Ch 25 – Electric Potential A difference in electrical potential

between the upper atmosphere and the ground can cause electrical discharge (motion of charge).

Ch 25 – Electric Potential So far, we’ve discussed electric

force

and

fields

.

Now, we associate a

potential energy function

with electric force.

This is identical to what we did with gravity last semester.

gravity 

F g

 

G m

1

m

2

r

2

r

ˆ 

g

 

G m r

2

r

ˆ

U g

 

G m

1

m

2

r

electricity

F e



k e



E



k e q

1

r

2

q

2

r q

2

r

ˆ ?

Ch 25.1 – Electric Potential and Potential Difference • Place a test charge,

q 0

, into an E-field. The charge will experience a force: 

F



q

0 

E

• This force is a conservative force.

• Pretend an

external agent

does

work

to move the charge through the E-field.

• The work done by the external agent

equals at least the negative of the work done by the E-field

.

Ch 25.1 – Electric Potential and Potential Difference • Let’s introduce a new symbol:

d s

 • We’re talking about moving charges through some displacement.

• The “ ds ”

vector

is a little tiny step of displacement along a charge’s path.

Ch 25.1 – Electric Potential and Potential Difference • If

q 0

moves through the E-field by a little step some work: ds , the E-field does

dW

E  field  

F

 

d s

• As the E-field performs this work, we say that the

potential energy

of the charge-field system changes by this amount.

• This is the basis for our

definition of the potential energy function

.

Ch 25.1 – Electric Potential and Potential Difference • If

q 0

moves through the E-field by a little step some work: ds , the E-field does

dW

E  field  

F

 

d s



dU

 

dW

E  field   

F



d s

  

q

0 

E



d s

  

U



U B



U A

 

q

0

A



B



E



d s



Ch 25.1 – Electric Potential and Potential Difference 

U



U B



U A

 

q

0

A



B



E



d s

 The change in electrical potential energy of a charge-field system as the charge moves from A to B in the field.

The integral accounts for the motion of the charge through a 1-D path. It’s called a “

path

” or “

line

”

integral

.

Ch 25.1 – Electric Potential and Potential Difference 

U



U B



U A

 

q

0

A



B



E



d s

 The change in electrical potential energy of a charge-field system as the charge moves from A to B in the field.

Because electric force is conservative, the value of the integral does not depend on the path taken between

A

and

B

.

Ch 25.1 – Electric Potential and Potential Difference

Potential Energy refresher:

Potential Energy measures the energy a system has

due to it’s configuration

.



U



U B



U A

 

q

0

A



B



E



d s

 The change in electrical potential energy of a charge-field system as the charge moves from A to B in the field.

We always care about

changes

in potential energy – not the instantaneous value of the PE.

The zero-point for PE is relative. You get to choose what configuration of the system corresponds to PE = 0.

Ch 25.1 – Electric Potential and Potential Difference • What we’re about to do is different than anything you saw in gravitation.

• In electricity, we choose to divide

q 0

out of the equation.



V

 

U q

0  

A



B



E



d s

 • We call this new function, Δ

V

, the “

electric potential difference

.”

Ch 25.1 – Electric Potential and Potential Difference 

V

 

U q

0  

A



B



E



d s

 Potential difference between two points in an Electric Field.

• This physical quantity only depends on the electric field.

• Potential Difference – the change in potential energy per unit charge between two points in an electric field.

• Units: Volts, [V] = [J/C]

Ch 25.1 – Electric Potential and Potential Difference 

V

 

U q

0  

A



B



E



d s

 Potential difference between two points in an Electric Field.

• Do not confuse “

potential difference

” with a change in “

electric potential energy

.” • A potential difference can exist in an E-field regardless the presence of a test charge.

• A change in electric potential energy can only occur if a test charge actually moves through the E-field.

Ch 25.1 – Electric Potential and Potential Difference • Pretend an external agent moves a charge,

q

, from

A

to

B

without changing its speed. Then:

W

 

U

But: 

V

 

U q

0

W



q



V

Ch 25.1 – Electric Potential and Potential Difference • Units of the potential difference are Volts: 1 V  1 J/C • 1 J of work must be done to move 1 C of charge through a potential difference of 1 V.

Ch 25.1 – Electric Potential and Potential Difference • We now redefine the units of the electric field in terms of volts.

1 N/C  1 V/m E-field units in terms of volts per meter

Ch 25.1 – Electric Potential and Potential Difference • Another useful unit (in atomic physics) is the electron-volt.

1 eV  1.60

 10 -19 C  V  1.60

 10 -19 J The electron-volt • One electron-volt is the energy required to move one electron worth of charge through a potential difference of 1 volt.

• If a 1 volt potential difference accelerates an electron, the electron acquires 1 electron-volt worth of kinetic energy.

Quick Quiz 25.1

Points

A

and

B

are located in a region where there is an electric field.

How would you describe the potential difference between

A

and

B

? Is it negative, positive or zero?

Pretend you move a negative charge from

A

to

B

. How does the potential energy of the system change? Is it negative, positive or zero?

Ch 25.2 – Potential Difference in a Uniform E-Field Let’s calculate the potential difference between

A

and

B

separated by a distance

d

.

Assume the E-field is uniform, and the path,

s

, between

A

and

B

is parallel to the field.



V

 

A



B



E

 

d s

Ch 25.2 – Potential Difference in a Uniform E-Field Let’s calculate the potential difference between

A

and

B

separated by a distance

d

.

Assume the E-field is

uniform

, and the displacement,

s

, between

A B

is

parallel to the field

.

and 

V

 

A



B



E

 

d s



V

 

A



B Eds

cos  1 

V

 

E A



B ds



V

 

Ed

Ch 25.2 – Potential Difference in a Uniform E-Field 

V

 

Ed

The negative sign tells you the potential at

B

is lower than the potential at

A

.

V B

<

V A

Electric field lines always point in the direction of decreasing electric potential.

Ch 25.2 – Potential Difference in a Uniform E-Field Now, pretend a charge

q 0

moves from

A

to

B

.

The change in the charge-field PE is: 

U



q

0 

V

 

q

0

Ed

If

q 0

is a positive charge, then Δ

U

is negative.

When a positive charge moves down field, the charge-field system loses potential energy.

Ch 25.2 – Potential Difference in a Uniform E-Field Electric fields accelerate charges… that’s what they do.

What we’re saying here is that as the E-field accelerates a positive charge, the charge-field system picks up kinetic energy.

At the same time, the charge-field system loses an equal amount of potential energy.

Why? Because in an isolated system without friction,

mechanical energy must always be conserved

.

Ch 25.2 – Potential Difference in a Uniform E-Field If

q 0

is negative then Δ from

A

to

B

.

U

is positive as it moves 

U



q

0 

V

 

q

0

Ed

When a negative charge moves down field, the charge-field system gains potential energy.

If a negative charge is released from rest in an electric field, it will accelerate against the field.

Ch 25.2 – Potential Difference in a Uniform E-Field Consider a more general case.

Assume the E-field is uniform, but the path,

s

, between

A

and

B

is

not

parallel to the field.



V

 

A



B



E



d s



Ch 25.2 – Potential Difference in a Uniform E-Field Consider a more general case.

Assume the E-field is uniform, but the path,

s

, between

A

and

B

is

not

parallel to the field.



V

 

B A

 

E

 

d s

  

E



B A



d s

   

E

 

s



U



q

0 

V

 

q

0 

E



s



Ch 25.2 – Potential Difference in a Uniform E-Field 

V

  

E



s

 If

s

is perpendicular to

E

(path

C

-

B),

the electric potential does not change.

Any surface oriented perpendicular to the electric field is thus called a

surface of equipotential

, or an

equipotential surface

.

Quick Quiz 25.2

The labeled points are on a series of equipotential surfaces associated with an electric field.

Rank (from greatest to least) the work done by the electric field on a positive charge that moves from

A

to

B

, from

B

to

C

, from

C

to

D

, and from

D

to

E

.

EG 25.1 – E-field between to plates of charge A battery has a specified potential difference Δ

V

between its terminals and establishes that potential difference between conductors attached to the terminals. This is what batteries do.

A 12-V battery is connected between two plates as shown. The separation distance is

d

= 0.30 cm, and we assume the E-field between the plates is uniform. Find the magnitude of the E-field between the plates.

EG 25.1 – Proton in a Uniform E-field A proton is released from rest at

A

in a uniform E-field of magnitude 8.0 x 10 4 V/m. The proton displaces through 0.50 m to point

B

, in the same direction as the E-field. Find the speed of the proton after completing the 0.50 m displacement.