectrics
As we saw earlier when we calculated the capacitance of
various geometries, the capacitance can always be
written as C = 0 L, where L is something with the
dimension of length. Dielectrics can change this:
C=
κ0 A
d
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
19 / 22
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
20 / 22
E field in a Dielectric
Wiley Plus Homework
Chapter 25: Questions 1, 3, 7. Problems: 1, 3, 24, 31,
32, 42, 58, 65.
Gauss’ Law for Dielectrics
I
0
~ = q = 0 κEA
κ~E · d A
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
21 / 22
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
February 23, 2016
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