Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been...

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Electric Potential

Transcript of Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been...

Page 1: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

Electric Potential

Page 2: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

CONSERVATIVE FORCES

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

Gravitational and electrostatic forces are conservativeFriction is NOT a conservative force

Page 3: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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’spotential energy is zero.

The gravitational force ‘gave back’ the work that we did when we lifted the mass.

Page 4: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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 willhave 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 thepotential energy of the charge.

Page 5: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

CONSERVATIVE FORCES

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

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

0F dr ����������������������������

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

B

A

F dr����������������������������

Page 6: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

POTENTIAL ENERGY

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

Page 7: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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 areference point (usually infinity).

We will always refer to, or imply, the change in potential energy(potential energy difference) between two points.

The change UAB in potential energy, associated with a conservativeforce 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

A

B

Page 8: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

POTENTIAL ENERGY IN A CONSTANT FIELD E

The potential energy difference between A and Bequals 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

UAB = q E L

E

• •A BL

Page 9: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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 = q E L

• •A BL

dL

UB - UA = q E L

Page 10: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

POTENTIAL ENERGY IN A CONSTANT FIELD E

The potential energy difference between A and Bequals 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

A

B

UAB = UB – UA = - FE L

Page 11: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

E

• •A BL

At which point (A or B) is the potential energy larger,a) For a positive charge +q ?b) For a negative charge –q ?

Page 12: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

• •B AL

x

An electric field E = a/x2 points towards +x.Calculate the potential energy difference UAB = UB – UA for a charge +q

D

Page 13: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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 = UAB / q

Electric Potential = Potential Energy per Unit Charge

VAB = VB – VA

Electric potential difference between the points A and B

Page 14: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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

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

B

Page 15: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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

VAB = E L

• •A BL

E

Page 16: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

ELECTRICAL POTENTIAL IN A CONSTANT FIELD E

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

VAB = E L

• •A BL

E

dLVAB = UAB / q

UAB = q E L

Page 17: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

ELECTRIC POTENTIAL IN A CONSTANT FIELD E

VAB

VAB = E L

• •A BL

E

VAB = UAB / q

POTENTIAL ENERGY IN A CONSTANT FIELD E

UAB

UAB = UB – UA = -WAB = -FE L VAB = VB – VA = -WAB /q = E L

UAB = q E L

Page 18: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

UNITS

Potential Energy U: [Joule] [N m](energy = work = force x distance)

Electric Potential V: [Joule/Coulomb] [Volt](potential = energy/charge)

Electric Field E: [N/C] [V/m](electric field = force/charge = potential/distance)

Page 19: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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

VAB = - E.dl A

B

•A

B

E

E . dl = E dl cos VAB = - E cos dl = - E L cos

Since F = q E is conservative, the field E is conservative.Then, the electrical potential difference does not dependon 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.

A

B

Page 20: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

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

• •A

B

E

C VAB = - E.dl A

BX

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

VAB = - E L cos

L

Page 21: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

Rank the points A, B, and C in order ofdecreasing potential energy, for a charge+q is placed at the point.

Page 22: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

Equipotential Surfaces (lines)

VAB = E L

Since the field E is constant

E

L

E

L

BA

X

VAX = E X

All the points along the dashed line,at X, are at the same potential.

The dashed line is an equipotential line

Then, at a distance X from plate A

Page 23: Electric Potential. CONSERVATIVE FORCES A conservative force “gives back” work that has been done against it Gravitational and electrostatic forces are.

Equipotential Surfaces (lines)

E

L

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