# Classical Meaning of Potential

Put simply, the potential is just the energy per unit **charge** which a test particle of negligible **charge** will
possess *due to its position in a particular* **field**. Two points should be noted concerning that last statement. First, I am using the word *charge* in its widest sense; for example, if one wants to
describe a falling ball, then the *field* in question is the gravitational field
and the corresponding *charge* is the ball's *mass*. In an
electric field the *charge* would be the familiar electromagnetic charge.
Secondly, I will restrict this
discussion to **conservative fields** where the work done in moving the test charge from one particular position to another is
independant of the path taken and of the velocity of the
test charge.

Let us consider a simple example. Since it requires energy to lift an object through a gravitational field it
follows that the potential of an object in the Earth's gravitational field
must increase with height. In fact, the gravitational potential close to the
Earth's surface is directly proportional to the height above any arbitrary
reference position.

Now, it is a fundamental law of nature that systems
always tend towards their lowest potential energy configuration, within
the constraint of conservation of energy.
This means that an object in any given
field will always feel a force in the direction of maximum decrease of the
potential. The magnitude of this force will be directly proportional to the
rate of decrease of potential in this direction. Thus a ball close to the
surface of the Earth will always experience a constant gravitational force
directly downwards (downwards because this is the direction of maximum
potential
decrease, and constant because the rate of decrease is constant). Electrical and other fields
work in just the same way, except they act on *charges* other than mass.
That is, in order to describe the motion of a classical particle it is only
necessary to know the potential in which it moves and its *charge*. Forces can be simply
worked out from the rate of change of the potential with position.

The
potential
is convenient for describing the environment of a quantum mechanical
particle because it allows us to assign a simple scalar quantity
to each point in space, rather than keeping track of vector forces acting
on a particle whose position may not be precisely known.

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Stephen Jenkins
Last modified: Fri Sep 13 14:53:09 BST 1996