d/Discussion question A.e/Discussion question B.Discussion Questions
A The figure shows a positive charge in the gap between two capacitor
plates. Compare the energy of the electric fields in the two cases. Does
this agree with what you would have expected based on your knowledge
of electrical forces?
B The figure shows a spherical capacitor. In the text, the energy stored
in its electric field is shown to beUe=
k q^2
2(
1
a
−
1
b)
.What happens if the difference betweenbandais very small? Does this
make sense in terms of the mechanical work needed in order to separate
the charges? Does it make sense in terms of the energy stored in the
electric field? Should these two energies be added together?
Similarly, discuss the cases ofb→∞anda→0.
C Criticize the following statement: “A solenoid makes a charge in the
space surrounding it, which dissipates when you release the energy.”
D In example 19 on page 607, I argued that for the charges shown
in the figure, the fields contain less energy when the charges are closer
together, because the region of cancellation expanded, while the region
of reinforcing fields shrank. Perhaps a simpler approach is to consider
the two extreme possibilities: the case where the charges are infinitely
far apart, and the one in which they are at zero distance from each other,
i.e., right on top of each other. Carry out this reasoning for the case of
(1) a positive charge and a negative charge of equal magnitude, (2) two
positive charges of equal magnitude, (3) the gravitational energy of two
equal masses.10.4.2 Gravitational field energy
Example B depended on the close analogy between electric and
gravitational forces. In fact, every argument, proof, and example
discussed so far in this section is equally valid as a gravitational
example, provided we take into account one fact: only positive mass
exists, and the gravitational force between two masses is attractive.
This is the opposite of what happens with electrical forces, which
are repulsive in the case of two positive charges. As a consequence of
this, we need to assign anegativeenergy density to the gravitational
field! For a gravitational field, we have
dUg=−1
8 πG
g^2 dv,whereg^2 =g·gis the square of the magnitude of the gravitational
field.
10.4.3 Magnetic field energy
So far we’ve only touched in passing on the topic of magnetic
fields, which will deal with in detail in chapter 11. Magnetism is
an interaction between moving charge and moving charge, i.e., be-
tween currents and currents. Since a current has a direction inSection 10.4 Energy in fields 609