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Consider a charge placed in an electric field generated by fixed charges.
Let us chose some arbitrary reference point in the field.
At this point, the electric
potential energy of the charge is defined to be zero. This
uniquely specifies the
electric potential energy of the charge at every other point in the field.
For instance, the electric potential energy at some point is
simply the work done in moving the charge from to along any path. Now,
can be calculated using Eq. (78). It is clear, from this equation, that
depends both on the particular charge which we place in the
field, and the magnitude and direction of the electric field along the chosen route between points and . However, it is also clear that
is directly proportional to the magnitude of the
charge . Thus, if the electric
potential energy of a charge at point is then the electric potential
energy of a charge at the same point is . We can exploit
this fact to define a quantity known as the electric potential. The
difference in electric potential between two points and in an electric
field is simply the work done in moving some charge between the two points
divided by the magnitude of the charge. Thus,

(80) 
where denotes the electric potential at point , etc. This definition
uniquely defines the difference in electric potential between points and
, but the absolute value of the potential at point remains arbitrary.
We can therefore, without loss of generality, set the potential at point equal to
zero. It follows that the potential energy of a charge at some point
is simply the product of the magnitude of the charge and the electric
potential at that point:

(81) 
It is clear, from a comparison of Eqs. (78) and (79), that the electric
potential at point (relative to point ) is solely a property of the
electric field, and is, therefore, the same for any charge placed at that point.
We shall see exactly how the electric potential is related to the electric
field later on.
The dimensions of electric potential are work (or energy) per unit charge.
The units of electric potential are, therefore, joules per coulomb (
).
A joule per coulomb is usually referred to as a volt (V): i.e.,

(82) 
Thus, the
alternative (and more conventional) units of electric potential are volts.
The difference in electric potential between two points in an electric field
is usually referred to as the potential difference, or even the difference
in ``voltage,''
between the two points.
A battery is a convenient tool for generating a difference in electric potential
between two points in space. For instance, a twelve volt (12V) battery
generates an electric field, usually via some chemical process, which is such
that the potential difference between its positive and negative
terminals is twelve volts. This means that in order to move a
positive charge of 1 coulomb from the negative to the positive terminal of
the battery we must do 12 joules of work against the electric field. (This is
true irrespective of the route taken between the two terminals).
This implies that the electric field must be directed predominately from the
positive to the negative terminal.
More generally, in order to move a charge through a potential difference we must do work
, and the electric potential energy
of the charge increases by an amount
in the
process. Thus, if we move an electron, for which
C,
through a potential difference of minus 1 volt then we must do
joules of work. This amount of work (or energy) is
called an electronvolt (eV): i.e.,

(83) 
The electronvolt is a convenient measure of energy in atomic physics.
For instance, the energy required to
break up a hydrogen atom into a free electron and a free proton is eV.
Next: Electric Potential and Electric
Up: Electric Potential
Previous: Electric Potential Energy
Richard Fitzpatrick
20070714