Capacitors and Dielectrics

Capacitance

A capacitor is basically two parallel conducting plates with air or insulating material in between.

V0 V1

E

L

A capacitor doesn’t have to look like metal plates.

Capacitor for use in high-performance audio systems.

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.

The battery in this circuit has some voltage V. We haven’t discussed what that means yet.

The symbol representing a capacitor in an electric circuit looks like parallel plates. Here’s the symbol for a battery, or an external potential. +-

-

-V

+-

If the external potential is disconnected, charges remain on the plates, so capacitors are good for storing charge (and energy).

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 wouldn’t bet my life on it.

Graphic from http://www.feebleminds-gifs.com/.

+

+

-

-V

conducting wires

On-line “toy” here.

assortment of capacitors

The magnitude of charge acquired by each plate of a capacitor is Q=CV where C is the capacitance of the capacitor.

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!)

QC

V C is always

positive.

+Q

+

-Q

-V

CHere’s this V again. It is the potential difference provided by the “external potential.” For example, the voltage of a battery. V is really a V.

V is really V.

Today’s agenda:

Capacitance.You must be able to apply the equation C=Q/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 C=Q/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.

Parallel Plate Capacitance

V0 V1

E

d

We previously calculated the electric field between two parallel charged plates:

0 0

QE .

A

This is valid when the separation is small compared with the plate dimensions. We also showed that E and V are related:

+Q-Q

A

d d

0 0V E d E dx Ed .

0

0

AQ Q QC

V Ed dQd

A

This lets us calculate C for a parallel plate capacitor.

Reminders:Q

CV

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.

V0 V1

E

d

+Q-Q

A

0ACd

Parallel plate capacitance depends “only” on geometry.

This expression is approximate, and must be modified if the plates are small, or separated by a medium other than a vacuum.

0ACd

Greek letter Kappa. For today’s lecture (and for exam 1), use Kappa=1.Do not use =9x109!

We can also calculate the capacitance of a cylindrical capacitor (made of coaxial cylinders).

L

Coaxial Cylinder Capacitance

The next slide shows a cross-section view of the cylinders.

Q

-Q

br

a

E

d

Gaussian surface

Q λ L λ LC = = =

bΔV ΔV2k λ ln

a

02πε LLC = =

b b2k ln ln

a a

Lowercase c is capacitance per unit length: 02πεCc = =

bLln

a

2kλE =

r

This derivation is sometimes needed for homework problems! (Hint: 24.10, 11, 12.)

Some necessary details are not shown on this slide!

b b

b a r

a a

ΔV = V -V = - E d = - E dr

b

a

dr bΔV = - 2k λ = - 2k λ ln

r a

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:

0

QV .

4 r

The potential at the surface of a charged sphere of radius R is

0

QV

4 R

so the capacitance at the surface of an isolated sphere is

0

QC 4 R.

V

Capacitance of Concentric Spheres

Let’s calculate the capacitance of a concentric spherical capacitor of charge Q. I’ll skip this calculation if there is no related homework assigned.

In between the spheres

20

QE

4 r

b

2a0 0

Q dr Q 1 1V

4 r 4 a b

04QC

1 1Va b

You need to do this derivation if you have a problem on spherical capacitors! (not this semester)

+Q

-Q

b

a

If there is related homework, details will be provided in lecture!

04QC

1 1Va b

Let aR and b to get the capacitance of an isolated sphere.

+Q

-Q

b

a

alternative calculation of capacitance of isolated sphere

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 = 0.001area = 0.2 x 0.03

If you keep everything in SI (mks) units, the result is “automatically” in SI units.

0ACd

128.85 10 0.2 0.03C

0.001

12C 53 10 F

C 53 pF

Example: what is the charge on each plate if the capacitor is connected to a 12 volt* battery?

0 V

+12 V

V= 12V

Q CV

12Q 53 10 12

10Q 6.4 10 C

*Remember, it’s the potential difference that matters.

If you keep everything in SI (mks) units, the result is “automatically” in SI units.

Example: what is the electric field between the plates?

0 V

+12 V

V= 12V

d = 0.001

E

VE

d

12VE

0.001 m

VE 12000 ,"up."

m

If you keep everything in SI (mks) units, the result is “automatically” in SI units.

“Quiz” time (maybe for points, maybe just for practice!)

Demo: Professor Tries to AvoidSpot-Welding His Fingers

to the Terminals of a CapacitorWhile Demonstrating Energy

Storage

Today’s agenda:

Capacitance.You must be able to apply the equation C=Q/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 C=Q/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.

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.

Capacitors connected in parallel:C1

C2

C3

+ -

V

The potential difference (voltage drop) from a to b must equal V.

a b

Vab = V = voltage drop across each individual capacitor.

Vab

Circuits Containing Capacitors in Parallel

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.

C2

C3

+ -

C1

C2

C3

+ -

V

a

Q = C V

Q1 = C1 V

& Q2 = C2 V

& Q3 = C3 V

Now imagine replacing the parallel combination of capacitors by a single equivalent capacitor.

By “equivalent,” we mean “stores the same total charge if the voltage is the same.”

Ceq

+ -

V

a

Qtotal = Ceq V = Q1 + Q2 + Q3

Q3

Q2

Q1

+ -

Q

Important!

Q1 = C1 V Q2 = C2 V Q3 = C3 V

Q1 + Q2 + Q3 = Ceq V

Summarizing the equations on the last slide:

Using Q1 = C1V, etc., gives

C1V + C2V + C3V = Ceq V

C1 + C2 + C3 = Ceq (after dividing both sides by V)

Generalizing:

Ceq = Ci (capacitors in parallel)

C1

C2

C3

+ -

V

a b

Capacitors connected in series:

C1 C2

+ -

V

C3

An amount of charge +Q flows from the battery to the left plate of C1. (Of course, the charge doesn’t all flow at once).

+Q -Q

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.

Circuits Containing Capacitors in Series

C1 C2

+ -

V

C3

+QA

-QB

The charges +Q and –Q attract equal and opposite charges to the other plates of their respective capacitors:

-Q +Q

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.

+Q -Q

C1 C2

+ -

V

C3

+QA

-QB

-Q +Q+Q -Q

Q = C1 V1 Q = C2 V2 Q = C3 V3

The charges on C1, C2, and C3 are the same, and are

But we don’t know V1, V2, and V3 yet.

a b

We do know that Vab = V and also Vab = V1 + V2 + V3.

V3V2V1

Vab

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. You may use this fact in homework solutions.

Ceq

+ -

V

+Q -QV

Let’s replace the three capacitors by a single equivalent capacitor.

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

Collecting equations:

Q = C1 V1 Q = C2 V2 Q = C3 V3

Vab = V = V1 + V2 + V3.

Q = Ceq V

Substituting for V1, V2, and V3:1 2 3

Q Q QV = + +

C C C

Substituting for V:eq 1 2 3

Q Q Q Q = + +

C C C C

Dividing both sides by Q:eq 1 2 3

1 1 1 1 = + +

C C C C

Important!

Generalizing:

OSE: (capacitors in series)ieq i

1 1 =

C C

Summary (know for exam!):

Series

eq ii

C C

same Q, V’s add

Parallel

same V, Q’s add

ieq i

1 1

C C

C1 C2 C3

C1

C2

C3

C3

C2

C1

I don’t see a series combination of capacitors, but I do see a parallel combination.

C23 = C2 + C3 = C + C = 2C

Example: determine the capacitance of a single capacitor that will have the same effect as the combination shown. Use C1 = C2 = C3 = C.

C1= CC23 = 2C

Now I see a series combination.

eq 1 23

1 1 1 = +

C C C

eq

1 1 1 2 1 3 = + = + =

C C 2C 2C 2C 2C

eq

2C = C

3

Example: for the capacitor circuit shown, C1 = 3F, C2 = 6F, C3 = 2F, and C4 =4F. (a) Find the equivalent capacitance. (b) if V=12 V, find the potential difference across C4.

I’ll work this at the blackboard.

C3

C2C1 C4

V

Homework Hint: each capacitor has associated with it a Q, C, and V. If you don’t know what to do next, near each capacitor, write down Q= , C= , and V= . Next to the = sign record the known value or a “?” if you don’t know the value. As soon as you know any two of Q, C, and V, you can determine the third. This technique often provides visual clues about what to do next.

You really need to know this:

Capacitors in series…all have the same chargeadd the voltages to get the total voltage

Capacitors in parallel…all have the same voltageadd the charges to get the total charge

(and it would be nice if you could explain why)

Homework Hint!

What does our text mean by Vab?

C3

C2C1 C4

V

a bOur text’s convention is Vab = Va – Vb. This is explained on page 762. This is in contrast to Physics 23 notation, where Vab = Vb – Va.

In the figure on this slide, if Vab = 100 V then point a is at a potential 100 volts higher than point b, and Vab = -100 V; there is a 100 volt drop on going from a to b.

When our text uses the notation Wab, it means the same thing as in Physics 23. The two sentences in bold type in the first two paragraphs on page 762 are correct becauseUab = q Bab = -Wconservative,ab .