Physics 1b Practical - Assignments - 2008

date	day	reading	tasks due
1/7	M:	handouts	--
1/9	W:	23.1-3	pick up tool kit, 3:30 - 5:30pm
			308 E. Bridge; have ID # with you
1/10	Th:	23.4-7	Probs. 23.3, 26, 38; QP1, QP2
			(Quiz 1 out)
1/11	F: Electric Fields
1/14	M:	24.1-3	Quiz 1 DUE
1/16	W:		Lab 0
1/17	Th:	24.4, 25.1-2	Probs. 24.11; 25.1; QP3, QP4
1/18	F: Gauss' Law & Potential
1/21	M: Martin Luther King Day		(no classes)
1/23	W:	27.1-4	Lab 1; NOTE: Save a copy of your light
			bulb data for homework problem EP1
1/24	Th:	27.5-6	Probs. 27.37, 56; EP1, QP5
			(Quiz 2 out)
1/25	F: Current & Resistance
1/28	M:	28.1-3	Quiz 2 DUE
1/30	W:		Lab 2
1/31	Th:	28.5-6	Probs. 28.3, 55; QP6, QP7
			(Quiz 3 out)
2/1	F: DC Circuits
2/4	M:	25.3-6	Quiz 3 DUE
2/6	W:		Lab 3
2/7	Th:	25.7-8	Probs. 25.42, 62; QP8, QP9
			(Quiz 4 out)
2/8	F: Electric Potential
2/11	M:	26.1-3	Quiz 4 DUE
2/13	W:		Lab 4
			Ombudsmeeting
2/14	Th:	26.4-5	Probs. 26.29, 46; QP10, QP11
2/15	F: Capacitors
2/18	M: Presidents' Day		(no classes)
2/20	W:	26.6-7	Lab 5
2/21	Th:	28.4	Probs. 26.56; 28.31; QP12, QP13
			(Quiz 5 out)
2/22	F: Dielectrics & RC Circuits

2/25	M:	29.1-2	Quiz 5 DUE
2/27	W:		Lab 6
2/28	Th:	29.3-6	Probs. 29.31, 61; QP14, QP15
			(Quiz 6 out)
2/29	F: Magnetic Force
3/3	M:	30.1-3	Quiz 6 DUE
3/5	W:		Lab 7
			(Lab Quiz out)
3/6	Th:	30.4-5	Probs. 30.17, 67; QP16, QP17
			(Quiz 7 out)
3/7	F: Magnetic Fields
3/10	M:	review	Quiz 7 DUE
3/12	W:		Lab Quiz DUE
			(Final out)
3/18	Tues: Final Exam DUE	at 1:00 pm	in 110 E. Bridge

Assigned Problems are specified by chapter and number. Answers to odd numbered Problems are in the back of the book; so be sure to write up your solutions showing all the work needed to get the answer.

The problems designated ``QP'' are taken from recent years' quizzes and finals. (The actual problems appear on the following pages.) As such, they may give you some idea of the nature and difficulty of questions that may appear this year. You should attempt them only after you have some confidence in the material, and you should initially work on them alone (at least for 1/4 to 1/2 hour).

To do problem EP1, you will need your light bulb data from Lab 1. If you turn in your lab notebook before doing EP1, you should save a copy of the light bulb data to work from.

All assignments and labs are to be handed in at class meetings. No late credit will be given except by permission of your TA.

We will meet with ombudsmen, representing each House, on Wednesday, 2/13, as noted in the syllabus. Please pass on to them your concerns and suggestions.

1/7/08

QP 1

An electric quadrupole is a particular configuration of charges which sum to zero but whose effects at a distance do not quite cancel. One example of electric quadrupole is formed by four charges located at the vertices of a square of side 2a. Point P lies a distance R from the center of the quadrupole on a line parallel to two sides of the square as shown.

(a) What is the direction of the electric field at point P?

(b) Write down an exact expression for the quadrupole electric field E at point P.

(c) Show that for $R\gg a$ the electric field reduces to the form $E\approx \alpha /R^4$ and give an expression for $\alpha$ in terms of the parameters specified.

QP 2

(a) Two point charges, each of charge $+Q$, are separated by a distance $d$. What is the set of points in space at which the total electric field due to these two charges is zero?

(b) Two point charges, one of charge $+Q$ and the other of charge $-Q$, are separated by a distance $d$. What is the set of points in space at which the total electric field due to these two charges is zero?

QP 3

Figure 2 shows a section through two long concentric cylinders having equal and opposite charge per unit length, $\lambda .$

(a) If the two cylinders are insulators with charge uniformly distributed throughout, what is the magnitude and direction of the electric field in all space, i.e., $r<a$, $a<r<b$, $b<r<c$, $r>c$?

(b) If the two cylinders are conductors, what is the magnitude and direction of the electric field in all space?

(c) If the two cylinders are conductors, how much charge per unit length is on each surface, labeled $S_1$, $S_2$, and $S_3$ in figure 2?

QP 4

Two large nonconducting sheets of positive charge, carrying the same surface charge density $\sigma$, face each other. Considering only points not near the edges and whose distance from the sheets is small compared to the dimensions of the sheets, what is the electric field E at points

(a) to the left of the sheets?

(b) between the sheets? and

(c) to the right of the sheets?

EP 1

Use your data from the I-V curve of the light bulb in Experiment 1 to estimate the relative change in resistance of the bulb filament between room temperature (around 0 V) and 6 V. Assume the filament is tungsten and estimate the temperature in degrees C of the filament at 6 V. Does your answer make sense relative to other ``red hot" temperatures (e.g., the surface of the sun, fires, glowing hot)?

QP 5

Two car bulbs, bulb A and bulb B, look superficially similar. However, when bulb B is connected to a 12 V car battery, its filament glows much brighter than did bulb A's when A was connected to the battery.

(a) Which bulb draws more current from the battery, A or B?

(b) Which bulb has a higher resistance?

A current-controlled power supply is set so that it delivers precisely 0.25 amps over a very wide range of load resistances.

(c) When attached to this current-controlled power supply set at 0.25 amps, which glows brighter, bulb A or bulb B?

QP6

Consider the circuit shown with resistors $R_1$, $R_2$, and $R_3$ and battery voltages ${\cal E}_1$ and ${\cal E}_2$ as indicated.

(a) What currents $i_1$, $i_2$, and $i_3$ flow through $R_1$, $R_2$, and $%% R_3$ respectively?

(b) What power is dissipated in each of the resistors?

(c) Compare the power supplied by the batteries to the power dissipated in the resistors. Is total energy conserved?

QP 7

A circuit consisting of a voltage source of magnitude $V_o$, three resistors with resistances $R_1$, $R_2$, and $R_3$, and a switch, $S$, is constructed as shown. We are interested in the voltages, $V_1$ and $V_2$, which are defined by the potential differences between the points indicated in the figure.

(a) What is $V_1+V_2$ with switch $S$ open?

(b) What is $V_1+V_2$ with switch $S$ closed?

(c) What is $V_1\div V_2$ with switch $S$ open?

(d) What is $V_1\div V_2$ with switch $S$ closed?

QP 8

Two isolated, concentric, spherical, conducting shells of radius r and R carry positive total net charges q and Q, respectively. The shells are very thin compared to the radii.

(a) What is the potential at the surface of the large sphere, assuming $V=0$ at infinity?

(b) What is the potential of the small sphere?

(c) Calculate the potential difference between the spheres, and indicate which one is at a higher potential.

(d) Now the spheres are connected with a fine conducting wire so that charge could redistribute between them. Will the charge move, and, if so, which direction will it flow? What will be the final charge on each sphere?

QP 9

An LED and a small light bulb are connected in series to a 3.0 V battery. (The battery internal resistance is negligible, and the bulb $I$-$V$ curve is reproduced below.)

(a) What would be the current, $I$, in the circuit if the $I$-$V$ curve of the LED were given by the simplest possible idealization of diode performance, indicated in the graph below labeled LED(a)?

(b) What would be the current, $I$, in the circuit if the $I$-$V$ curve of the LED were given by the slightly more sophisticated idealization of diode performance, indicated in the graph labeled LED(b)?

(c) Estimate the actual current, $I$, in the circuit using the measured $I$-$V$ curve of the LED, represented in the graph above labeled LED(c).

QP 10

When the switch S is thrown to the left, the plates of the capacitor $C_1$ acquire a potential difference $V_0$. $C_2$ and $C_3$ are initially uncharged.

(a) What is the initial charge $Q_1$ on capacitor $C_1$?

(b) The switch is now thrown to the right. What are the final charges $q_1$, $q_2$, and $q_3$ on the corresponding capacitors? Express your answers in terms of $V_0$, $C_1$, $C_2$, and $C_3$.

QP 11

A parallel plate capacitor with plates of area A is filled with two dielectrics (of dielectric constants $\kappa _1$ and $\kappa _2$. Each dielectric slab has thickness d/2.

(a) Can this arrangement be equivalently viewed a two capacitors in series or parallel? If so, which?

(b) Find the capacitance of the arrangement in figure 7a.

(c) Now consider the arrangement in figure 7b, where each dielectric occupies half the volume. Can this arrangement equivalently be viewed as two capacitors in series or parallel? If so, which?

(d) Find the capacitance of the arrangement in figure 7b.

QP 12

Rufus and Dufus, two barefoot lads, are fascinated by a Van de Graaff generator. Its upper metal sphere has a radius of 20cm. On this particular day, they note that the generator is capable of producing sparks to a similar size grounded sphere that are as much as 15cm in length. (That number depends on the humidity and the cleanliness and precise mechanical adjustment of the machine.) They both want to ``feel'' the charge, like electricians of old (i.e., the 18$^{\text{th}}$Century).

Rufus takes his 1000$\mu$F capacitor (``big blue'') from his ZAP! kit, attaches long wire leads to each end, connects one lead (the proper one) to ground and approaches the maximally charged Van de Graaff sphere with the other. There's a spark between the sphere and the capacitor lead, and then he makes proper metallic contact between that lead and the sphere.

Dufus points out the all of the original charge is still there, now just distributed somehow between the sphere and the attached side of the capacitor.

(a) Estimate the fraction of the original Van de Graaff charge that still resides on the sphere.

``OK,'' says Rufus, ``Right you are," and he touches the sphere with his finger.

For the present purposes, model the electrical properties of each boy from finger tip to feet as a 20k$\Omega$ resistor to ground.

(b) What is the maximum current that flows through Rufus' body?

That current has a characteristic exponential decay for its time dependence.

(c) What is the 1/$e$ time for that current?

Dufus, observing there wasn't all that much zap to Rufus' experience, charges up the Van de Graaff again to its maximum and tries something else. He foregos the 1000$\mu$F capacitor and simply approaches the metal sphere with his bare finger. Again, there is a spark, this time from sphere to finger. For the present purposes you should assume that once the spark forms, the intervening air is a conductor with negligible resistance compared to that of Dufus' body. In other words, he establishes a conductive path from sphere to ground with a resistance of 20k$\Omega$ (essentially all of which is skin).

(d) What is the maximum current that flows through Dufus' body?

(e) What is the characteristic exponential decay 1/$e$ time for that current?

QP 13

The Leiden jar, named after the city of its origin and developed in the mid-18$^{\text{th}}$ Century, was the first really effective design for storing charge, i.e., a capacitor. It consisted of a wide-mouth glass jar, lined with lead foil and wrapped on the outside with another piece of lead foil. The two pieces of foil served as the metallic capacitor ``plates.'' Contact with the outer foil was direct, while contact with the inner foil was achieved via a metal chain that hung from the center of an insulating wooden stopper down to the (inner) bottom of the jar (not shown in the accompanying sketches).

In the following, please use these dimensions: The glass is 2mm thick. The foil covered area is cylindrical in shape, of height 15cm and radius 6cm. Note also that the bottom of the jar is foiled inside and out.

(a) Estimate the capacitance of the Leiden jar as described.

(b) What is the maximum charge that can be stored -- before there is electrical breakdown, i.e., sparking between the plates?

To see how much of an improvement this is over storing charge on an isolated conductor, consider an isolated, hollow, metal sphere, such as the sphere on a Van de Graaff generator.

(c) Estimate the capacitance of an isolated, hollow, metal sphere of radius 10cm.

(d) What is the maximum charge that can be stored on the sphere -- before there is corona discharge to the air?

QP 14

In the mass spectrometer shown, protons (mass $m_p$ and charge e) are injected at point P and accelerated by the voltage V across a pair of pair of parallel plates, $P_1$ and $P_2$. The protons then enter a region of uniform magnetic field B, directed into the page. The top of the spectrometer is defined by a third plate $P_3$, spaced a distance d above the second plate.

(a) Will the protons be deflected to the right or the left as they enter the magnetic field?

(b) What is the velocity of the protons as they enter the magnetic field, expressed in terms of the constants given? (Their pre-accelerated velocity is negligible.)

(c) What maximum voltage, V, can be applied between the plates and have the protons not strike the top of the spectrometer?

(d) If the particles were deuterons (twice as massive as protons but with the same charge, does the voltage found in part (c) increase, decrease, or stay the same?

QP 15

Over the past 25 years, a great many surprising discoveries and major advances in the understanding of electrons in solids have come from the study of a fabricated system in which electrons are confined to move in a 2-dimensional plane at the interface of two (insulating) crystals. Making the systems small, very clean, and very cold allows individual electrons to move in that plane essentially as if they were free particles, oblivious to the atoms of the solid. Typically a strong magnetic field is applied perpendicular to the plane of motion. (The interesting complications -- that we will ignore for this problem -- arise from quantum mechanics and the presence of many other electrons in the plane that exert Coulomb forces on each other.)

When an electron hits the edge of its planar area of motion, it bounces elastically off the edge, with angle of incidence equaling angle of reflection, just like an ideal ball bouncing off a wall.

In the spirit of these experiments, consider a single particle of mass $m$ and charge $-Q$ (i.e., with $-Q<0$) moving on a square, 2-dimensional surface of dimensions $L\times L$. If the particle encounters the boundary, it rebounds elastically, with equal angles of incidence and reflection. A magnetic field of magnitude $B$ is applied, pointing perpendicular to the plane -- and out of the page as drawn in the figure. The particle is initially located at the point labeled $P$, which is $d$ from one edge and $L/2$ from the other. It has an initial velocity v which is parallel to the nearby side as indicated.

The parameters in the problem satisfy the following relations:

$L=8d$

$v=\frac {QBd} {m}$

Draw the $L\times L$ square on your paper, mark the initial point $P$, and then sketch the complete orbit of the particle. Include an occasional arrowhead on the orbit to indicate the direction of motion along the line.

QP 16

A hollow cylindrical conductor of inner radius a and outer radius b (cross section shown above) carries a total current I uniformly distributed over its cross section.

(a) Find an expression for the magnitude of the magnetic field B for points inside the body of the conductor, i.e., $a<r<b$.

(b) What is the direction of the magnetic field?

(c) Make a rough plot of the general behavior of $B(r)$ from $r=0$ to infinity.

QP 17

An insulating rod of length $L$ and diameter $d$ (with $d\ll L$) has a uniform, positive charge density and total charge $Q$ (with $Q>0$). As suggested by the accompanying sketch, the length of the rod is parallel to the $x$-axis (across the page) and the rod has a velocity v which points along the positive $y$-axis (up the page). This all takes place within a uniform magnetic field B, which points along the positive $z$-axis (out of the page).

(a) What is the total magnetic force F $_{%% \text{B}}$ (magnitude and direction) on the rod?

As the rod moves, it sweeps out an area in the $x$-$y$ plane. For example, in a time interval $\Delta T$, it sweeps out an area $L\times v\times \Delta T$. We say that the (time) rate of sweeping area is $L\times v$, i.e., the area swept per unit time. There is a magnetic flux $\Phi _{\text{B}}$ associated with such an area.

(b) What is the rate of flux, $\frac{d\Phi _{%% \text{B}}}{dt}$, swept by the rod?

Consider now a neutral rod made out of a metallic conductor, with the same dimensions and velocity in the same magnetic field as described above. Since it is neutral, the total magnetic force on the rod is zero. However, the magnetic field certainly has some effect. In particular, it pushes plus charges one way and negative charges the other. Since the charges are constrained to remain within the rod, their distribution reaches an equilibrium configuration such that the net force on each charge, i.e., the sum of the magnetic force and the electric forces due to all the other charges, is zero.

(c) What is the electric field E (magnitude and direction) at the geometric center of the rod due to the magnetically induced charge distribution?

(d) What is the electric potential difference between the two ends of the rod?