Capacitors

Capacitance:

When a capacitor is charged, its plates have charges of equal magnitudes but opposite signs: q+ and q-. However, we refer to the charge of a capacitor as being q, the absolute value of these charges on the plates.

The charge q and the potential difference V for a capacitor are proportional to each other:

The proportionality constant C is called the capacitance of the capacitor. Its value depends only on the geometry of the plates and not on their charge or potential difference.

The SI unit is called the farad (F): 1 farad (1 F)= 1 coulomb per volt =1 C/V.

Calculating the Capacitance - Specifics of Geometry!!

And the charge enclosed by our Gaussian Surface is

The potential difference between the plates is

Calculating the Capacitance - Cylindrical Geometry

What Gaussian surface here??

Calculating the Capacitance - Spherical Geometry:

This example is stepped through in the book – how is it different from the cylindrical example we did on the previous slide?

For a single, isolated sphere? What would be your next step?

The plates of an isolated parallel plate capacitor with a capacitance C carry a charge Q. What is the capacitance of the capacitor if the charge is increased to 4Q?

a) C/2

b) C/4

c) 4C

d) 2C

e) C

Since the battery stays connected, the

voltage must remain constant ! Since

, when the spacing d

is doubled the capacitance C is halved.

And since Q = C V, that means the

charge must decrease.

+Q –Q

dAC 0

A parallel-plate capacitor

initially has a voltage of 400 V

and stays connected to the

battery. If the plate spacing is

now doubled, what happens?

1) the voltage decreases

2) the voltage increases

3) the charge decreases

4) the charge increases

5) both voltage and charge change

ConcepTest 20.9b Varying Capacitance II

Follow-up: How do you increase the charge?

Once the battery is disconnected, Q has to

remain constant, since no charge can flow

either to or from the battery. Since

, when the spacing d is doubled

the capacitance C is halved. And since Q =

C V, that means the voltage must double.

A parallel-plate capacitor initially has

a potential difference of 400 V and is

then disconnected from the charging

battery. If the plate spacing is now

doubled (without changing Q), what

is the new value of the voltage?

1) 100 V

2) 200 V

3) 400 V

4) 800 V

5) 1600 V

+Q –Q

dAC 0

ConcepTest 20.9c Varying Capacitance III

Capacitor with a Dielectric:

A dielectric, is an insulating material such as mineral oil or plastic, and is characterized by a numerical factor k, called the dielectric constant of the material.

Some dielectrics, such as strontium titanate, can increase the capacitance by more than two orders of magnitude.

Two parallel conducting plates are connected to a battery for a long time and become fully-charged. How does the potential difference across the plates change, if at all, when a conducting slab is inserted in between the plates without touching either plate?

a) The potential difference will increase.

b) The potential difference will decrease.

c) The potential difference will remain unchanged.

Two parallel conducting plates are connected to a battery for a long time and become fully-charged. How does the charge on the plates change, if at all, when a conducting slab is inserted in between the plates without touching either plate?

a) The charge will increase.

b) The charge will decrease.

c) The charge will remain unchanged.

Energy Stored in an Electric Field:

The work required to bring the total capacitor charge up to a final value q is

This work is stored as potential energy U in the capacitor, so that, OR

U=1/2 qV (potential energy)

A parallel plate capacitor is connected to a battery that maintains a constant potential difference across the plates. Initially, the space between the plates contains only air. Then, a Teflon ( = 2.1) sheet is inserted between, but not touching, the plates. How does the stored energy of the capacitor change as a result of inserting the Teflon sheet?

a) The energy will decrease.

b) The energy will not be affected.

c) The energy will increase.

d) The energy will be zero joules.

Capacitors in circuits – Parallel and Series

V is applied across each capacitor in parallel. The total charge q stored on the capacitors is the sum of the charges stored on all the capacitors.Capacitors connected in parallel can be replaced with an equivalent capacitor that has the same total charge q and the same potential difference V as the actual capacitors.

When a potential difference V is applied across several capacitors connected in series, the capacitors have identical charge q. The sum of the potential differences across all the capacitors is equal to the applied potential difference V.

Capacitors that are connected in series can be replaced with an equivalent capacitor that has the same charge q and the same total potential difference V as the actual series capacitors.

Capacitors in circuits – Parallel and Series

+

-12 V

1 nF

C1

2 nF

C2

3 nF

C3

1 nF 2 nF 3 nF

12 V

Series:

Parallel:

nFCCC

Ceq 11

6

6

11

3

1

2

1

1

1111111

321

nFCCCCeq 6321321

Capacitor B has one-half the capacitance of capacitor A. How does the charge on capacitor A compare to that on B when the two are connected in series to a battery for a long time?

a) The charge on capacitor A is one-fourth the charge on capacitor B.

b) The charge on capacitor A is one-half the charge on capacitor B.

c) The charge on capacitor A is the same as the charge on capacitor B.

d) The charge on capacitor A is twice the charge on capacitor B.

e) The charge on capacitor A is four times the charge on capacitor B.

Capacitor B has one-half the capacitance of capacitor A. How does the charge on capacitor A compare to that on B when the two are connected in parallel with a battery for a long time?

a) The charge on capacitor A is one-fourth the charge on capacitor B.

b) The charge on capacitor A is one-half the charge on capacitor B.

c) The charge on capacitor A is the same as the charge on capacitor B.

d) The charge on capacitor A is twice the charge on capacitor B.

e) The charge on capacitor A is four times the charge on capacitor B.

Capacitor B has one-half the capacitance of capacitor A. How does the potential difference on capacitor A compare to that on B when the two are connected in series with a battery for a long time?

a) The potential difference on capacitor A is one-fourth the charge on capacitor B.

b) The potential difference on capacitor A is one-half the charge on capacitor B.

c) The potential difference on capacitor A is the same as the charge on capacitor B.

d) The potential difference on capacitor A is twice the charge on capacitor B.

e) The potential difference on capacitor A is four times the charge on capacitor B.

Roughly how much charge can this ultra capacitor store?

A. 10,000,000 C B. 400 nCC. 8000 CD. Not enough info