PHY 142!
Assignment 1!
Summer 2013
Reading: Electrostatics 1.
Key concepts: Electric charge, conservation of charge, Coulomb’s law, definition of E, force on
charge in E-field, electric dipoles.
1.!
2.!
3.!
4.!
Calculations showing the strengths of the various fundamental forces.
a.!
Two identical small objects of mass 10 g and charge 1 µC are 1 m apart.
What is the ratio of the Coulomb repulsion to the gravitational attraction?
b.!
Consider two protons in a nucleus, at distance 1 fm ( = 10−15 m) apart.
What is the Coulomb force of repulsion between them? [The “strong”
force that binds the nucleus must be stronger than this.]
Two positive point charges are a distance r apart. The total charge of the pair is
Q. If each charge is Q/2, the force of repulsion is F0 .
a.!
Show that dividing the charge unequally gives a force smaller than F0 .
b.!
What are the two charges if the force is
3
F ?
4 0
Two positive point charges of magnitude Q are held in place on the x-axis at
x = 10 cm and x = −10 cm. A small bead carrying charge –q is threaded on a nonconducting fiber that runs along the y-axis, constraining the bead to move only
along that axis. Initially the bead is at rest at y = 0 .
a.!
Show that the origin is a point of stable equilibrium for the bead, i.e., that
if the bead is displaced slightly the forces on it will bring it back toward
the origin. (No calculation needed, only an argument based on directions.)
b.!
Now suppose the fiber runs along the x-axis instead. Is the origin still a
point of stable equilibrium for the bead? Explain.
Two point charges, q and Q, are separated by distance  . The magnitudes obey
q = 4 Q . We are interested in the total E-field at points on the line passing
through the charges.
a.!
If q and Q have the same sign, is there a point on that line and between
them where E = 0 ? If so, how far is it from q?
b.!
Is there also a point on that line but not between the charges where E = 0 ?
If so, where is it in relation to q?
c.!
Answer the same two questions if q and Q have opposite signs.!
PHY 142!
5.!
Assignment 1!
A positive point charge and a small electric dipole
are arranged as shown.
a.!
Summer 2013
−q
Q
Find the exact formula for the total force
exerted on the dipole by the charge.
x
+q
d
!
⎛ 1
1 ⎞
Ans: F = kQq ⎜ 2 −
⎟ , toward Q.
⎝x
(x + d)2 ⎠
b.!
Derive an approximate formula for d << x . You should find that the force
varies as x −3 . [Use the binomial approximation.]
c.!
6*.!
What would be different if the dipole were oriented the other way?
An electron (charge −e , mass m) enters the region between
the two oppositely charged plates shown. The field E
θ
between the plates is vertical and uniform in the
•
space between the plates. The electron enters with
velocity v at the angle θ shown, at the midpoint
d
between the plates. When it reaches the top plate its velocity is horizontal.
a.!
Which plate is positively charged?
b.!
What is θ in terms of the given quantities? Ans: sin θ =
1 eEd
.
v m
Reading: Electrostatics 2
Key concepts: Field equations for E, properties of conductors in electrostatics, field lines.
7.!
A thin rod of length  carries charge +Q , spread
uniformly. Point a is at distance r above the middle;
point b is at distance r from the right end.
a•
r
Q

r
a.!
What are the directions of the E-fields at a and
at b. How do you know?
b.!
If r >>  , what is the formula for the E-field magnitude at each point?
•
b
PHY 142!
Assignment 1!
Summer 2013
8.!
A positively charged rod is brought toward a small ball at rest, hanging
suspended by a thread; the ball swings toward the rod. What, if anything, can
you definitely conclude about the net charge on the ball just from this
information? What are the possibilities?
9.!
Questions about the field equations and field lines.
10*.!
a.!
Argue from Gauss’s law that lines of the electrostatic field start on positive
charges and end on negative charges.
b.!
What general property of the electrostatic field forbids the field lines to
close on themselves? Give the argument.
Two large conducting plates arranged as shown
carry uniformly distributed charges. An area A of
the positive plate has charge +σ ⋅ A and the
+
−
corresponding area of the negative plate has charge −σ ⋅ A .
a.!
Where does the charge lie on the positive plate: equally on both surfaces,
entirely on one surface (which?), or uniformly spread throughout the
plate? Explain how you know. Repeat for the negative plate.
b.!
What is the field just above and below both plates? How do you know?
c.!
What is the E-field (magnitude and direction) in the space between the
plates? How do you know? [Use Gauss’s law or another argument.]
d.!
What is the field within the material of the plates? How do you know?
11.!
Let the plates in #10 be non-conductors of thickness t with charge distributed
uniformly throughout them. For the positive plate find E in terms of the distance
x from the top surface. Ans: E(x) = (σ / ε 0 ) ⋅ (x /t) .
12.!
Spherical symmetry and Gauss’s law.
a.!
A non-conducting sphere of radius R has charge Q distributed uniformly
through its volume. Use Gauss’s law to find the magnitude of E at
distance r from the sphere’s center for r > R . Ans: E = kQ/r 2 .
b.!
Repeat for r < R . Ans: E = kQr /R 3 .
c.!
Let this sphere be surrounded by an uncharged conducting concentric
shell, of inner radius R1 and outer radius R2 (both > R ). Use Gauss’s law
to find the charge on the inner surface at r = R1 . Ans: –Q.
d.!
For this combination, plot E vs. r from r = 0 to r = R2 + h , where h is a
small distance beyond outer surface of the shell.
PHY 142!
13.!
Assignment 1!
Summer 2013
An isolated atom can be modeled as consisting of a positive point charge Ze
representing the nucleus, surrounded by a sphere of negative charge
representing the electrons, with total charge −Ze . For simplicity assume the
electron “cloud” charge is uniformly distributed through a sphere of radius a.
a.!
Use #12(b) to show that in the absence of forces from outside the atom the
equilibrium position of the nucleus is at the center of the electron cloud.
b.!
Now suppose there is a uniform external field E0 (to the right) that affects
the atom. Draw a picture to show where the new equilibrium position of
the nucleus is relative to the electron cloud’s sphere. [How is it displaced?]
c.!
14.!
Show that the at points outside the atom the E-field created by the atom is
now that of an electric dipole, with moment parallel to E0 .
A dipole of moment p is in an external field E0 . If the field is uniform the total
force on the dipole is zero because the forces on the dipole charges cancel.
a.!
Show that if p is parallel to E0 there is no torque on the dipole, so the
equilibrium is stable in terms of both translation and rotation.
b.!
If p is parallel to E0 but the field is not uniform, show that the net force on
the dipole is toward the region where E0 has greater magnitude.
15*.!
A small non-conducting spherical ball is
suspended inside a larger conducting
spherical shell of inner radius R. The small
ball carries uniformly distributed positive
charge +Q; the total charge on the shell is
zero. The small sphere is closer to the right
side of the shell as shown. The dots indicate
equally spaced points on the surface of the
small sphere.
r
a.!
On the figure, draw eight lines of the
P
E-field that begin at the points
•
represented by the dots. Show clearly
how they leave the surface of the small sphere and how they intersect the
inner surface of the shell.
b.!
Give an argument using Gauss’s law to determine the total charge on the
inner surface of the shell.
c.!
Consider point P outside the shell at distance r from its center as shown.
What is the E-field (magnitude and direction) at P? How do you know?