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Electric Potential. CONSERVATIVE FORCES. A conservative force “gives back” work that has been done against it. Gravitational and electrostatic forces are conservative Friction is NOT a conservative force. CONSERVATIVE FORCES.

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Conservative forces
CONSERVATIVE FORCES

A conservative force “gives back” work that has been done against it

Gravitational and electrostatic forces are conservative

Friction is NOT a conservative force


Conservative forces1
CONSERVATIVE FORCES

A conservative force “gives back” work that has been done against it

When we lift a mass m from ground to a height h,

the potential energy of the mass increases by mgh.

If we release the mass, it falls, picking up kinetic

energy (or speed). As the mass falls, the potential

energy is being converted into kinetic energy.

By the time it reaches the ground, the mass has

acquired a kinetic energy ½ mv2 = mgh, and it’s

potential energy is zero.

The gravitational force ‘gave back’ the work that

we did when we lifted the mass.


Conservative forces2
CONSERVATIVE FORCES

A conservative force “gives back” work that has been done against it

The gravitational force is a conservative force.

The electric force is a conservative force as well.

We will be able to define a potential energy

associated with the electric force. A charge will

have potential energy when in an electric field.

Work done on the charge (by an external agent,

or by the field) will result in changes in the

potential energy of the charge.


Conservative forces3
CONSERVATIVE FORCES

When the total work done by a force F, moving an object over a

closed loop, is zero, then the force is conservative

 F is conservative

The circle on the integral sign indicates that the integral is taken over a closed path

The work done by a conservative force, in moving and object

between two points A and B, is independent of the path taken

is a function of A and B only

is NOT a function of the path selected

A conservative force “gives back” work that has been done against it


The change UAB in potential energy,

associated with a conservative force,

is the negative of the work done by that force,

as it acts from point A to point B

UAB = -WAB

UAB = UB – UA = potential energy difference between A and B

POTENTIAL ENERGY


The change UAB in potential energy, associated with a conservative

force F, is the negative of the work done by that force, as it acts

(over any path) from point A to point B

UAB = -WAB = -  F.dr

UAB = UB – UA = potential energy difference between A and B

B

A

POTENTIAL ENERGY

Potential energy is a relative quantity, that means, it is always the

difference between two values, or it is measured with respect to a

reference point (usually infinity).

We will always refer to, or imply, the change in potential energy

(potential energy difference) between two points.


E

L

A

B

UAB =q E L

POTENTIAL ENERGY IN A CONSTANT FIELD E

The potential energy difference between A and B

equals the negative of the work done by the field

as the charge q is moved from A to B

UAB = UB – UA = -WAB = -FE L = q E L


L

A

B

dL

UB - UA =q E L

UAB =q E L

POTENTIAL ENERGY IN A CONSTANT FIELD E

E

Potential energy difference between A and B

UAB = UB – UA = -  q E.dl

But E = constant, and E.dl = -1 E dl, then:

UAB = -  q E.dl =  q E dl = q E  dl = q E L


UAB = UB – UA = - FE L

B

A

POTENTIAL ENERGY IN A CONSTANT FIELD E

The potential energy difference between A and B

equals the negative of the work done by the field

as the charge q is moved from A to B

UAB = q E L when the +q charge is moved against the field


E

L

A

B

  • At which point (A or B) is the potential energy larger,

  • For a positive charge +q ?

  • For a negative charge –q ?


L

B

A

x

D

An electric field E = a/x2 points towards +x.

Calculate the potential energy difference

UAB = UB – UA for a charge +q


VAB = UAB / q

Electric Potential = Potential Energy per Unit Charge

ELECTRIC POTENTIAL DIFFERENCE

The potential energy U depends on the charge being moved.

In order to remove this dependence, we introduce the concept

of electric potential V

VAB = VB – VA

Electric potential difference between the points A and B


B

VAB = UAB / q = - (1/q)  q E . dL = -  E . dL

A

ELECTRICAL POTENTIAL DIFFERENCE

The potential energy U depends on the charge being moved.

In order to remove this dependence, we introduce the concept

of electrical potential V

VAB = UAB / q

Electrical Potential = Potential Energy per Unit Charge

VAB = Electrical potential difference between the points A and B


E

L

A

B

VAB = E L

ELECTRIC POTENTIAL IN A CONSTANT FIELD E

The electric potential difference between A and B equals the negative of the work per unit charge, done by the field,

as the charge q is moved from A to B

VAB = VB – VA = -WAB /q = qE L/q = E L


E

L

A

B

dL

VAB = E L

ELECTRICAL POTENTIAL IN A CONSTANT FIELD E

VAB = UAB / q

The electrical potential difference between A and B equals the work per unit charge necessary, for an external agent, to move a charge +q from A to B

VAB = VB – VA = -WAB /q = -  E.dl

But E = constant, and E.dl = -1 E dl, then:

VAB = -  E.dl =  E dl = E  dl = E L

UAB =q E L


E

L

A

B

UAB = UB – UA = -WAB = -FE L

VAB = VB – VA = -WAB /q = E L

VAB = E L

UAB =q E L

POTENTIAL ENERGY

IN A CONSTANT FIELD E

UAB

ELECTRIC POTENTIAL

IN A CONSTANT FIELD E

VAB

VAB = UAB / q


UNITS

Potential Energy U: [Joule]  [N m]

(energy = work = force xdistance)

Electric Potential V: [Joule/Coulomb]  [Volt]

(potential = energy/charge)

Electric Field E: [N/C]  [V/m]

(electric field = force/charge = potential/distance)


B

VAB = -  E.dl

A

Cases in Which the Electric Field E is not Aligned with dL

E

A

B

Since F = q E is conservative, the field E is conservative.

Then, the electrical potential difference does not depend

on the integration path.

One possibility is to integrate along the straight line AB.

This is convenient in this case because the field E is constant, and the angle  between E and dL is constant.

B

E . dl = E dl cos   VAB = - E cos   dl = - E L cos 

A


B

VAB = -  E.dl

A

VAB = - E L cos 

Cases in Which the Electric Field E is not Aligned with dL

E

X

A

C

L

B

Another possibility is to choose a path that goes from A to C, and

then from C to B

VAB = VAC + VCB VAC = E X VCB = 0 (E  dL)

Thus, VAB = E X but X = L cos  = - L cos 


Rank the points A, B, and C in order of

decreasing potential energy, for a charge

+q is placed at the point.


A

B

E

E

VAB = E L

VAX = E X

X

L

L

Equipotential Surfaces (lines)

Since the field E is constant

Then, at a distance X from plate A

All the points along the dashed line,

at X, are at the same potential.

The dashed line is an

equipotential line


E

L

Equipotential Surfaces (lines)

X

It takes no work to move a charge

at right angles to an electric field

E dL   E•dL = 0  V = 0

If a surface (line) is perpendicular to

the electric field, all the points in

the surface (line) are at the same

potential. Such surface (line) is called

EQUIPOTENTIAL

EQUIPOTENTIAL  ELECTRIC FIELD


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