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Electric Potential Energy

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Electric Potential Energy

Consider a charge $q$ placed in a uniform electric field
${\bf E}$ (e.g., the field between two oppositely charged, parallel
conducting plates). Suppose that we very slowly displace the charge by a vector displacement ${\bf r}$ in a straight-line. How much work must we perform
in order to achieve this? Well, the force ${\bf F}$ we must exert on the charge is
equal and opposite to the electrostatic force $q\,{\bf E}$ experienced by the charge
(i.e., we must overcome the electrostatic force on the charge before we are
free to move it around).
The amount of work $W$ we would perform in displacing the charge is simply the product of the force
${\bf F}=- q\,{\bf E}$ we
exert, and the displacement of the charge in the direction of this force.
Suppose that the displacement vector subtends an angle $\theta$ with
the electric field ${\bf E}$. It follows that

Thus, if we move a positive charge in the direction of the electric field then we
do negative work (i.e., we gain energy). Likewise,
if we move a positive charge in the opposite direction to the electric field
then we do positive work (i.e., we lose energy).

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 ${\bf E}$ 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 $q$ in the field, at point $A$, say,
and
then slowly
move it along some curved path to a different point $B$. How much work must we perform in order to
achieve this? Let us split up the charge’s path from point $A$ to point
$B$ into a series of $N$ straight-line segments, where the $i$th segment
is of length
${\mit\Delta}r_i$ and subtends an angle $\theta_i$ with the
local electric field $E_i$. If we make $N$ sufficiently large then we can
adequately represent any curved path between $A$ and $B$, and we can also ensure
that $E_i$ is approximately uniform along the $i$th path segment. By a simple
generalization of Eq. (76), the work $W$ we must perform in moving
the charge from point $A$ to point $B$ is

Finally, taking the limit in which $N$ goes to infinity, the right-hand side of
the above expression becomes a line integral:

Let us now consider the special case where point $B$ is identical with point
$A$. 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 $E_i$ on the $i$th path segment is unaffected, but,
since the charge is moving along the segment in the opposite direction,

$\theta_i \rightarrow 180^\circ +\theta_i$, and, hence,
$\cos\theta_i\rightarrow -\cos\theta_i$. 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 $A$ and $B$ 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 $A$ and $B$.
Let the work done in taking the charge from $A$ to $B$ along
path 1 be $W_1$, and the work done in taking the charge from $A$ to $B$ along
path 2 be $W_2$. Let us take the charge
from $A$ to $B$ along path 1, and then from $B$ to $A$ along path 2. The net
work done in taking the charge around this closed loop is $W_1-W_2$.
Since we know this work must be zero, it immediately follows that $W_1=W_2$. 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 $q$ very slowly from point $A$ to point $B$
in an electric field generated by fixed charges. The work $W$ which we must perform in order to
achieve this can be calculated
using Eq. (78). Since we lose the energy $W$ as the charge moves from $A$ to
$B$, something must gain this energy. Let us, for the moment, suppose that this
something is the charge. Thus, the charge gains the energy $W$ when
we move it from point $A$ to point $B$. What is the nature of this energy gain?
It certainly is not a gain in kinetic energy, since we are moving the particle
slowly:
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 $A$, the work $W$
done in taking the charge from point $A$ to point $B$ depends only on the
position of point $B$, and not, for instance, on the route taken between
$A$ and $B$. 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 $q$ is taken from point $A$ to point $B$ in an electric field generated by fixed charges its
electric potential energy $P$ increases by an amount $W$:

Here, $P_A$ denotes the electric potential energy of the charge at point $A$,
etc. This definition uniquely defines the difference in the potential energy
between points $A$ and $B$ (since $W$ is independent of the path taken
between these points), but the absolute value of the potential energy
at point $A$ remains arbitrary.

We have seen that when a charged particle is taken from point $A$ to point $B$ 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 $W$ done in taking the
charge from point $A$ to point $B$, 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 $W$. 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 $A$ to point $B$ then the energy of the field increases by an amount $W$
(the work which we perform in moving the particle from $A$ to $B$), but we can
safely associate
this energy increase with the particle because we know that if the particle is
moved back to point $A$ 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.

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Richard Fitzpatrick
2007-07-14

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