Plotting of Potential Fields
Eric Tao
January 28th, 2013
Abstract
Electrical fields caused by a charge will vary based on, theoretically, only the
charge and distance. However, at close enough distances to a surface charge,
the geometry of the charge will start to dominate. This study determines a
number of the electrical equipotential lines around charges with a given
voltage difference, using a voltmeter and power supply. From this, we can
extrapolate the potential lines around most geometric figures.
1
Introduction
From the formula of Coulomb’s Law, we know that the force felt between two
charges is equal to
F
=
q
1
q
2
r
2
. However, if we then consider the force felt on one
of the particles, when its charge
q
2
is
q
1
, then we find that the force exerted
on it goes as
q
2
r
2
. This provides the basis, then, for the concept of an electrical
field. We define the electrical field as
F
e
q
2
, eliminating the arbitrary second
charge from our equation. Thus, we may think of an electrical field as a
measurement of how any charge will behave when placed nearby, due to the
field exerted by our measured charged particle. This can be likened to a
gravitational field, replacing charges with masses. However, this view works
only for a test charge
q
2
sufficiently small as to not create a significant
electrical field of its own; in that case, the two fields would interact, creating a
more complex field arrangement. One method of measuring the field would be
to drop test charges into the field and then observing with what accleration it
moves, and in what direction. This assumes that test charges have a minimal
charge, so as to not disturb the overall integrity of the field. Alternatively, we
may think of an electrical field as having some energy contained within it.
Thus, to move with or against the electrical field will constitute a change in
energy. Then, by plotting the potential fields, we may then extrapolate the
electrical fields, as they must lie perpendicularly. (Mathematically, we may
represent this as
E
∝ ∇
φ
) In that case then, by measuring the lines of equal
potentials, that is, the lines in which it takes no extra work to move along, we
may then find the electrical field lines by finding the perpendiculars. This is