Download Week 2 – Continuous charge have a lot of

Week 2 – Continuous charge have a lot of potential
The world is continuous, but the mind is
discrete.
David Mumford
Exercise 2.1: Sticky Electricity
If you peel two strips of transparent tape off the same roll and immediately let them hang near each
other, they will repel each other. If you then put the sticky side of one to the shiny side of the other
and rip them apart, they will attract each other. Discuss with your fellow students and come up with a
plausible explanation for this behaviour.
Exercise 2.2: Semicircular charge distribution
Consider a semicircle of radius a lying in the first and second quadrant in the xy-plane with its center
at the origin as shown in figure 1. A charge Q is smeared out uniformly over the semicircle, giving rise
to a constant charge density λ(θ) = Q/(πa).
a) Find the magnitude and direction of the electric field at the origin.
Answer: E = − 2π2Qε0 a2 ĵ
1
Figure 1: Semicircle with charge smeared out on it. What is the electric field at the origin?
b) Suppose now that we spread the charge Q such that the distribution becomes non-uniform and varies
with θ like λ(θ) = (Q/2a) sin θ. Sketch a plot of the charge distribution and confirm that the total
charge density on the semicircle is indeed Q.
c) Find the magnitude and direction of the electric field in the origin with the new charge distribution.
Which distribution has the greatest magnitude and why? Why does the electric field in both cases
point in the same direction?
Answer: E =
Q
16ε0 a2 ĵ
Exercise 2.3: Discussion Questions
a) Imagine a certain region of space bound by a surface which contains no charge. Is the electric field
always zero everywhere on the surface? If not, in what circumstances is it zero on the surface?
b) You find a sealed box on your doorstep. You are suspicious that it might contain some sort of
charged material. Is it possible to determine whether or not it contains charge without opening it?
If so, explain how.
c) Some modern aircrafts are made primarily of composite materials that do not conduct electricity. The
U.S. Federal Aviation Administration requires that such aircraft have conducting wires embedded on
the surface to provide protection when flying through thunderstorms. Explain the physics behind
this requirement.
Figure 2: A charged sphere hanging close to a charged sheet. What is the angle of the thread?
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Exercise 2.4: A charge from a thread
A small sphere with a mass of 0.002 g carrying a charge of 500 µC hangs from a thread near a very large,
conducting sheet, as shown in figure 2. The sheet is charged with a charge density of 2.5 nC/m2 . Find
the angle of the thread.
Answer:
θ = 89.99◦
Exercise 2.5: The Coaxial Cable
A long coaxial cable consists of an inner cylindrical conductor with radius a and an outer coaxial,
conducting cylinder with inner radius b and outer radius c. The outer cylinder is mounted on insulating
supports and has no net charge. The inner cyldinder has a uniform positive charge per unit length λ.
a) Calculate the electric field at any point between the cylinders a distance r from its axis and at any
point outside the outer cylinder. I.e. for a ≤ r ≤ b and r ≥ c. What is the value of the field inside
the cylindrical conductors?
Answer:
E=
λ
r̂
2πε0 r
E=
for a ≤ r ≤ b.
(1)
for r ≥ c.
(2)
λ
r̂
2πε0 r
and inside the conductors E = 0.
b) Graph the magnitude of the electric field as a function of the distance r from the axis of the cable,
from r = 0 to r = 2c.
c) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.
Answer: Inner conductor has +λ distributed on its surface. Outer conductor has −λ on the inner
surface. Outer conductor has +λ distributed on the outer surface.
Exercise 2.6: A nonuniform charge distribution
A nonuniform, but spherically symmetric distribution of charge is specified as
ρ(r)
= ρ0 (1 − r/R)
for r ≤ R
ρ(r)
=
for r ≥ R
0
(3)
where ρ0 = 3Q/πR3 is a positive constant.
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a) Find the electric field in both the region r ≤ R and the region r ≥ R. Check that they agree at r = R
Does your result for r ≥ R agree with the statement that an extended spherically symmetric charge
distribution can be treated as though all its charge were located at its center?
Answer:
E=
and
E=
Q
r̂
4πε0 r2
r > R.
Qr r 4
−
3(
) r̂.
4πε0 R3
R
r<R
b) Sketch a graph of the magnitude of the electric field as a function of r. At which point is its value a
maximum and what is the value there?
Answer: We have maximum at
r=
with value
E=
2
R
3
Q
.
3πε0 r2
Exercise 2.7: A charged rod
Let us study the electric field from a rod placed along the x-axis of length 2L and with charge density
λ. The situation is shown in figure 3.
a) Determine the direction of the field in the point P , a distance z normal to the middle of the rod (on
the z-axis). Show that the strength of the electric field can be written as
Z L
2λz
1
Ez =
dx.
4πε0 0 (x2 + z 2 )3/2
z
dE
P
r-r'
r
r'
-L
dx
L
x
Figure 3: A finite line segment. Notice the symmetry across the z-axis.
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where Ex = Ey = 0.
Hint: Pair up one contribution at −x with another at x. Is there some cancellation going on here?
b) Evaluate the integral. You might want to use the substitution x = z tan θ.
Answer:
Ez =
L
2λz 1
√
2
2
4πε0 z
L + z2
c) What do you expect the field to look like when you move very far away from the rod, i.e. for z L?
Does the expression you have for the field make sense in this limit?
Answer: Looks like the field from a point charge.
Ez =
1 2Lλ
4πε0 z 2
d) The expression you found in (b) was only valid along the z-axis. What does the field look like for an
infinite rod, L → ∞? Where is this expression valid?
Answer:
lim Ez =
L→∞
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1 2λ
.
4πε0 z
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