Chapter 18Electrical energy and

Capacitance

Today’s Topics

• Electric Potential Energy• Electric Potential• Electric Equi-potential Lines

Work

• You do work when you push an object up a hill• The longer the hill the more work you do: more

distance• The steeper the hill the more work you do: more

forceThe work W done on an object by an agent exerting a constant force is the product of the component of the force in the direction of the displacement and the magnitude of the displacement

dFW ||

Work done by gravity

dFW cos

m

mg

d

cosF

Energy is capacity to do work

• Gravitational Potential Energy• Kinetic Energy• Energy can be converted into other forms of

energy • When we do work on any object we transfer

energy to it • Energy cannot be created or destroyed

mghUG 2

21 mvK

GU

GUKW

• A person lifts a heavy box of mass m a vertical distance h

• They then move a distance d, carrying the box

• How much work is done?

Quiz

Conversion of Gravitational Potential Energy to Kinetic

Energym

mg

m

v

mghUG 2

21 mvK

ghv

ghv

mghmv

2

221

2

2

Work done on object

h

Electric Potential Energy

EEF Q

+Q

+Q

d

FdW

QEd

QEdU e

v

Electric Potential Energy

• Work done (by electric field) on charged particle is QEd

• Particle has gained Kinetic Energy (QEd)

• Particle must therefore have lost Potential Energy U=-QEd

Electric PotentialThe electric potential energy depends on the charge present

We can define and electric potential V which does not depend on charge by using a “test” charge

EdQU 0

0QUV

Change in potential is change in potential energy for a test charge per unit charge

EdQUV

0

for uniform field

Electric Potential

0QUV

compare with the Electric Field and Coulomb Force

0QFE

If we know the potential field this allows us to calculate changes in potential energy for any charge introduced

VQU EF Q

Electric Potential

Electric Potential is a scalar field

it is defined everywhere

but it does not have any direction

it doesn’t depend on a charge being there

Electric Potential, units

SI Units of Electric Potential

0QUV

EdV

Units are J/C

Alternatively called Volts (V)

We have seen

dVE / Thus E also has units of V/m

Potential in Uniform fieldE

+Q +Q

+Q

A B

C

0|| dFWBC

|||| QEddFWAB

BCABAC WWW

||QEd

||QEdU AC d||

||EdVAC

Electric Potential of a single charge

+

r

E

B

A

Advanced

Equi-potential Lines

Like elevation potential can be displayed as contours

A contour diagram

Like elevation potential requires a zero point, potential V=0 at r=

Like slope & elevation we can obtain the Electric Field from the potential field

rVE

Potential Energy in 3 chargesrQV

041Q2

Q1

Q3

12

1

02212 4

1rQQVQU

12

21

012 4

1rQQU

23

2

13

1

03123312 4

1rQ

rQQUVQUU

231312 UUUU

23

32

13

31

12

21

041

rQQ

rQQ

rQQU

Capacitors

A system of two conductors, each carrying equal charge is known as a capacitor

-

Capacitance of charged sphere

+Q

r=

VQC

definition

R

rQ V

0 41

potential due to isolated charge

Capacitors

-

+ +Q -Q

e.g. 1: two metal spherese.g. 2: two parallel sheets

Each conductor is called a plate

CapacitanceCapacitance…….. is a measure of the amount of charge a capacitor can store (its “capacity”)

Experiments show that the charge in a capacitor is proportional to the electric potential difference (voltage) between the plates.

Units

VQC

Thus SI units of capacitance are:

C/V

This unit is also known as the farad after Michael Faraday

Remember that V is also J/C so unit is also C2J-1

1F=1C/V

Capacitance

The constant of proportionality C is the capacitance which is a property of the conductor

VQ VCQ VQC

Experiments show that the charge in a capacitor is proportional to the electric potential difference (voltage) between the plates.

Capacitance of parallel plates

+Q -Q

IntutivelyThe bigger the plates the more surface area over which the capacitor can store charge C A

E

Moving plates togeth`er Initially E is constant (no charges moving) thus V=Ed decreases charges flows from battery to increase V C 1/d Never Ready+

V

Batteries, Conductors & Potential

Never R

eady

+A battery maintains a fixed potential difference (voltage) between its terminals A conductor

has E=0 within and thus V=Ed=0

VV= 0

Capacitance of parallel plates

+Q -Q

Physically

E

Never Ready+

EdV

VQC

property of conductorV