Capacitance Chapter 26. Electric Potential of Conductors The electric field E is zero within a conductor at equilibrium. E=0 V1V1 V2V2

Capacitance

Chapter 26

Electric Potential of Conductors

• The electric field E is zero within a conductor at equilibrium.

E=0 E=0

V1V2


• The electric field E is zero within a conductor at equilibrium.• Consequently the conductor has the same electric potential V everywhere inside.

E=0 E=0

V1V2



E=0 E=0

V1V2

€

rE = −

r ∇V



•The electric potential depends on how much excess charge has been added to the metal. The relationship between these quantities defines the capacitance.

E=0 E=0

V1V2

€

rE = −

r ∇V

Capacitance

A capacitor consists of two pieces of metal carrying equal and opposite charges Q and –Q.

Capacitance


An electric potential difference V develops between the two pieces of metal.

Capacitance



It turns out that Q and V are proportional. The constant of proportionality is called the capacitance C.

Q = CV [Units: Coulomb /Volt = Farad]

Capacitance




Q = CV [Units: Coulomb /Volt = Farad]

Capacitance




The capacitance C depends on the size and shape of the capacitor, and the material (if any) between them.

Parallel Plate Capacitor

-Q

+Q

dA

The charge on the top and bottom plates attract so that the it all ends up on the inner facing surfaces. These have surface charge density with =Q/A.


-Q

+Q

dA



-Q

+Q

dA

E


Gauss’s law


-Q

+Q

dA

E


Gauss’s law


-Q

+Q

dA

E


Gauss’s law gives EA=Q/0 so E=Q/0A.


-Q

+Q

dA

E



The potential difference between the two plates is ∆V=∫E·dl = Ed or ∆V = Qd/0A.


-Q

+Q

dA

E



The potential difference between the two plates is ∆V=∫E·dl = Ed or ∆V = Qd/0A.

Turning this about gives Q = (0A/d)∆V i.e. Q=C∆V with C=0A/d.


-Q

+Q

dA

E

Q=C∆V with C= 0A/d

The capacitance is like the capacity of the object to hold charge.

Bigger C means the object can hold more charge (at a given ∆V).

The capacitance is proportional to the area A (more capacity) and inversely proportional to the separation d (smaller ∆V=Ed for given Q).


-Q

+Q

dA

E

How Do You Get Q and -Q?

Q -Q

Move Q from one plate to the other.

How?


Q -Q


How? With a gadget that pushes charge – for instance, a battery.


+ -

Q -Q


How? With a gadget that pushes charge – for instance, a battery.

Hooked to metal plates, a 1.5 Volt battery moves charge until the potential difference between the plates is also 1.5 V.


+ -

Q -Q

A battery is like a charge escalator.

With just a wire between the plates, the charges making up Q would repel each other and run through the wire to neutralize -Q.

But the battery pushes the charges “uphill.”


+ -

Q -Q Q -Q

Thinking About Capacitors

+

-

12

3 46V

The black lines are metal wires attached to metal rods.

Suppose the battery has been hooked up for a long time so that it has finished pushing charge and the system has come to equilibrium.

What is ∆V12?

What is ∆V34?

What is ∆V23?


+

-

12

3 46V



What is ∆V12? Zero


What is ∆V23? 6V


+

-

12

3 46V



More About Capacitors

+-

What is ∆V12?

What is ∆V23?

What is ∆V34?

12

5

4

6V

Now hook the battery to a parallel plate capacitor.

3

3 mm


+-



What is ∆V34? 6V

12

5

4

6V


3

3 mm


+-



What is ∆V34? 6V

What is E at point 5?

12

5

4

6V


3

3 mm


+-



What is ∆V34? 6V

What is E at point 5? E = 6V/(3x10-3m) = 2000 V/m

12

5

4

6V


3

3 mm


+-

12

5

4

6V

3

3 mm

Now suppose you pull the plates apart slightly while keeping the battery attached. What happens to ∆V, Q, and E?


+-

∆V depends on the battery: it stays at 6V.

12

5

4

6V

3

3 mm



+-


Use Q=C∆V. Here C=0A/d decreases so Q decreases.

12

5

4

6V

3

3 mm



+-


Use Q=C∆V. Here C=0A/d decreases so Q decreases.

E= ∆V / (new length) gets smaller.

12

5

4

6V

3

3 mm


What Does a Capacitor Do?

• Stores electrical charge.• Stores electrical energy.

Capacitors are used when a sudden release of energy isneeded (such as in a photographic flash).

Capacitors are basic elements of electrical circuits bothmacroscopic (as discrete elements) and microscopic (asparts of integrated circuits).

What Does a Capacitor Do?

• Stores electrical charge.• Stores electrical energy.

The charge is easy to see. If a certain potential, ∆V, is applied to a capacitor C, it must storea charge Q=C∆V:

∆V

-Q +Q

C

(Symbol for a capacitor)

Energy Stored in a Capacitor

• Build the charge up a little at a time, letting the charge q on the plate grow from 0 to Q.



• When the charge is q the potential is V=q/C.



• When the charge is q the potential is V=q/C.• Now transfer a little more charge dq.



• When the charge is q the potential is V=q/C.• Now transfer a little more charge dq.• This requires a work dW = Vdq = (1/C) q dq.



• When the charge is q the potential is V=q/C.• Now transfer a little more charge dq.• This requires a work dW = Vdq = (1/C) q dq.• Integrating q from 0 to Q gives the total stored (potential)

electric energy:



• When the charge is q the potential is V=q/C.• Now transfer a little more charge dq.• This requires a work dW = Vdq = (1/C) q dq.• Integrating q from 0 to Q gives the total stored (potential)

electric energy:

€

U =q

Cdq

0

Q

∫ =Q2

2C=

1

2CV 2

• Look at an energy density, i.e., energy per unit volume.

• For the parallel plate capacitor the volume is Ad, souE= U/(Ad) = (1/2 C∆V2)/Ad

• Now also use C = A/d. Then

Energy Density

+Q

-Q

∆V

uE = (∆V/d)2 = (

• This leads to another way to understand the energy.• We can think of the energy as stored in the FIELD,

rather than in the plates.• If an electric field exists, then you can associate an

electric potential energy density of (

Energy Density

+Q

-Q

V

uE = (∆V/d)2 = (

Documents

Capacitance Chapter 26. Electric Potential of Conductors The electric field E is zero within a conductor at equilibrium. E=0 V1V1 V2V2