Consider a set of point charges,
distributed in space, which are rigidly clamped in position so that they cannot
move. We already know how to calculate the electric field generated by such a
charge distribution (see Sect. 3). In general, this electric
field is going to be
non-uniform. Suppose that we place a charge in the field, at point , say,
move it along some curved path to a different point . How much work must we perform in order to
achieve this? Let us split up the charge's path from point to point
into a series of straight-line segments, where the th segment
is of length
and subtends an angle with the
local electric field . If we make sufficiently large then we can
adequately represent any curved path between and , and we can also ensure
that is approximately uniform along the th path segment. By a simple
generalization of Eq. (76), the work we must perform in moving
the charge from point to point is
Let us now consider the special case where point is identical with point . In other words, the case in which we move the charge around a closed loop in the electric field. How much work must we perform in order to achieve this? It is, in fact, possible to prove, using rather high-powered mathematics, that the net work performed when a charge is moved around a closed loop in an electric field generated by fixed charges is zero. However, we do not need to be mathematical geniuses to appreciate that this is a sensible result. Suppose, for the sake of argument, that the net work performed when we take a charge around some closed loop in an electric field is non-zero. In other words, we lose energy every time we take the charge around the loop in one direction, but gain energy every time we take the charge around the loop in the opposite direction. This follows from Eq. (77), because when we switch the direction of circulation around the loop the electric field on the th path segment is unaffected, but, since the charge is moving along the segment in the opposite direction, , and, hence, . Let us choose to move the charge around the loop in the direction in which we gain energy. So, we move the charge once around the loop, and we gain a certain amount of energy in the process. Where does this energy come from? Let us consider the possibilities. Maybe the electric field of the movable charge does negative work on the fixed charges, so that the latter charges lose energy in order to compensate for the energy which we gain? But, the fixed charges cannot move, and so it is impossible to do work on them. Maybe the electric field loses energy in order to compensate for the energy which we gain? (Recall, from the previous section, that there is an energy associated with an electric field which fills space). But, all of the charges (i.e., the fixed charges and the movable charge) are in the same position before and after we take the movable charge around the loop, and so the electric field is the same before and after (since, by Coulomb's law, the electric field only depends on the positions and magnitudes of the charges), and, hence, the energy of the field must be the same before and after. Thus, we have a situation in which we take a charge around a closed loop in an electric field, and gain energy in the process, but nothing loses energy. In other words, the energy appears out of ``thin air,'' which clearly violates the first law of thermodynamics. The only way in which we can avoid this absurd conclusion is if we adopt the following rule:
The work done in taking a charge around a closed loop in an electric field generated by fixed charges is zero.
One corollary of the above rule is that the work done in moving a charge between two points and in such an electric field is independent of the path taken between these points. This is easily proved. Consider two different paths, 1 and 2, between points and . Let the work done in taking the charge from to along path 1 be , and the work done in taking the charge from to along path 2 be . Let us take the charge from to along path 1, and then from to along path 2. The net work done in taking the charge around this closed loop is . Since we know this work must be zero, it immediately follows that . Thus, we have a new rule:
The work done in taking a charge between two points in an electric field generated by fixed charges is independent of the path taken between the points.
A force which has the special property that the work done in overcoming it in order to move a body between two points in space is independent of the path taken between these points is called a conservative force. The electrostatic force between stationary charges is clearly a conservative force. Another example of a conservative force is the force of gravity (the work done in lifting a mass only depends on the difference in height between the beginning and end points, and not on the path taken between these points). Friction is an obvious example of a non-conservative force.
Suppose that we move a charge very slowly from point to point
in an electric field generated by fixed charges. The work which we must perform in order to
achieve this can be calculated
using Eq. (78). Since we lose the energy as the charge moves from to
, something must gain this energy. Let us, for the moment, suppose that this
something is the charge. Thus, the charge gains the energy when
we move it from point to point . What is the nature of this energy gain?
It certainly is not a gain in kinetic energy, since we are moving the particle
i.e., such that it always possesses negligible kinetic energy.
In fact, if we think carefully, we can see that the gain in energy of the
charge depends only on its position. For a fixed starting point , the work
done in taking the charge from point to point depends only on the
position of point , and not, for instance, on the route taken between
and . We usually call energy a body possess by virtue of its position
potential energy: e.g., a mass has a certain gravitational potential energy
which depends on its height above the ground. Thus, we can say that when
a charge is taken from point to point in an electric field generated by fixed charges its
electric potential energy increases by an amount :
We have seen that when a charged particle is taken from point to point in an electric field its electric potential energy increases by the amount specified in Eq. (79). But, how does the particle store this energy? In fact, the particle does not store the energy at all. Instead, the energy is stored in the electric field surrounding the particle. It is possible to calculate this increase in the energy of the field directly (once we know the formula which links the energy density of an electric field to the magnitude of the field), but it is a very tedious calculation. It is far easier to calculate the work done in taking the charge from point to point , via Eq. (78), and then use the conservation of energy to conclude that the energy of the electric field must have increased by an amount . The fact that we conventionally ascribe this energy increase to the particle, rather than the field, via the concept of electric potential energy, does not matter for all practical purposes. For instance, we call the money which we have in the bank ``ours,'' despite the fact that the bank has possession of it, because we know that the bank will return the money to us any time we ask them. Likewise, when we move a charged particle in an electric field from point to point then the energy of the field increases by an amount (the work which we perform in moving the particle from to ), but we can safely associate this energy increase with the particle because we know that if the particle is moved back to point then the field will give all of the energy back to the particle without loss. Incidentally, we can be sure that the field returns the energy to the particle without loss because if there were any loss then this would imply that non-zero work is done in taking a charged particle around a closed loop in an electric field generated by fixed charges. We call a force-field which stores energy without loss a conservative field. Thus, an electric field, or rather an electrostatic field (i.e., an electric field generated by stationary charges), is conservative. It should be clear, from the above discussion, that the concept of potential energy is only meaningful if the field which generates the force in question is conservative.
A gravitational field is another example of a conservative field. It turns out that when we lift a body through a certain height the increase in gravitational potential energy of the body is actually stored in the surrounding gravitational field (i.e., in the distortions of space-time around the body). It is possible to determine the increase in energy of the gravitational field directly, but it is a very difficult calculation involving General Relativity. On the other hand, it is very easy to calculate the work done in lifting the body. Thus, it is convenient to calculate the increase in the energy of the field from the work done, and then to ascribe this energy increase to the body, via the concept of gravitational potential energy.
In conclusion, we can evaluate the increase in electric potential energy of a charge when it is taken between two different points in an electrostatic field from the work done in moving the charge between these two points. The energy is actually stored in the electric field surrounding the charge, but we can safely ascribe this energy to the charge, because we know that the field stores the energy without loss, and will return the energy to the charge whenever it is required to do so by the laws of Physics.