Chapter 32
Calculating the Electric Field from the Electric Potential
284
32
Calculating the Electric Field from the Electric Potential
The plan here is to develop a relation between the electric field and the corresponding electric
potential that allows you to calculate the electric field from the electric potential.
The electric field is the forcepercharge associated with empty points in space that have a force
percharge because they are in the vicinity of a source charge or some source charges.
The
electric potential is the potential energypercharge associated with the same empty points in
space.
Since the electric field is the forcepercharge, and the electric potential is the potential
energypercharge, the relation between the electric field and its potential is essentially a special
case of the relation between any force and its associated potential energy.
So, I’m going to start
by developing the more general relation between a force and its potential energy, and then move
on to the special case in which the force is the electric field times the charge of the victim and the
potential energy is the electric potential times the charge of the victim.
The idea behind potential energy was that it represented an easy way of getting the work done by
a force on a particle that moves from point A to point B under the influence of the force.
By
definition, the work done is the force along the path times the length of the path.
If the force
along the path varies along the path, then we take the force along the path at a particular point on
the path, times the length of an infinitesimal segment of the path at that point, and repeat, for
every infinitesimal segment of the path, adding the results as we go along.
The final sum is the
work.
The potential energy idea represents the assignment of a value of potential energy to
every point in space so that, rather than do the path integral just discussed, we simply subtract
the value of the potential energy at point A from the value of the potential energy at point B.
This gives us the change in the potential energy experienced by the particle in moving from point
A to point B.
Then, the work done is the negative of the change in potential energy.
For this to
be the case, the assignment of values of potential energy values to points in space must be done
just right.
For things to work out on a macroscopic level, we must ensure that they are correct at
an infinitesimal level.
We can do this by setting:
Work as Change in Potential Energy = Work as ForceAlongPath times Path Length
s
d
F
harpoonrightnosp
harpoonrightnosp
⋅
=
−
dU
where:
dU
is an infinitesimal change in potential energy,
F
harpoonrightnosp
is a force, and
s
d
harpoonrightnosp
is the infinitesimal displacementalongthepath vector.
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 Fall '08
 RABE
 Physics, Electric Potential, Potential Energy, Electric charge, Fundamental physics concepts

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