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
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 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 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  dr  0
 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
B
 F  dr
A
is a function of A and B only
is NOT a function of the path selected
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
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.
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
B
UAB = -WAB = -  F.dr
A
UAB = UB – UA = potential energy difference between A and B
POTENTIAL ENERGY IN A CONSTANT FIELD E
E
L
A
•
•B
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
UAB = q E L
POTENTIAL ENERGY IN A CONSTANT FIELD E
E
L
A
• dL
•B
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
UB - UA = 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 = - FE L
B
A
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,
a) For a positive charge +q ?
b) 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
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
Electric Potential = Potential Energy per Unit Charge
VAB = UAB / q
VAB = VB – VA
Electric potential difference between the points A and B
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
B
VAB = UAB / q = - (1/q)  q E . dL = -  E . dL
A
VAB = Electrical potential difference between the points A and B
ELECTRIC POTENTIAL IN A CONSTANT FIELD E
E
L
A
•
•B
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
ELECTRICAL POTENTIAL IN A CONSTANT FIELD E
E
L
A
• dL
•B
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
VAB = E L
UAB = q E L
POTENTIAL ENERGY
IN A CONSTANT FIELD E
UAB
E
ELECTRIC POTENTIAL
IN A CONSTANT FIELD E
VAB
L
A
•
• B
VAB = UAB / q
UAB = UB – UA = -WAB = -FE L
VAB = VB – VA = -WAB /q = E L
UAB = q E L
VAB = E L
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)
Cases in Which the Electric Field E is not Aligned with dL
E
B
A
VAB = -  E.dl
•

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
Cases in Which the Electric Field E is not Aligned with dL
E
X

A
•

L
•
•
B
VAB = -  E.dl
C
A
B
Another possibility is to choose a path that goes from A to C, and
then from C to B
VAB = VAC + VCB
Thus, VAB = E X
VAC = E X
VCB = 0 (E  dL)
but X = L cos  = - L cos 
VAB = - E L cos 
Rank the points A, B, and C in order of
decreasing potential energy, for a charge
+q is placed at the point.
Equipotential Surfaces (lines)
A
E
B
Since the field E is constant
VAB = E L
X
L
E
Then, at a distance X from plate A
VAX = E X
All the points along the dashed line,
at X, are at the same potential.
The dashed line is an
equipotential line
L
Equipotential Surfaces (lines)
X
E
It takes no work to move a charge
at right angles to an electric field
E  dL   E•dL = 0  V = 0
L
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