24.9 - A comparison of electric and gravitational fields
In this section, we compare the uniform electric field
produced by a large negatively charged plate with the
gravitational field created by the Earth. The goal is to
use the gravitational field í one you experience all the
time í to help you better understand the electric field.
In the gravitational field, the test objects in the
illustrations to the right are people: Individuals of
varying masses can walk up or down the stairs of a
building to change their height above the Earth’s
surface. In the electric field, the test objects are
positive test charges that can be farther from or closer
to the charged plate. Their distance from the plate is
indicated by d. In this discussion, the surface of the
plate, and the Earth, are assumed to be points of zero potential energy.
Let’s now use the analogy. First, consider two people on the stairs. One person has a
mass of 50 kg, the other 100 kg. Say they are both on the 12th floor of a building. Each
person’s potential energy is a function of his mass and height above the ground, and
the acceleration due to Earth’s gravitational field. Specific examples of this are
illustrated in the Concept 2 diagram to the right: A 100 kg person on the 12th floor has
twice as much gravitational potential energy as a 50 kg person on the same floor.
An analogous situation exists for a positive test charge above the center of the large
negatively charged plate. The plate attracts the charge because it creates a field with
the same downward orientation as the gravitational field. The electric potential energy of
the charge is a function of its magnitude, its distance from the plate, and the strength of
the field. In the Concept 2 diagram, two charges of different magnitudes are the same
distance from the plate. The stronger charge will have more electric potential energy,
just as the more massive person at a given height has more gravitational potential
energy.
Now we will observe how position relates to potential energy. This is shown in the
Concept 3 diagram. The same person is shown on the 12th and 6th floors. (To simplify
the discussion, we use the European convention that the ground level is the “zeroth”
floor, not the first, so that the 12th floor is twice as far above the ground as the 6th.) The
person’s gravitational potential energy is twice as much on the 12th floor as on the 6th
floor. We also show two test charges of the same magnitude at different distances from
the plate. The test charge at twice the distance from the plate has twice the potential
energy.
Finally let’s consider the role of the field. Potential energy is a function of the strength of
the field-producing object. For example, if we moved the building to Jupiter, the same
people on the same floors would have greater gravitational potential energy than they
do on Earth. Jupiter’s stronger gravitational field would account for the greater
gravitational PE. If the plate had more charge and therefore a stronger field, test
charges at the same positions would have greater electric potential energy. With a
mass like Earth or with an electrically charged plate, you can think of fields as
surrounding these entities, waiting to exert forces on nearby objects.
In both fields, the work done on an object equals its change in potential energy. For
instance, if an elevator picks up a person at the 6th and drops him off at the 12th floor,
the work done by the elevator equals the change in his potential energy. Similarly, if you
exert a force to move a test charge farther from the plate, the work you do equals the
change in the system’s potential energy. (The person and charge are both stationary at
the beginning and end of their journeys, so the work only changes their PEe. Their KE
at the beginning and end is zero.)
Now consider the concept of gravitational potential. As illustrated in the Concept 4
diagram, gravitational potential depends only on height above the Earth’s surface. The
gravitational potential is twice as great on the 12th floor as it is on the 6th floor.
We can also compare positions 12 floors above the ground on Jupiter and on the Earth,
two planets of different masses. The gravitational potential 12 floors above the ground
is greater on Jupiter than it is on the Earth. To put it another way, any object you place
on the 12th floor on Jupiter will have more potential energy than the same object would
on the 12th floor on the Earth. This is because Jupiter’s gravitational field is stronger
than the Earth’s. The important distinction between potential energy and potential is that
the potential is independent of the test object. Jupiter’s mass generates more
gravitational potential at a given height above its surface than the Earth’s mass does.
The concept of gravitational potential is analogous to electric potential. You see this also in the Concept 4 diagram. A location one centimeter
above the plate has twice the electric potential of a location only half a centimeter above the plate. The distance above the plate determines thePoints in space around Jupiter have more potential in its gravitational field
than would points the same distance from the Earth in its gravitational field.Electric and gravitational fields
Compare electric and gravitational
effects
·Use charged plate, tall building
·Consider distance
·Use test object’s charge or massPotential energy
Depends on charge or mass
Potential energy
Also depends on location of test object
(^446) Copyright 2000-2007 Kinetic Books Co. Chapter 24