There will be a quiz next class period, Feb 1, covering Ch 22 and the beginning of Ch 23 (what we cover in class today)

Definitions

• Electric potential—Potential energy per unit charge at a point in an electric field

• Path integral (line integral)—An integral performed over a path such as the path a charge q follows as it moves from one point to another

• Volt—The unit of electric potential. 1V = 1 J/C

• Electron volt (eV)—the energy that an electron (or proton) gains or loses by moving through a potential difference of 1 V.

• Equipotential surface—A surface consisting of a continuous distribution of points having the same electric potential

Electric Potential

• Electric force is a conservative force, therefore there is a potential energy associated with it.

• We can define a scalar quantity, the electric potential, associated with it.

€

V =U

q= −

r E • d

r l

A

B

∫

€

WEfield =r F E • d

r l = q

r E • d

r l

dU = −qr E • d

r l

ΔU = −qr E • d

r l

A

B

∫

Electric Potential Energy

Concepts of work, potential energy and conservation of energy

For a conservative force, work can always be expressed in terms of potential energy difference

( )b

a b b aa

W F d l U U U

Energy Theorem

For conservative forces in play,total energy of the system is conserved

a a b bK U K U

• The line integral used to calculate V does not depend on the path taken from A to B; therefore pick the most convenient path to integrate over

Electric Potential

• We can pick a 0 for the electric potential energy

• U is independent of any charge q that can be placed in the Electric field

• U has a unique value at every point in the electric field

• U depends on a location in the E field only

rU 0

0a bW Fd q Ed 0U q Ey 0 ( )a b a bW U q E y y

Potential energy U increases as the test charge q0 moves in the

direction opposite to the electric force : it decreases as it moves in the same direction as the force acting on the charge

0F q E

Electric Potential Energy of Two Point Charges

02

cosb

a

rb

a b ea r

qqW F d l k dl

r

01 1

a b ea b

W k qqr r

0e

qqU k

r Electric potential energy of two point charges

Example: Conservation of energy with electric forces

A positron moves away from an – particle

-particle

positron

0

31

100

60

9.1 10

7000

2

10

3 10 /

p

p

m kg

m m

q e

r m

V m s

What is the speed at the distance ?What is the speed at infinity?Suppose, we have an electron instead of positron. What kind of motion we would expect?

1002 2 10r r m

Conservation of energy principle

0 0 1 1K U K U

Electric Potential Energy of the System of Charges

Potential energy of a test charge q0

in the presence of other charges0

04i

ii

q qU

r

Potential energy of the system of charges(energy required to assembly them together)

0

1

4i j

iji j

q qU

r

Potential energy difference can be equivalently described as a work done by external force required to move charges into the certain geometry (closer or farther apart). External force now is opposite to the electrostatic force

( )a b b a extW U U F d l

Electric potential is electric potential energy per unit charge

Finding potential (a scalar) is often much easier than the field (which is a vector). Afterwards, we can find field from a potential

0

UV

q Units of potential are Volts [V]

1 Volt=1Joule/Coulomb

If an electric charge is moved by the electric field, the work done by the field

0 0

( )a ba b

W UV V

q q

Potential difference if often called voltage

Two equivalent interpretations of voltage:

1.Vab is the potential of a with respect to b, equals the work done by the electric force when a UNIT charge moves from a to b.

2. Vab is the potential of a with respect to b, equals the work that must be done to move a UNIT charge slowly from b to a against the electric force.

Potential due to the point charges

0

1

4

dqV

r Potential due to a continuous

distribution of charge

Finding Electric Potential through Electric Field

0

ba b

a ba

WV V E d l

q

Some Useful Electric Potentials

• For a uniform electric field

• For a point charge

• For a series of point charges

€

V = −r E • d

r l = −

r E • d

r l = −

r E •

r l ∫∫

rq

kV e

i

ie r

qkV

Potential of a point charge

Moving along the E-field lines means moving in the direction of decreasing V.

As a charge is moved by the field, it loses it potential energy, whereas if the chargeis moved by the external forces against the E-field, it acquires potential energy

• Negative charges are a potential minimum

• Positive charges are a potential maximum

Positive Electric Charge Facts

• For a positive source charge– Electric field points away from a positive source charge

– Electric potential is a maximum

– A positive object charge gains potential energy as it moves toward the source

– A negative object charge loses potential energy as it moves toward the source

Negative Electric Charge Facts

• For a negative source charge– Electric field points toward a negative source charge

– Electric potential is a minimum

– A positive object charge loses potential energy as it moves toward the source

– A negative object charge gains potential energy as it moves toward the source