Lecture 4Capacitance and Capacitors

Chapter 16.6 16.10

Outline

• Definition of Capacitance• Simple Capacitors• Combinations of Capacitors• Capacitors with Dielectrics

Capacitance

Introduction

The capacitance C of a capacitor is the ratio of the charge (Q) on either conductor plate to the potential difference (V) between the plates.

QC V

General definition

Units of capacitance are farads (F)1F 1C/1V

C(Earth) ~ 1 FC(adult) ~ 150 pF = 150 1012 F

History

The Parallel-Plate Capacitor

The capacitance of a parallel-plate capacitor whose plates are separated by air is:

AC = є0 d

A is the area of one of the platesd is the distance between the platesє0 is permittivity of free space

More about capacitors

Capacitors

Problem: A parallel-plane capacitor has an area of A=5cm2 and a plate separation of d=5mm. Find its capacitance.

Unit conversion: A = 5 cm2 = 5 104 m2

d = 5 mm = 5 103 mC = є0 A/d = 8.85 1012 C2 / (N m2) 5 104 m2 / 5 103 m =

8.85 1013 C2/(N m)= 8.85 1013 F = 0.885 pFN/C = V/m C/N = m/V, F=C/V

C2/(N m)=C (C/N)/m = C (m/V)/m = C/V = F

Combinations of Capacitors

In real electric circuits capacitors can be connected in various ways.In order to design a circuit with desired capacitance, equivalent capacitance of certain combinations of capacitors can be calculated.

There are 2 typical combinations of capacitors:• Parallel combination• Series combination

Parallel Combination

Parallel Combination

• The left plate of each capacitor is connected to the positive terminal of a battery by a wire

• the left plates are at the same potential • the potential differences across the capacitors are

the same, equal to the voltage of the battery (V).• The charge flow ceases when the voltage across

the capacitors equals to that of the battery and the capacitors reach their maximum charge.

Q = Q1 + Q2

Q1 = C1 VQ2 = C2 V

Q = Ceq VCeq V = C1 V + C2 VCeq = C1 + C2

Examples

Series Combination

Series Combination

The magnitude of the charge is the same on all the plates.The equivalent capacitor must have a charge –Q on the right plate and +Q on the left plate.

QV = Ceq

V = V1 + V2

V1 = Q/C1

V2 = Q/C2

Q Q Q = + Ceq C1 C2

1 1 1 = + Ceq C1 C2

Examples

Energy Stored in a Capacitor

The work required to move a charge Q through a potential difference V is W = V Q.

V = Q/C, Q is the total charge on the capacitor.The voltage on the capacitor linearly increases with the magnitude of the charge.Additional work increases the energy stored.

W = ½ Q V = ½ (C V) V = ½C (V)2 = Q2/2C

Capacitors with Dielectrics

A dielectric is an insulating material.The dielectric filling the space between the plates completely increases the capacitance by the factor > 1, called the dielectric constant.

If V0 is the potential difference (voltage) across a capacitor of a capacitance C0 and a charge Q0 in the absence of a dielectric. Filling the capacitor with a dielectric reduces the voltage by the factor to V, so that V = V0/.

C = Q0/V = Q0/V0/ = Q0/V0 = C0

Dielectric Strength

For a parallel-plate capacitor: C = є0 A/d

The formula shows that the capacitance can be made very large by decreasing the plate separation.In practice, the lowest value of d is limited by the electric discharge through the dielectric.The discharge occurs when the electric field in the dielectric material reaches its maximum, called dielectric strength.Dielectric strength of air is 3 106 V/m.

Summary• Capacitance is defined as the charge over the

potential difference• Capacitance of parallel-plate capacitor is directly

proportional to the plate area and inversely proportional to the plate separation

• The equivalent capacitance of a parallel combination of capacitors equals to the sum of individual capacitances

• The inverse equivalent capacitance of a series combination of capacitors equals to the sum of the inverse individual capacitances

• Placing a dielectric between the plates of a capacitor increases the capacitance by a factor , called the dielectric constant