Electrical potential energy of point charges
One point charge has a potential energy relative to another one, since we know that, if we
let one of the charges go, it will move; that is, it will gain kinetic energy. Imagine a very
massive collection of negative charges. If a positive charge some distance away is released,
it will move toward the negative charge. In this case, its electrical potential energy is
converted into kinetic energy. Were it instead a negative charge that was some distance
away, our negative charge would flee the vicinity upon release. Its potential energy, too,
is converted into kinetic energy.
If we imagine a pair of charges to have an electrical potential energy of 100 joules with
both charges fixed in place, we may get 100 J of kinetic energy from the charge we release.
If the charge that moves does work as it moves from a position with potential energy of
100 J to one with potential energy of 50 J, then 50 J of work may be obtained.
Remember, in determining the net final kinetic energy or the net work done, only the
difference between the original value of the potential energy and the final value of the
potential energy is important.
Since only differences in potential energy are of any interest, we may set the “scale” of
this potential energy any way we want. This means we can conspire together to set the
electrical potential energy to zero (or any other desired value) anywhere we please. For
collections of point charges, it is usual to take an infinite separation of charges to
correspond to zero electrical potential energy. This is a reasonable choice since the
charges should not know of one another’s existence if they are infinitely separated.
Once this has been done, there is an electrical potential energy difference between each
point in space and the point infinitely far away. Each point can then be said to have a
Energy, Ch. 4, extension 2 Electrical potential energy of point charges
particular electrical potential energy. As before, only differences are important, but we
have set the scale by making this decision. With this choice, we may write for the
electrical potential energy UE of two point charges of size q1 and q2 whose centers are
separated by a distance R as
UE = (9 x 109 N m2 /C2 )
q1 q2
.
R
Work is done when objects having forces exerted on them move. If we have a charge at
some position, it takes work to move another charge of the same sign (positive with
positive, or negative with negative) in from infinity. To move a charge of opposite sign
(negative with positive) in from infinity would liberate work. The work required or
liberated can, in principle, be calculated from the basic definition of work.
By doing work on charges by moving them in a region of electric field, we can build up
electrical potential energy. By letting them move together or apart, we may get the work
back.
How big is UE? In principle, it can be any value. Let’s consider two examples. We will
take for the first example two pith balls, each carrying charge 0.05 C and separated by 0.5
m. The second example will be the potential energy of the hydrogen atom. In the first
case, we have
UE = (9 x 109 N m2 /C2 )
[0.05 C][0.05 C]
0.5 m
= (9 x 109 N m2 /C2 ) x [0.0025 C2]/0.50 m = 4.5 x 107 J = 45 MJ.
In the second case, the potential energy between these tiny charges, each one of size 1.6 x
10-19 C but opposite in sign, separated by a distance of 5.29 x 10-11 m is
UE = (9 x 109 N m2 /C2 )
[1.6 x 10-19 C][-1.6 x 10-19 C]
5.29 x 10-11 m
= -(9 x 109 N m2 /C2 ) x [2.56 x 10-38 C2]/5.29 x 10-11 m
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Energy, Ch. 4, extension 2 Electrical potential energy of point charges
= -4.36 x 10-18 J = -4.36 aJ.
Clearly, the pith balls have a much greater-sized potential energy than the hydrogen atom.
If we let them convert all their electrical potential energy to kinetic energy, though, the
speed of the electron (say) would be much, much greater than that of the pith ball because
the electron’s mass is so small, 9.11 x 10-31 kg, in comparison to a typical pith ball mass
of 25 g.
Consider that minus sign. The negative potential energy, which we can (but will not)
show to be twice as great as the positive kinetic energy, means that the total energy of the
hydrogen atom is -2.18 aJ; since the total energy is negative, the hydrogen atom is a
bound system. The constituents have given up energy to form this bound state, so this
energy will have to be paid back to separate them. The two pith balls, with total energy
positive, will produce kinetic energy if released.
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