PHY 142!
Assignment #4!
Summer 2013
Reading: DC Circuits
Key concepts: Current and current density, emf, resistance, Joule heating, power to a circuit
element, Kirchhoff’s rules.
1.!
2.!
3.!
Questions about incandescent light bulbs.
a.!
A 100 W light bulb and a 1000 W space heater both operate on 120 V. What
is the ratio of the resistance of the bulb to that of the heater?
b.!
Why is the glass enclosure for a 100 W bulb larger than for a 25 W bulb?
c.!
The “drift” speed of electrons in a current is about 1 m/hr. Why does a
light bulb begin to glow almost immediately when the switch is closed?
More questions about light bulbs.
a.!
Why are lamps in ordinary household wiring connected in parallel?
b.!
Why are the small bulbs for holiday lighting wired in series?
c.!
A large bank of batteries connected in series has a total emf of 120 V. When
a light bulb rated 25 W at 120 V is connected across this bank it glows with
normal brightness. When a bulb rated 250 W at 120 V is connected across
the bank, its brightness is much less than normal. Explain.
Questions about circuit elements.
a.!
Is the current through a circuit element always proportional to the
potential difference across it? If not, give a counter-example.
b.!
The circuit element shown contains a 6 V
battery and a 0.2 Ω resistor. The current is 5 A,
and at the terminals Va = 5 V, Vb = 10 V. What is
I
a
b
the power supplied to the element?
c.!
4*.!
Draw a circuit diagram showing how the battery and resistor might be
configured within in the element shown.
Each bulb in a string of 48 holiday lights (connected in series) has
total resistance 10 Ω . It consists of a filament of resistance R and a
130 Ω shunt resistor, connected in parallel as shown.
a.!
What is the purpose of the shunt resistor?
b.!
What is the filament resistance R?
c.!
What power does the string consume if all bulbs are working?
d.!
What power does the string consume if one bulb’s filament burns out?
PHY 142!
5.
6.!
Summer 2013
Questions about Ohm’s law.
a.!
Distinguish clearly between the definition of resistance and Ohm’s law,
since both involve the equation ΔV = IR .
b.!
The equations that arise from Kirchhoff’s rules are linear equations for the
currents only if all the resistances are ohmic. Does this mean that
Kirchhoff’s rules are only valid in that case? Explain.
In the circuit, the ammeter (negligible resistance) reads 2 A.
a.!
7.!
Assignment #4!
5Ω
Find the current I1 (magnitude and direction of flow)
through the 15 V battery.
b.!
Find the current I2 (magnitude and direction of flow)
through the other battery.
c.!
Find the value of E.
15 V
5Ω
A
2Ω
To determine the value of a resistance, one can measure both
the voltage ΔV across it and the current I through it, and use
R = ΔV /I . Shown is a circuit containing an unknown
E
V
A
R
resistance. The voltmeter (denoted by V) measure accurately
the potential difference ΔV between its terminals, and the
E
ammeter (denoted by A) measures accurately the current I
passing through it. If one uses these values to determine R,
does one obtain a value larger or smaller than the actual resistance? Explain how
you know (without doing a lot of algebra).
8.!
The “flasher” circuit shown consists of a battery, a resistor,
a capacitor, and a gas discharge bulb. The bulb becomes
a low resistance conductor if the voltage across it
E
exceeds the threshold V0 , which is less than E . In that
R
C
case the capacitor will discharge very quickly through the
bulb, making it give off a bright flash. This drops the capacitor voltage nearly to
zero, and the bulb ceases to be a conductor. The charging of the capacitor then
starts over and continues until its voltage reaches V0 again.
a.!
Assume the charging starts at t = 0 . At what time does the voltage across
⎛ E ⎞
the capacitor reach V0 for the first time? Ans: t = RC ⋅ ln ⎜
.
⎝ E − V0 ⎟⎠
b.!
If E = 90 V, V0 = 60 V, and C = 1 µF , for what R will it flash every 5 s?
!
Ans: R = (5 × 106 )/ln 3 Ω.
PHY 142!
Assignment #4!
Summer 2013
Reading: Magnetostatics 1
Key concepts: Magnetic fields, Lorentz force, motion of charge in uniform field.
9.!
Questions about charged particle motion in a B-field.
a.!
A proton is moving in a circle under the influence of a uniform B-field. A
uniform E-field in the same direction as B is turned on for a short time Δt ,
then turned off. Describe the motion of the proton after the time Δt .
b.!
A proton from the sun enters the earth’s B-field at a point high above the
equator, where the field lines are parallel to the surface and to the north.
Its velocity v makes a 45° angle with the field lines, with components
toward the surface and to the north. Take the B-field to be uniform. Draw
a picture and describe the proton’s (initial) motion.
c.!
We have used Newtonian mechanics to derive the formula r = mv /qB for
the radius of the circle followed by a charged particle moving
perpendicular to a uniform B-field. If the particle’s speed is close to the
speed of light c we must replace the Newtonian formula for momentum
p = mv by the correct formula p = mvγ , where γ = 1/ 1 − (v /c)2 . How
would this affect the “cyclotron frequency” of the circular orbit?
10.!
11.!
Electron #1, with speed v, enters a velocity
selector consisting of a uniform E field in the
plane of the page and directed vertically, and
•
Field region
a uniform B field perpendicular to the page.
2
•
This electron enters from the left and passes
1
through without deflection. Electron #2
enters from the right, also with speed v, and
is deflected upward as shown. Comment on the following possibilities, stating
which are right and which are wrong and why.
a.!
E is upward and B is out of the page.
b.!
E is downward and B is out of the page.
c.!
E is upward and B is into the page.
d.!
E is downward and B is into the page.
The behavior of a magnetic dipole in a B-field is much like that of an electric
dipole in an E-field: there is a torque τ = µ × B and a potential energy of
alignment U = −µ ⋅ B . We are interested in the force on the dipole in a non-uniform
field.
PHY 142!
12.!
13*.!
Assignment #4!
a.!
Suppose the dipole is aligned with the field. Show that the force will be
toward the region where the field is stronger. [Use the general rule that
systems left to themselves move toward lower potential energy.]
b.!
Now suppose that (somehow) the dipole remains aligned opposite to the
field. Show that the force is toward the region of weaker field.
An electromagnetic pump (which could be used to pump blood in
artificial hearts) has crossed E and B fields as shown in cross-section.
The B-field is static, but the E-field can be turned on and off at regular
intervals. The ions in the blood move in response to these fields.
E
B
a.!
Show that soon after the E-field is turned on a positive ion will experience
a force with a component out of the page.
b.!
What happens if the ion is negative?
A device to measure isotope masses is shown. Ions of
charge +e and mass m are accelerated from rest through the
potential difference between the plates. After passing
through a small hole in the grounded plate they enter a
region of uniform B-field into the page as shown. The
radius R of the circle they describe is then measured.
a.!
R
⊗B
V
Find the speed v of the ions as they emerge into the
B-field region. Ans: v =
14*.!
Summer 2013
2eV
.
m
mv
.
eB
b.!
Find R in terms of v and the other given quantities. Ans: R =
c.!
e(BR)2
Find the formula for m, in terms of e, V, B, and R. Ans: m =
.
2V
A proton (charge e, mass m, a deuteron (charge e, mass 2m) and an α -particle
(charge 2e, mass 4m) are accelerated through the same potential V0 and then
enter a uniform field B perpendicular to their velocities.
a.!
What are the kinetic energy and orbit radius for the proton?
b.!
Repeat for the deuteron.
c.!
Repeat for the α -particle .
PHY 142!
15.!
!
Assignment #4!
Summer 2013
A current I is moving in the direction shown through a
rectangular slab of conducting material. Assume that
the current is actually a flow of microscopic particles
B
of charge q. Since overall the conductor is neutral
there must be stationary objects of equal and opposite charge
in the material.
I
a.!
Assume q is positive, so the direction of the current is the same as the
direction of the drift velocity of the moving charges. Which way will the
B-field deflect these charges (toward the top or the bottom of the slab)?
b.!
This deflection will result in an accumulation of positive charge at the top
or bottom of the slab, with a corresponding accumulation of negative
charge (of the stationary objects) at the opposite side. The creates a
potential difference across the thickness of the slab. Which side (top or
bottom) will be at higher potential?
c.!
How would your answers change if q is negative, so the direction of the
current is opposite to that of the velocity of the moving charges?
d.!
Explain how this situation can be used to determine the sign of the
moving charges in a conductor.
[This is the Hall effect, which was used in the 1880‘s — well before the discovery
of the electron — to show that the moving charges in metals are negative.]