capacitance, dielectrics, electric energy...

Capacitance, Dielectrics,

Electric Energy Storage

Capacitors A capacitor is a device which stores electric charge. Capacitors vary in shape and size, but the basic configuration is two conductors carrying equal but opposite charges.

The two conductors forming a capacitor are close but not touching. In the uncharged state, the charge on either one of the conductors in the capacitor is zero. During the charging process, due to voltage !V across the plates, a charge Q is moved from one conductor to the other one, giving one conductor a charge +Q , and the other one a charge "Q .

Parallel-plate Capacitor

A: surface area of each plate Q: Charge on a surface h: Separation of two plates

!V+Q

- Q

Q = C !V

Experiments show that the amount of charge Q stored in a capacitor is linearly proportional to !V , the electric potential difference between the plates.

C is called capacitance. The SI unit of capacitance is farad (F):

1 F = 1 farad = 1 Coulomb/Volt =1 C/V

Ref: Wikipedia

Calculating Capacitance (Vacuum) Using the principle of superposition and Gauss’s law we found that

E = !"0

where ! is the magnitude of the surface charge density on each plate

E = !"0

= Q"0A

Vab = Ed =1"0QdA

C = QVab

#

$

%%%

&

%%%

'C = "0Ad

!0 = 8.85!10"12 F

m

Problem (1) A parallel-plate capacitor has plates with equal and opposite charges ±Q, separated by a distance d, and is not connected to a battery. The plates are

pulled apart to a distance D > d. What happens to V and Q?

1)! V increases, Q increases

2)! V decreases, Q increases

3)! V is the same, Q increases

4)! V increases, Q is the same

5)! V decreases, Q is the same

6)! V is the same, Q is the same

7)! V increases, Q decreases

Spherical Capacitor consider a spherical capacitor which consists of two concentric spherical shells of radii a and b . The inner shell has a charge +Q uniformly distributed over its surface, and the outer shell an equal but opposite charge –Q. What is the capacitance of this configuration?

The potential difference between the two conducting shells is:

!V =Va "Vb =Vab =Q

4#$0ra" Q4#$0rb

Vab =Q4#$0

1ra" 1rb

%&'

()*= Q4#$0

rb " rararb

%&'

()*

C = QVab

= 4!"0rarbrb ! ra

"#$

%&'

Combination of Capacitors Capacitors are manufactured with certain standard and working voltage. However, these standard values may not be the ones we actually need in a particular application. We can combine them to obtain what we need.

Also, in circuits with multicomponent, we always try to reduce the circuit to single components. To do so, we can combine all capacitors to one capacitor. Hence one can reduce most circuits to a simple circuit of an equivalent capacitor Ceq.

Capacitors can be combined in: ①! Parallel. ②! Series.

The potential difference across the capacitors in a parallel circuit, (1), are the same and each is equal to the battery’s voltage.

Capacitors in Parallel

!V = !V1 = !V2 " C1 =Q1

!V , C2 =

Q2

!VThese two capacitors can be replaced by a single equivalent capacitor Ceq , (2), with a total charge Q supplied by the battery. Q is shared by the two capacitors, so

Q =Q1 +Q2 = C1 !V +C2 !V = C1 +C2( ) !V

(1)

(2)

The equivalent capacitance is then seen to be given by

Thus, capacitors that are connected in parallel add. The generalization to any number of capacitors is

Ceq =Q!V

= C1 +C2

Ceq = C1 +C2 +C3 +!+Cn = Ci

i=1

n

! (Parallel)

The magnitude of charge must be the same at all plates for a series combination, (1),

Capacitors in Series

!V1 = QC1

, !V2 = QC2

The potential difference across capacitors C1 and C2 are:

Q1 =Q2 =Qi =Q

The total potential difference is simply the sum of the two individual potential differences:

The generalization to any number of capacitors connected in series is

!V = !V1 + !V2

1Ceq

= 1C1

+ 1C2

+ 1C3

+!+ 1Cn

= 1Cii=1

n

! (Series)

(1)

(2)

The total potential difference across any number of capacitors in series is equal to the sum of potential differences across the individual capacitors. Replacing two capacitors by a single equivalent capacitor Ceq, (2), and using the fact that the potentials add in series, Q

Ceq

= QC1

+ QC2

! 1Ceq

= 1C1

+ 1C2

①! Make sure units are all in SI — C in farads, lengths in meters, etc..

②! Make equivalent capacitors from capacitors in the circuit. Choose either sets of parallel capacitors or series capacitors first, depending on which is more obvious to chose.

③! Continue on making these equivalent capacitors until you only have one left.

④! To find the charge on, or the potential difference across, one of the capacitors in a complicated circuit, start with the final (i.e., reduced) circuit of step (3) and gradually work your way back through the circuits using

Problem-Solving Strategy

C = Q!V

Find the equivalent capacitance for the combination of capacitors shown in the figure.

Problem (2)

Find the equivalent capacitance between points a and b for the group of capacitors connected as shown the figure, if

Problem (3)

C1 = 5.0µF, C2 = 10.0µF, C3 = 2.0µF

!! !

Storing Energy in a Capacitor capacitors can be used to store electrical energy. The amount of energy stored is equal to the work done to charge it. During the charging process, the battery does work to remove charges from one plate and deposit them onto the other. Work is done by an external agent in bringing +dq from the

negative plate and depositing the charge on the positive plate.

Suppose the amount of charge on the top plate at some instant is +q , and the potential difference between the two plates is V=q/C. To add charge +dq to the top plate, the amount of work done to overcome electrical repulsion is dW =Vdq . If at the end of the charging process, the charge on the top plate is +Q , then the total amount of work done in this process is:

W = dqV0

Q

! = dq qC0

Q

! = 12Q2

C

This is equal to the electrical potential energy U of the system. U = 1

2Q2

C= 12QV = 1

2CV 2

Energy Density of the Electric Field

One can think of the energy stored in the capacitor as being stored in the electric field itself. In the case of a parallel-plate capacitor we have,

C = !0Ad

V = Ed

"#$

%$ U = 1

2CV 2 = 1

2!0Ad

Ed( )2 = 12!0E

2 Ad( )

u = UVolume

= 12!0E

2

Since the quantity (Ad ) represents the volume between the plates, we can define the electric energy density as

Dielectric In many capacitors there is an insulating material such as paper or plastic between the plates. Such material, called a dielectric, can be used to maintain a physical separation of the plates. Since dielectrics break down less readily than air, charge leakage can be minimized, especially when high voltage is applied.

Experimentally it was found that capacitance C increases when the space between the conductors is filled with dielectrics. To see how this happens, suppose a capacitor has a capacitance C0 when there is no material between the plates. When a dielectric material is inserted to completely fill the space between the plates, the capacitance increases to

Where K is called the dielectric constant.

C = KC0

!! For a capacitor with no dielectric, the voltage drop across the capacitor is

!! If a dielectric is inserted between the plates of a capacitor, the voltage drop is reduced by a scale factor K (note that K >1)

!! Charge on the capacitor does not change when a dielectric is introduced, however, the capacitance in the presence of a dielectric changes to the value

!! As it can be seen the potential difference between the plates decreases by a factor K. Therefore the electric field between the plates decreases by the same factor.

V0 =Q0

C0

V = V0K

C = QV

= Q0

V0K

= K Q0

V0= KC0

Dielectric

E = E0

K (Q is constant)

Problem (4)

A parallel-plate capacitor is charged to a total charge Q and the battery is removed. A slab of material with dielectric constant k is inserted

between the plates. The charge stored in the capacitor

1)! Increases

2)! Decreases

3)! Stays the same

Problem (5)

A parallel-plate capacitor is charged to a total charge Q and the battery is removed. A slab of material with dielectric constant k is inserted

between the plates. The energy stored in the capacitor

1)! Increases

2)! Decreases

3)! Stays the same

Summary

charges move, and is now being provided to the agent moving the charge at constant speed along the electric field of the other charges. The energy provided to that agent as we destroy the electric field is exactly the amount of energy that the agent put into creating the electric field in the first place, neglecting radiative losses (such losses are small if we move the charges at speeds small compared to the speed of light). This is a totally reversible process if we neglect such losses. That is, the amount of energy the agent puts into creating the electric field is exactly returned to that agent as the field is destroyed.

There is one final point to be made. Whenever electromagnetic energy is being created, an electric charge is moving (or being moved) against an electric field ( ). Whenever electromagnetic energy is being destroyed, an electric charge is moving (or being moved) along an electric field (

0q v E

0q v E ). When we return to the creation and destruction of magnetic energy, we will find this rule holds there as well.

5.7 Summary

A capacitor is a device that stores electric charge and potential energy. The capacitance C of a capacitor is the ratio of the charge stored on the capacitor plates to the the potential difference between them:

| |

QC

V

System Capacitance

Isolated charged sphere of radius R 04C R

Parallel-plate capacitor of plate area A and plate separation d 0Cd

Cylindrical capacitor of length , inner radius a and outer radius b 02ln( / )

LC

b a

Spherical capacitor with inner radius a and outer radius b 04ab

Cb a

The equivalent capacitance of capacitors connected in parallel and in series are

eq 1 2 3 (parallel)C C C C

eq 1 2 3

1 1 1 1 (series)

C C C C

28



0q v E


5.7 Summary


| |

QC

V

System Capacitance




LC

b a


Cb a



eq 1 2 3

1 1 1 1 (series)

C C C C

28



0q v E


5.7 Summary


| |

QC

V

System Capacitance


Parallel-plate capacitor of plate area A and plate separation d 0A

Cd

Cylindrical capacitor of length L , inner radius a and outer radius b 02ln( / )

LC

b a


Cb a



eq 1 2 3

1 1 1 1 (series)

C C C C

28

The equivalent capacitance of capacitors connected in parallel and in series are:



0q v E


5.7 Summary


| |

QC

V

System Capacitance




LC

b a


Cb a



eq 1 2 3

1 1 1 1 (series)

C C C C

28



0q v E


5.7 Summary


| |

QC

V

System Capacitance




LC

b a


Cb a



eq 1 2 3

1 1 1 1 (series)

C C C C

28

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