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PH504 – Part 3
Electric potential, potential energy.
1. 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?
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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:
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.
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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
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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.
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Relationship between U and electric force
If the electric force is non-zero along one axis only (e.g. Fx) then
.
More generally in three dimensions
In words 'the electric force is equal to the negative of the
gradient of the potential energy'.
2. 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 Efield and electric force.
The units of electric potential are J C-1 V (Volt)
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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
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.
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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.
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.
3. 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.
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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).
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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