Electricity and Magnetism: PHY-204
Fall Semester 2014
Tutoring Session on Electric Circuits
1. The drift speed in a copper wire is 7×10−5 m/s for a typical electron current. Calculate
the magnitude of the electric field E inside the copper wire. The mobility of mobile
electrons in copper is 4.5 × 10−3 (m/s)(N/C). (Note that though the electric field in
the wire is very small, it is adequate to push a sizable electron current through the
copper wire.)
2. Suppose that wire A and wire B are made of different metals and are subjected to the
same electric field in two different circuits. Wire B has 6 times the cross-sectional area,
1.3 times as many mobile electrons per cubic centimeter, and 4 times the mobility of
wire A. In the steady state, 2 × 1018 electrons enter wire A every second. How many
electrons enter wire B every second?
3. In the circuit in figure below the narrow resistor is made of the same material as the
thick connecting wires.
1
6
2
5
3
4
Resistor
(a) Sketch a graph of the electric field’s magnitude at different locations.
(b) Sketch a graph of of the drift speed of the electrons at different locations. Graph
will look something like the following.
E
1
2
3
4
5 6
(location)
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Electricity and Magnetism: PHY-204
Fall Semester 2014
(c) Sketch the potential at different locations along the circuit.
4. A solid metal sphere of radius R carries a uniform charge of +Q. Another solid metal
sphere of radius r carries a uniform charge −q. The amount of charge is not enough
to cause breakdown in air. The two spheres are very far apart (distance ≫ R and
distance ≫ r). At time t = 0 a very thin wire of length L is connected to the two
spheres (as shown in figure). The mobility u of mobile electrons in this wire is very
small, and the wire conducts electrons so poorly that it takes about an hour for the
system to reach equilibrium. In a short time ∆t (a few seconds) how many electrons
leave the sphere of radius r? There are n mobile electrons per cubic meter in the
wire, and the wire has a constant cross-sectional area A. Explain your work and any
approximations you need to make.
+Q
+
+
+
+
R
+
+ + +
L
-q - - -r -
-- -
5. The diagram below shows a circuit in which a steady state current is running. All wires
are made of the same material, but some are thinner than others. Circle two locations
in which the distribution of surface charge shown in the diagram is inconsistent with
the existence of a steady state current. For each location, explain briefly but clearly
why the surface charge shown is wrong.
6. The wires in the circuit shown in diagram are made of different materials. The mobile
electron density of both materials is the same. The thicker wire has cross-sectional
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Electricity and Magnetism: PHY-204
Fall Semester 2014
area A1 and electron mobility u1 . The thinner wire (dark) has cross-sectional area
A1 /2, and electron mobility 2u1 .
(a) At each location inside the wire, marked by “X”, draw and label an arrow representing the electric field at that location. Relatives magnitudes of the arrows
should be correct. Label the arrows “E”.
(b) At locations B and C, draw and label arrows representing the drift speed of an
electron at that location. Label the arrows ”v”.
(c) Draw an approximate distribution of charge on the surface of the wires. Make
sure the charge distribution you show is consistent with your answer to (a).
7. A circuit contains one battery and three wires in series as shown in diagram. The
wires have different lengths and are made of different materials. Two of the wires are
identical: they are each L1 = 7 cm long, and made of material having n1 = 6 × 1025
mobile electrons per cubic meter. The central wire is L2 = 5 cm long, and has
n2 = 2.5 × 1025 mobile electrons per cubic meter.
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Electricity and Magnetism: PHY-204
Fall Semester 2014
Both wires have cross-sectional area 4 × 106 m2 and the electron mobility of each
material is 5 × 104 (m/s)/(V/m). The emf of the battery is 2.9 volts. In the steady
state, the potential drop across the length L2 of the wire 2 is 1.33 V.
(a) What is the magnitude of the electric field in wire 2? Show all steps in your work.
(b) In the steady state, what is the magnitude of the electric field in the left wire, of
length L1 ? Show all steps in your work.
(c) How long would it take a single electron to travel from the negative end of the
battery all the way around the circuit to the positive end of the battery? Show
all steps in your work.
(d) How many electrons per second cross the junction between the two wires? Show
your work.
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