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9/13/2008 3:07:00 PM
PH504 – Part 3
Electric potential, potential energy.
Introduction: energy
In the previous lecture we considered electrostatics in terms of the
electric force. A different approach is in terms of energy. This is
particularly useful for situations where conversion to different forms of
energy (e.g. kinetic) occurs.
In addition, for a number of situations, it is easier to find the electric
potential (which is a scalar quantity) due to a charge distribution than
the E-field which is a vector quantity. The E-field can subsequently be
determined once the electric potential is known (see below) .
Remember: curl E = 0 .
The mutual potential energy of a charge system
The potential energy of a system of charges depends upon its spatial
configuration. The difference in potential energy U between two
configurations is given by the work done by external forces to change
the system from one configuration to the other (this is done
infinitesimally slowly so that there is no change in the kinetic energy).
If U is positive then the new configuration has a greater potential
energy than the old configuration.
Sometimes the (absolute) potential energy (U) is given; this is relative to
some standard configuration for which U=0 is assumed.
U and U for two point charges
Charges Q1
and Q2 are initially separated by a distance r1. An external force alters
their separation to r2. What is U?
The force needed to push the charges together is equal but opposite in
direction to the electric force between them.
Using work = force x distance
If Q1 and Q2 have the same sign and r2<r1 then U is positive: energy is
required to push the charges closer together against the repulsive
electric force.
In terms of U a suitable choice for U=0 is when the charges are an
infinite distance apart (r1=). Hence U for two charges separated by a
distance r is given by
Path-independence:
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As the electric force is a radial or central one, work is only done for
movement along the line joining the two charges (U=0 for any
tangential displacement). Hence U is independent of the path taken in
moving between two configurations.
No work is done along the arc segments AB, CD, EF and GH.
Hence U for path ABCDEFGH is U(BC)+U(DE)+U(FG)
=U(AH).
Any line from AH can be made up from a (possibly infinite) sequence of
arcs and radii.
Superposition: U for >2 point charges
Because of the superposition of forces the total potential energy is given
by the summation of the individual potential energies. e.g. for three point
charges:
For a collection of N point charges
where rij is the distance between charges i and j.
The factor 1/2 compensates for each pair of charges being counted
twice in the summation.
An alternative way of writing the above result is in terms of the electric
potential Vi produced at the site of charge i by the other (N-1) charges
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1 N
U   qV
i i
2 i 1
where again the factor of ½ avoids counting the same interaction twice.
Moving a charge from A to B:
U  U B  U A
B
 
   q o E  ds
A
U
q0
B 

   E  ds
V 
A
The equation can be modified to account for the case where there is a
continuous charge distribution given by 
U
1
Vd
2 
This form is particularly useful when calculating the mutual potential
energy of a charged body. Substituting
div 0E = 
and integrating by parts, the energy per unit volume is
ue =0E2 /2
….integrated over all space gives U! No superposition principle.
Relationship between U and electric force
If the electric force is non-zero along one axis only (e.g. Fx) then
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.
More generally in three dimensions
In words 'the electric force is equal to the negative of the gradient of
the potential energy'.
Electric potential (NOT potential energy)
Since curl E = 0, field is irrotational. Hence potential, a scalar field,
exists.
E = – grad V
If the potential energy of a system varies by UAB as a test charge Qt is
moved from point A to point B then the potential difference VAB between
points A and B is defined by
VAB is related to UAB in a similar way to the relationship between E-field
and electric force.
The units of electric potential are J C-1 V (Volt)
The previous equation gives the potential difference between the
points B and A. The potential at a point can also be given assuming the
zero point is known or specified.
If a charge Q is moved between points A and B then its potential
energy will change by UAB=QVAB (Q should be sufficiently small so as
not to perturb the charges which cause VAB).
Relationship between E-field and V
We have
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and
and also F = -U .
Hence E = -V
… it is very easy to derive E from V !
Alternatively,
where the integral is a line along a path from point A to point B. For
electrostatic fields VAB is independent of the path taken from A to B.
Potential due to a point charge
Find the potential at a distance r1 from a point charge Q where the
potential at infinity is taken as zero.
where the integral is performed in a radial direction so that E is parallel
to r (cos=1)
The potential difference between two points at distances r1 and r2 from
the point charge is
For a collection of point charges the potential at a given point is the
algebraic sum of the individual potentials (electric potential obeys the
superposition principle)
where V is the total potential a distance r1 from Q1, r2 from Q2 etc.
If a charge system contains continuous distributions of charge then the
potential may be found using a suitable integration.
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This is an alternative, and possibly simpler, method for finding the E-field
as V is simply the algebraic sum of the individual potentials (not a
vector sum as for E-field). E can be determined from the relationship E=V once V has been calculated.
Equipotential surfaces and E-Field lines
Equipotential surfaces are those which connect points at the same
potential. In practice we can only draw two-dimensional cross-sections
of the equipotential surfaces.
For a point charge the lines of force point radially outwards and the
equipotential lines form a series of concentric circles. At all points the
two types of lines are normal to each other.
Proof that lines of E (or force) are always perpendicular to
equipotentials
In a direction tangential (along) an equipotential surface there can be
no change in V. Hence there can be no component of E tangential to
the surface (as E=-V) and hence the only component of E must be
normal to the surface (important).
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Conclusions
 Electric potential energy (U) and difference (U)
 U and U for two or more point charges
 Relationship between electric force and electric potential energy
 Path independence of electric potential energy
 Electric potential (definition and units)
 Electric potential for a single point charge and multiple point
charges
 Electric potential due to continuous charge distributions
 Relationship between electric potential and E-field E=-V

Equipotential surfaces (relationship to E-field)
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