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Problem Set 3 Physics 2220 (Schroeder)
Name
fall, 2005
Problem Set 3
(due Thursday, Sept. 15)
1.
Suppose that in a particular lightning ash, the voltage dierence between a cloud
and the ground is one billion volts and the quantity of charge transferred is 30 C. (a)
What is the change in energy of that transferred charge? (b) If all this energy could
be used to accelerate a 1000 kg car from rest, what would be the cars nal speed?
2.
In a Van de Graa accelerator a proton is accelerated through a voltage dierence
of 14 MV (megavolts). Assuming that the proton starts from rest, calculate its nal
kinetic energy, rst in joules and then in electron-volts. (1 eV equals 1.6
� 10
19
J.)
3.
In the illustration below, the horizontal lines represent electric eld lines and the
vertical lines represent equipotential surfaces. (a) Does the voltage (potential) increase
toward the right or toward the left? How can you tell? (b) The voltage at the rightmost
equipotential surface is
100 V, and the adjacent surfaces dier by 10 V. What is the
voltage at the leftmost surface? (c) Suppose that you move an electron from left to
right through this region, in such a way that it is always moving very slowly. Is the
work that you do on the electron positive or negative? (d) In the same situation, is
the work done on the electron by the electric eld positive or negative?
4.
Consider a positively charged conducting sphere, in static equilibrium. The charge is
uniformly distributed over the surface, and the electric eld inside is zero. Does this
imply that the voltage inside is zero? Explain carefully.
5.
Two large, horizontal, parallel conducting plates are 12 cm apart and have charges of
equal magnitude and opposite sign on their facing surfaces. An downward electrostatic
force of 3.9
� 10
15
N acts on an electron placed anywhere between the two plates
(except near the edges). (a) Find the electric eld at the position of the electron
(magnitude and direction). (b) What is the voltage dierence between the two plates?
Which is at the higher voltage?
6.
In a certain region of space there is a uniform electric eld of 5900 V/m, pointing in
the +x direction. (a) Suppose we take V = 0 at the origin. Where else does V = 0?
(b) Where does V = 100 V? (c) Where does V =
100 V? 7.
The illustration below shows a region in which the electric eld strength is 2000 V/m.
The illustration is shown actual size, so you can measure distances on it with a ruler.
(a) Calculate the dot product E
� s for each of the three paths AC, CB, and AB. (b)
If the voltage at point A is zero, what is the voltage at points B and C?
E
E
A
B
C
8.
(a) Referring again the illustration above, calculate the electrostatic force felt by a
point charge of
15 nC located in this region. (b) From your answer to part (a),
calculate the work you would have to do to slowly push this charge from A to B.
(c) From your answer to part (b), calculate the voltage dierence between points A
and B.
9.
Consider a region of space in which the electric eld everywhere points in the +x
direction. In the region where x < 0, the magnitude of the eld is 1000 V/m, but
in the region where x > 0, the magnitude of the eld is 3000 V/m. (a) Sketch this
electric eld. (b) Consider a small particle whose charge is 1 nC. You wish to move this
particle backwards along the x-axis from the point x = 2 m to the point x =
2 m.
Calculate the work required for this operation, by determining the electrostatic force
on the particle in each region and multiplying by the displacement. (c) Use your
answer to part (b) and the denition of voltage to calculate the voltage dierence
between the starting and ending points.
10. In a repeat of the historic Millikan oil drop experiment, you use a pair of plates
measuring 6 cm-square, separated by a distance of 1 cm. By observing the rate
at which a particular oil drop falls (acted upon by gravity and air resistance), you
determine that its mass is 8.1
�10
14
kg. You then turn on the voltage, and note that
this droplet is suspended motionless when the voltage between the plates is 5000 V.
What is the charge on the droplet? How many fundamental units of charge is this?
11. The gure on the following page shows a CRT oscilloscope tube. The voltage dierence
between the cathode and anode, which accelerates the electrons, is 1000 V. To the
right of the anode are two deection plates, measuring 3 cm along the direction of
the beam and separated by a distance of .5 cm. You wish to use the oscilloscope as a
voltmeter, so you connect these plates (from points A and B) to an unknown voltage
source, and see that the electron beam is deected by an angle of 5 . What is the
voltage dierence between the plates? (Hints: You may assume that the horizontal
velocity of the electrons is constant after they pass through the anode. Treat the
motion between the plates as a projectile motion problem, but with gravity replaced
by the electrostatic force. Express the deection angle in terms of the components of
the nal velocity of the electrons.) battery
cathode anode
electron beam
A
B
12. Suppose you have a roll of aluminum foil and a roll of wax paper. Estimate, very
roughly, how large a capacitor you could make by sandwiching them together. You
may neglect the eect of the wax paper, except as a separator between sheets of foil.
13. Suppose that you have two capacitors, one with known capacitance (such as a parallel-
plate capacitor) and one with unknown capacitance. You also have a battery and a
working voltmeter. You can determine the unknown capacitance by hooking all three
components in series, as shown below. You measure a voltage dierence of 11.45 V
between points A and B, and .55 V between points B and C. If the known capacitor
has a capacitance of 160 pF, what is the capacitance of the other capacitor?
A
B
C
12 V
unknown capacitor
known capacitor (160 pF) Study Guide for Quiz 3
Voltage (or electric potential) is dened as electrostatic energy (of a hypothetical test
charge in some given environment) per unit charge:
V (x, y, z) = U
e
q
0
.
Here U
e
is electrostatic potential energy, and q
0
is the amount of charge on the test charge.
Like the potential energy U
e
, the voltage is always relative to some arbitrary reference
point (ground).
Relation between V and E:
V =
E � ds.
In other words, E points from high voltage to low voltage, and the magnitude
|E| is the
change in voltage per unit distance as you move in that direction. This relation completely
analogous to that between U
e
and F
e
.
Voltage is a useful quantity because batteries (as well as many grid-connected power sup-
plies) supply a xed voltage. In a battery, this is because the chemical reaction going on
inside supplies a xed amount of energy to each electron (that is, a xed energy per unit
charge). Voltage dierences are also relatively easy to measure, unlike charges and electric
eld strengths.
In his famous oil drop experiment, Robert Millikan showed that electric charge is quantized,
coming only in multiples of
�1.6 � 10
19
C. An electron carries exactly one such unit of
negative charge, while a proton carries exactly one such unit of positive charge.
A capacitor is a device that stores positive charge Q and negative charge
Q, separated
from each other, when a voltage dierence V is applied. The capacitance is dened as
C = Q
V ,
charge per unit voltage. The simplest capacitor consists of two parallel plates separated
by a small gap. Using the formula for the eld near a plane of charge, you can show that
in this case the capacitance is
0
A/d, where A is the plate area and d is the separation.