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Electromagnetism
Topic 11.1 Electrostatic Potential
Electric Potential due to a Point
Charge
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The electric potential at a point in an electric field
is defined as being numerically equal to the work
done in bringing a unit positive charge from
infinity to the point.
Electric potential is a scalar quantity and it
has the volt V as its unit.
Based on this definition, the potential at infinity
is zero.
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Let us take a point r metres from a charged
object. The potential at this point can be
calculated using the following
Electric Field Strength and
Potential
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Suppose that the
charge +q is
moved a small
distance by a
force F from A to
B so that the force
can be
considered
constant.
The work done is given by:
 ΔW = Fx Δx
 The force F and the electric field E are
oppositely directed, and we know that:
 F = -q x E
 Therefore, the work done can be given as:
 ΔW = -qE x Δ x = qV
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Therefore E = - ΔV / Δx
 This is the potential gradient.
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Electric Field and Potential due to
a charged sphere
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When the sphere becomes charged, we know that the
charge distributes itself evenly over the surface.
Therefore every part of the material of the conductor is
at the same potential.
As the electric potential at a point is defined as being
numerically equal to the work done in bringing a unit
positive charge from infinity to that point, it has a
constant value in every part of the material of the
conductor,
Since the potential is the same at all points on the
conducting surface, then Δ V / Δx is zero. But E = - Δ V
/ Δ x.
Therefore, the electric field inside the conductor is
zero. There is no electric field inside the conductor.
Equipotentials
Regions in space where the electric
potential of a charge distribution has a
constant value are called equipotentials.
 The places where the potential is constant
in three dimensions are called
equipotential surfaces, and where they
are constant in two dimensions they are
called equipotential lines.
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They are in some ways analogous to the contour
lines on topographic maps.
Similar also to gravitational potential.
In this case, the gravitational potential energy is
constant as a mass moves around the contour
lines because the mass remains at the same
elevation above the earth's surface.
The gravitational field strength acts in a direction
perpendicular to a contour line.
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Similarly, because the electric potential on an
equipotential line has the same value, no work
can be done by an electric force when a test
charge moves on an equipotential.
Therefore, the electric field cannot have a
component along an equipotential, and thus it
must be everywhere perpendicular to the
equipotential surface or equipotential line.}
This fact makes it easy to plot equipotentials if
the lines of force or lines of electric flux of an
electric field are known.
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For example, there are a series of equipotential
lines between two parallel plate conductors that
are perpendicular to the electric field.
There will be a series of concentric circles that
map out the equipotentials around an isolated
positive sphere.
The lines of force and some equipotential lines
for an isolated positive sphere are shown in the
next figures.
Analogies exist between electric
and gravitational fields.
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(a) Inverse square law of force
Coulomb's law is similar in form to Newton's law of universal
gravitation.
Both are inverse square laws with 1/(4πε) in the electric case
corresponding to the gravitational constant G.
The main difference is that whilst electric forces can be
attractive or repulsive, gravitational forces are always attractive.
Two types of electric charge are known but there is only one
type of gravitational mass.
By comparison with electric forces, gravitational forces are
extremely weak.
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(b) Field strength
The field strength at a point in a gravitational field is
defined as the force acting per unit mass placed at
the point.
Thus if a mass m in kilograms experiences a force F
in newtons at a certain point in the earth's field, the
strength of the field at that point will be F/m in
newtons per kilogram.
This is also the acceleration a the mass would have
in metres per second squared if it fell freely under
gravity at this point (since F = ma).
The gravitational field strength and the acceleration
due to gravity at a point thus have the same value
(i.e. F/m) and the same symbol, g, is used for both. At
the earth's surface g = 9.8 N kg-' = 9.8 m s-2
(vertically downwards).
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(c) Field lines and equipotentials
These can also be drawn to represent
gravitational fields but such fields are so
weak, even near massive bodies, that there
is no method of plotting field lines similar to
those used for electric (and magnetic) fields.
Field lines for the earth are directed towards
its centre and the field is spherically
symmetrical.
Over a small part of the earth's surface the
field can be considered uniform, the lines
being vertical, parallel and evenly spaced.
(d) Potential and p.d.
 Electric potentials and pds are
measured in joules per coulomb (J
C-1) or volts;
 gravitational potentials and pds are
measured in joules per kilogram (J
kg-1).
 As a mass moves away from the earth
the potential energy of the earth-mass
system increases, transfer of energy
from some other source being
necessary.
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If infinity is taken as the zero of gravitational
potential (i.e. a point well out in space where
no more energy is needed for the mass to
move further away from the earth)
then the potential energy of the system will
have a negative value except when the
mass is at infinity.
At every point in the earth's field the
potential is therefore negative (see
expression below), a fact which is
characteristic of fields that exert attractive
forces.