MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2007
Problem Set 3 Solutions
Problem 1: Charges on a Square
Three identical charges +
Q
are placed on the corners of a square of side
a
, as shown in
the figure.
(a) What is the electric field at the fourth corner (the one missing a charge) due to the first
three charges?
We’ll just use superposition:
()
3
2
3
33
2
00
ˆˆ
12
44
2
Qa
a
a
a
Q
aa
a
a
πε
−
⎛⎞
+
⎜⎟
=+
+
+
⎝⎠
ii
jj
Ei
G
j
(b) What is the electric potential at that corner?
A common mistake in doing this kind of problem is to try to integrate the
E
field we just
found to obtain the potential.
Of course, we can’t do that we only found the
E
field at a
single point, not as a function of position.
Instead, just sum the point charge potentials
from the 3 points:
0
11
2
4
22
i
ij
ij
q
QQQ
Q
V
ra
a
a
a
≠
⎛
==
+
+
=
+
⎜
⎝
∑
1
⎞
⎟
⎠
(c) How much work does it take to bring another charge, +
Q
, from infinity and place it at
that corner?
The work required to bring a charge +
Q
from infinity (where the potential is 0) to the
corner is:
2
0
1
2
4
2
Q
WQV
a
=Δ=
+
(d) How much energy did it take to assemble the pictured configuration of three charges?
The work done to assemble three charges as pictured is the same as the potential energy
of the three charges already in such an arrangement. Now, there are two pairs of charges
situated at a distance of
a
, and one pair of charges situated at a distance of
2
a
, thus we
have
PS03 Solution  1
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View Full DocumentProblem 1: Charges on a Square
continued…
22
2
00
0
11
1
21
2
44
4
2
QQ
Q
W
aa
a
πε
⎛⎞
⎛
⎞
=+
=
+
⎜⎟
⎜
⎟
⎝⎠
⎝
⎠
2
2
Alternatively we could have started with empty space, brought in the first charge for free,
the second charge in the potential of the first and so forth.
We’ll get the same answer.
Problem 2:
Electric Potential
Suppose an electrostatic potential has a maximum at point P and a minimum at point M.
(a) Are either (or both) of these points equilibrium points for a negative charge?
If so are
they stable?
The electric field is the gradient of the potential, which is zero at both potential minima
and maxima.
So a negative charge is in equilibrium (feels no net force) at both P & M.
However, only the maximum (P) is stable.
If displaced slightly from P, a negative charge
will roll back “up” hill, back to P.
If displaced from M a negative charge will roll away
from the potential minimum.
(b) Are either (or both) of these points equilibrium points for a positive charge?
If so are
they stable?
Similarly, both P & M are equilibria for positive charges, but only M is a stable
equilibrium because positive charges seek low potential (this is probably the case that
seems more logical since it is like balls on mountains).
Problem 3: A potential for finding charge
The electric potential
V(r)
for a distribution of charge
with spherical symmetry is:
2
0
0
2
Region I:
( )=
for
0
2
Region II:
( )
for
2
2
R egion III:
( )
0
for
2
Vr
V
r R
R
VR V
R
r
R
r
r
R
−+
<
<
=−
<
<
=>
Where
V
0
is a constant of appropriate units. This function is plotted above.
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 Fall '07
 Hudson
 Charge, Electrostatics, Electric charge

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