March 30, 2009

Physics 1c Practical - Assignments - 2009

date	day	reading	tasks due
3/30	M:	--	--
4/1	W:	30.6-7	Probs. 29.23; 30.3, 43, 45, 57; QP1
4/2	Th:		(Quiz 1 out)
4/3	F: Magnetic Materials & the
	Mystery of Phase Transitions
4/6	M:	31.1-2	Quiz 1 DUE
4/8	W:	31.3-4	Probs. 31.11, 16, 25; QP2, QP3, QP4
4/9	Th:		(Quiz 2 out)
4/10	F: Faraday's Law
4/13	M:	31.5-6	Quiz 2 DUE
4/15	W:	32.1-2	Probs. 31.42; 32.8, 10, 61; QP5, QP6
4/16	Th:
			(Quiz 3 out)
4/17	F: Induction
4/20	M:	32.3-4	Quiz 3 DUE
		op-amps*
		*in ZAP! (the Ph8 manual) p. 74-6	or Ph1c on-line
4/22	W:	32.5-6	Probs. 32.27, 36, 42, 47; QP7, QP8
4/23	Th:		(Quiz 4 out)
4/24	F: Inductance
4/27	M:	33.1-3	Quiz 4 DUE
4/29	W:	33.4-5	Probs. 33.57, 65; QP9, QP10
			Ombudsmeeting
4/30	Th:
5/1	F: AC Circuits
5/4	M:	33.6-7	--
5/6	W:	33.8-9	Probs. 33.40, 49; QP11, QP12, QP13, QP14
5/7	Th:		(Quiz 5 out)
5/8	F: AC Circuits & EM Waves
5/11	M:	34.1-5	Quiz 5 DUE
5/13	W:	34.6-7	Probs. 34.14, 57; QP15, QP17, QP18
5/14	Th:		(Quiz 6 out)
5/15	F: EM Waves

5/18	M:	39.1-4	Quiz 6 DUE
5/20	W:	39.1-4	Probs. 39.4, 5, 8, 12; QP19, QP20
5/21	Th:
5/22	F: Simultaneity &
	Moving Clocks
5/25	M: Memorial Day		(no classes)
5/27	W:	39.5-9	Probs. 39.24, 36, 40, 41, 43; QP21
5/28	Th:		(Quiz 7 out)
5/29	F: Relativity & $E\&M$;
	$E=mc^2$
6/1	M:		Quiz 7 DUE
6/3	W:	39.10	Probs. 39.46, 55, 60, 63, 64; QP22
6/4	Th:
6/5	F: The Twin Paradox		(Final out)
	-- Explained (!?)
6/11	Th: Final Exam DUE	at 1:00 pm	in 110 E. Bridge

Assigned Problems and Questions 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 on the Final Exam. 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).

All assignments 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, during the term, as noted in the syllabus. Please pass on to them your concerns and suggestions.

Here are the ``point values'' for each part of the course. Note that TAs will award 0-4 points for section participation (as distinct from ``attendance," which is necessary but not sufficient).

Section participation: 4

Written homework: 24

Quizzes: 42

Final exam: 30

Physics 1c Prac is a continuation of Physics 1b Prac. All principles and policies remain the same -- as described in the Ph 1b - Prac - Information - 2008 sheets and the several other handouts. If you are joining the course in the Spring term and have not completed Ph 1b Prac, you MUST pick up copies of each of these memos in 110 E. Bridge or retrieve them from the Ph 1b Prac Web page; also, see the note on the following page. Continuing students, if you no longer have those sheets for reference but have some questions, feel free to get copies for this term.

TO STUDENTS SWITCHING FROM THE ANALYTIC TRACK:

Ph 1c Prac picks up where 1b Prac left off. At the end of the winter term we did magnetic forces and the magnetic field due to currents (e.g., Ampere's Law and Biot-Savart). We begin Spring term with magnetic properties of materials, and then there is induction, RC circuits, light, and finally special relativity. (This last item occupies the final two or three weeks of the term.) The textbooks are the same for both terms, and you can get the Ph 1b syllabus off the Web or in 110 E. Bridge.

QP 1

Oscilloscope display tubes use voltages applied to parallel plates both to accelerate a beam of electrons up to high speed and to steer the beam vertically and horizontally to produce the desired display. In contrast, television picture tubes and computer monitors (the fat ones - not flat screen") use magnetic fields (produced by currents in coils) to do the steering.

The inner surface of the screen is coated with phosphor pixels (not more than approximately 1 mm wide), which glow red, green, or blue (depending on the phosphor) when struck. The Earth's magnetic field will also contribute to the deflection of the beam -- but by a direction and an amount that depends on the location and orientation of the tube relative to the Earth.

Assuming a tube length or electron path length of $L'$ = 50 cm from high voltage plates to the screen, estimate the maximum possible total deflection $d$ of the beam at the screen due to the Earth's field.

QP 2

A Los Angeles County civil engineer, looking over plans for the light rail system (before it was finally built) was concerned about a particular aspect of one of the design proposals. The high voltage line that was to power the electric locomotives lay between the tracks, which were grounded and served as the circuit return for the locomotive power. But buried just beneath the high voltage line were sensing and signal control devices. Remembering her freshman ${\cal E}\&{\cal M}$, she was concerned about induced ${\cal EMF}$'s due to the AC power line effecting the sensors and controls. So she proposed the following simple model to estimate the kinds of voltages that could be expected to be induced.

Let the current in the power line be

$I(t) = I_o \cos \omega t$

and let the device of interest be modeled by a square loop of wire of area $A$ located a distance $R$ from the power line. $R \gg A^{1/2}$ so that the magnetic field due do $I(t)$ can be approximated as a constant at any given time over the whole area $A$. Assume that the power line lies in the plane of $A$ to maximize its effect.

What is the emf ${\cal E}$ induced in the loop due to $I(t)$, in terms of the parameters given?

The power line runs at 2 kV (RMS) at 60 Hz. The typical rms power drawn by the locomotive is 1.5 MW (i.e., $1.5 \times 10^6$ watts). In the design, $R$ was about 30 cm, and $A$ is typically no bigger than 10 cm $\times$ 10 cm (i.e., $10^{-2}$ m$^2$).

What is the numerical estimate of the emf ${\cal E}$ induced in a typical device due to the power line, for the values given?

QP 3

$\parbox{3.3 in}{ At the Caltech Athletics Department 1978 awards din... ...ters of the problem? \bigskip \par c) What is $f(\varphi )$? \par \medskip\ }$ $\parbox{2.5 in}{ \begin{figure}[h] \centerline{{\epsfxsize=2.5in \epsfbox{MaryMack.eps}}} \end{figure}}$

Held tightly in sweaty hands, with full hand contact, the rope-to-two-hands (in series) resistance was only 500$\Omega$ in total. Compared to that contact resistance, the rope and her internal resistances were negligible.

d) Estimate the largest possible peak current induced by the Earth's field that passed through her arms when she first tried out this rope.

e) Taking account of the fact that the Earth's field in Pasadena has a vertical component as well as a horizontal component, was there an orientation for which Miss Mack would have generated essentially no current while she jumped? If so, what would have been the relation of her hands, e.g., the straight line that connected her two hands, to the direction of the B field?

f) What were the color of Miss Mack's buttons?

QP 4

A conducting bar is sliding at velocity v to the right on a V-shaped conducting rail, as shown in figure 1. There is a uniform magnetic field B out of the page. The rails are frictionless and resistanceless. The bar has a resistance $\lambda$ per unit length. The half-angle of the V is $%% \theta$.

(a) What is the current I as a function of time (taking the position of the bar to be $x=0$ at time $t=0$)?

(b) What is the direction of the current?

(c) What is the magnitude and direction of the force required to maintain a constant velocity versus time?

QP 5

In a standard design of an AC voltage transformer, two wire coils are wound around the central post of an iron ``core'' (which is really more of a frame) as shown in the sketches below. Iron is chosen for its outstanding magnetic permeability. Note that in the sketches, the dashed lines denote edges that are not visible from the outside in that particular view.

When an AC current passes through the first of the coils, it induces electric fields in the iron which, in turn, generate currents in the iron. These currents are undesirable because they lead to Ohmic heating and consequent power loss and because, by Lenz's Law, they reduce the desired magnetic field.

To reduce the effects of these undesired currents the iron is laminated. That means that it is actually made up of thin sheets of iron that are insulated from each other with a layer of varnish.

a) Of the four lamination geometries, $A$, $B$, $C$, or $D$, suggested by the accompanying sketches, which one would be most effective at minimizing the currents induced in the iron?

b) Of the three lamination geometries, $A$, $B$, or $C$, suggested by the accompanying sketches, which one would be least effective at minimizing the currents induced in the iron?

c) (for thought, not for points -- ) So what's with geometry $D$? To answer parts a) and b) above, you have to distiguish between eddy currents with diameters of order 1 cm and those restricted to be less than one lamination thickness. Would there be eddy currents in $D$? Where? How big? What limits their magnitude?

QP 6

An iron toroid of rectangular cross section has an inner radius $r_1$, an outer radius $r_2$, and a thickness $d$, as shown in figure 2. A large number of turns, $N$, of wire are wound uniformly around the iron core, which has magnetic permeability $\mu$. (You should ignore the resistance of the wire.)

(a) A steady current $I$ is flowing in the coil, which generates a magnetic field B in the toroid. Derive an expression for B within the iron toroid as a function of the radial distance from the symmetry axis ($r_1<r<r_2$) in terms of the quantities given.

(b) Using the result of part (a), obtain an expression for the self-inductance $L$ of this coil. You should obtain a result of the form $%% L=N^2\cdot L_0$, where $L_0$ is the single-turn inductance. What is $L_0$ in terms of the properties of the toroid given above (i.e., $\mu$, $r_1$, $%% r_2$, and $d$)?

QP 7

Consider a long air-core solenoidal coil with $N$ turns and total inductance $L_0$. A constant current $I$ is flowing in the coil. Answer the following in terms of the quantities $L_0$, $N$, and $I$.

(a) What is the magnitude of the magnetic flux $\Phi _B$ through each turn?

(b) What is the total energy stored in the coil?

Now a rod of soft iron with magnetic permeability $\mu _m$ is inserted into the solenoid, completely filling its interior volume. The current through the coil is held fixed at the value $I$. Answer the following in terms of the quantities $L_0$, $N$, and $I$ (as above) and $\mu _m.$

(c) What is the new magnetic flux $\Phi _B^{\prime }$ through each turn?

(d) What is the new value of inductance $L$?

(e) What is the total energy now stored in the coil? If the energy is different, discuss the origin of the increase or decrease in terms of the principle of conservation of energy.

QP 8 The formula derived in ZAP! and used in Experiment 10 for the op-amp amplifier circuit shown below is correct in the limit that the ``open loop'' gain, $G$, of the op-amp itself is enormous.

Open loop gain $G$ is defined by

$V_{\text{pin 6}}-V_{\text{ground}}=G(V_{\text{pin 3}}^{(+)}-V_{\text{pin 2}}^{(-)})$

where the four voltages are defined in the next figure.

Your op-amp has a value $G\simeq 5\times 10^5$ for DC applications. For slowly varying sinusoidal inputs this value is maintained, but at some high frequency it begins to drop -- eventually reaching values that aren't big at all. Hence, it may be of interest to know how the amplifier circuit above behaves for $G\neq \infty$. a) Find the formula for the amplification factor $V_{%% \text{out}}/V_{\text{in}}$, for finite $G$, in terms of $R_1$, $R_2$, and $G$. You should assume that no current flows in or out of the $+$ or $-$ op-amp inputs even though voltages are applied.

This last idealization concerning op-amp input currents is not precisely valid either. The current that actually flows in or out of the $+$ or $-$ op-amp inputs can be roughly characterized as follows. The real inputs behave as if there were a large resistance $R_{\text{in}}$ to ground which precedes an ideal op-amp that allows no input current flow. This is represented in the following figure.

b) Find the formula for the amplification factor $V_{%% \text{out}}/V_{\text{in}}$ of the original amplifier circuit in terms of $R_1$, $R_2$, and $R_{\text{in}}$, assuming $G=\infty$. c) Describe in words (a few is OK, not more than four sentences) what happens to $V_{\text{out}}$ in the amplifier circuit with a real op-amp when the $+$ and $-$ inputs are inadvertently reversed, i.e., as shown below.

QP9

In your laboratory kit you find an unlabeled inductor. Let $R$ be its internal resistance and $L$ be its inductance. You quickly determine $R$ to be 35 Ohms using your ohmmeter. Curiosity drives you to go to the help lab where you find a 1000 Hz signal generator and a 1 microfarad capacitor. Your affinity for the smell of melting solder then drives you to construct the circuit shown in figure 4.

Using your AC voltmeter, you measure the RMS voltage between points $X$ and $%% Z$ to be 10.1 volts. The RMS voltage between $Y$ and $Z$ is then measured to be 15.5 volts.

(a) What is the RMS current in the circuit?

(b) What are the two values of $L$ (in henries) that are consistent with these data?

(c) For each value of $L$ from part (b) predict the voltmeter reading between $X$ and $Y$. Therefore, one can make this final measurement to deduce the correct value of $L$.

QP 10

Consider a battery connected to an inductor through a switch, as shown. The point of this problem is to begin to figure out what actually happens when the switch is opened.

The instant the switch is opened, it actually becomes a capacitor, with capacitance $C$, and $C\simeq 100$ pF = 10$^{-10}$ F. (It is, after all, two pieces of metal, separated a small distance by an insulator.) The largest resistance in the circuit is $R_{\text{int}}$, the internal resistance of the battery, and $R_{\text{int}}\simeq 0.6$ $\Omega$, while the unloaded voltage of the battery is $V_{\text{o}}$, with $V_{\text{o}}\simeq 3$ V. The inductor has an inductance $L$, with $L\simeq 0.02$ H. (While there certainly are other resistances, capacitances, and inductances in the circuit, these others are negligible in magnitude compared to the ones just described.)

Let $t=0$ be the time of opening the switch. The steady-state behavior established before $t=0$ serves as initial conditions on the $t\geq 0$ system. In particular, if $Q(t)$ is the charge on the capacitor (i.e., the open switch), then $Q(0)=0$. And, if $I(t)$ is the current in the circuit, then $I(0)=V_{\text{o}}/R_{\text{int}}$.

[In your answers for parts a) and b), please use the symbols $V_{\text{o}}$, $R_{\text{int}}$, $C$, and $L$ rather than their numerical values.]

a) What is the differential equation that governs the $t$-dependence of $Q(t)$ for $t\geq 0$? (Hint: draw the effective circuit diagram and follow the voltage drops and rises all the way around á là Kirchhoff's loop rule.)

b) The answer to part a) can be cast into the form of the equation for the $RLC$ circuit (i.e., without any $V_{\text{o}}$) by a change in variables that involves shifting $Q(t)$ by a time-independent constant. What is that constant shift (in terms of the parameters of this problem)?

c) If there were absolutely no resistance at all in this circuit (e.g., $R_{\text{int}}\equiv 0$), it would oscillate indefinitely as $t\rightarrow \infty$. (Imagine, however, that $I(0)$ were still some finite initial value.) What would be the period of those oscillations (in seconds, using the numerical values as provided)?

d) Taking into account the actual value of $R_{%% \text{int}}$, estimate the decay time of the current, i.e., the time it takes to drop roughly to $1/e$ of its $t=0$ value (in seconds, using the numerical values as provided).

QP 11

Recall that a well-designed circuit with an op-amp can be analyzed using two properties of ideal op-amps:

1. The current into the + and - inputs is 0.

2. The + and - inputs are at the same voltage relative to ground.

Consider the circuit shown in figure 5. An AC signal is applied at $v_{IN}$. This circuit forms a frequency-dependent amplifier. It has the property that the input impedance is very large. In other words, no matter how large a voltage is applied at $v_{IN}$, very little current is drawn through the input.

(a) Draw a phasor diagram for the two resistors, the capacitor, and the inductor, indicating $i$, $v_{R_1}$, $v_{R_2}$, $v_C$, $v_{IN}$, and $v_{OUT}$ on the diagram.

(b) What is the ratio of voltage amplitudes $V_{OUT}/V_{IN}$ for this circuit?

(c) In the region of frequency where $\omega L>1/(\omega C)$, does the input voltage $v_{IN}$ lead or lag the output voltage $v_{OUT}$?

(d) What is the ratio of voltage amplitudes $V_{OUT}/V_{IN}$ for values of the frequency $\omega$ which are very large? What is the ratio for $\omega$ very small?

(e) At what frequency $\omega _0$ do you expect the signal to be a maximin? (Note: it is not necessary to differentiate to write down this answer!) What is the value of $V_{OUT}/V_{IN}$ at $\omega =\omega _0$?

QP 12

Traditional electric guitar design goes back well before the invention of op-amps and transistors, but it does include built-in combinations of capacitors and variable resistors that allow the player to adjust the volume and tone of the output with knobs on the face of the guitar. The initial electrical signal is the voltage induced in the ``pick-up'' coil by the oscillatory motion of the magnetized strings. ``Volume'' is altered by feeding the signal through a variable resistor. Tone control is achieved with a variety of variable $R$ filters.

a) If we consider this as a system with a specified $v_{\text{in}}(t)$ and a desired $v_{\text{out}}(t)$, sketch a circuit that could serve as a low-pass filter using a single capacitor $C$ and a single variable resistor $R$, i.e., it would pass from ``in'' to ``out'' all frequencies well below some adjustable point and seriously attenuate signals well above that point. I.e., draw the appropriate connections to a $C$ and an $R$ in place of the ``?" box in the following diagram.

For historical reasons, the industry standard for such variable resistors is a 250 k$\Omega$ pot, i.e., $0\leq R\leq 250$ k$\Omega$.

b) What is the minimum value of $C$ that ensures that the roll-over frequency, i.e., the point where the filter gives $%% V_{\text{out}}/V_{\text{in}}=1/2$ for the respective amplitudes of sinusoidal voltages, can be varied to include the range of 1000 to 10,000 Hz when combined with a 250 k$\Omega$ pot? (Don't forget to distinguish between the angular frequency $\omega$ in radians per second and the oscillation cycle frequency $\upsilon$ in Hz.)

While the considerations above give a reasonable estimate of the appropriate $C$, real guitars are not wired this way. A relevant consideration is that the pick-up is not an ideal voltage source. While it is true that the vibrating magnetized strings induce voltages in the pick-up coil, the coil's voltage output is influenced by the actual current and its time derivative. Just like the battery and power supply internal resistances that you measured, a magnetic pick-up coil has an internal resistance that can be as high as 10 k$\Omega$ -- simply because it is an enormous length of very fine wire. Furthermore, it has its self-inductance, which, at a sizeable fraction of a Henry, can have a significant influence on the circuit at audio frequencies.

Hence a better model of the pick-up coil is a voltage source $v_{\text{in}}$ (due to the action of the strings) in series with the coil's self-inductance $L$ and its internal resistance $R_{\text{int}}$. A typical hook up is then described by the diagram below, where $R_{\text{pot}}$ is the 0 to 250 k$%% \Omega$ variable resistor and $C$ is the accompanying capacitor.

c) Assuming an input voltage of angular frequency $\omega$, find the formula for the voltage amplitude ratio $V_{\text{out}}/V_{\text{in}}$ for the circuit above in terms of $L$, $R_{\text{int}}$, $%% R_{\text{pot}}$, $C$, and $\omega$.

QP 13

A variable inductor (inductance $L_{\text{adj}}$) can be used as a dimmer for lights in an AC circuit, and it's far more efficient than using a variable resistor. Consider a 100 W bulb plugged into a wall socket (which provides 60 Hz 120 V $_{\text{AC}}^{\text{RMS}}$).

First, consider an adjustable resistor as a dimmer, as indicated below.

If $R_{\text{adj}}$ is adjusted so that the bulb runs at 25 W, what is the total (time-averaged) power being drawn from the wall socket? (You should assume that the bulb is approximately ``Ohmic'' over the range 25 to 100 W, i.e., has a fixed resistance $R_{\text{bulb}}$ such that it runs at 100 W when connected directly into the wall socket.)

And compare to using an adjustable $L_{\text{adj}}$, wired in series with the bulb, as shown below.

For what value of $L_{\text{adj}}$ will the bulb run at 25 W? For that value of $L_{\text{adj}}$, what is the total (time-averaged) power being drawn from the wall socket?

QP 14

Consider a 10:1 step-down transformer that plugs into a 60 Hz, 120 V (rms) wall socket and provides 12 V (rms, 60 Hz) for low voltage applications. The primary (high voltage) and secondary (low voltage) coils are wound around a common iron core with a winding number ratio of 10:1. The relevant inductances are $L_{2}=L=0.010$ H, $L_{1}=100L=1.0$ H, and (mutual) $%% M=10L=0.10$ H so that $L_{1}/M=10$, $L_{2}/M=1/10$, and $L_{1}L_{2}=M^{2}$. The resistances of the coils themselves are negligible and are to be ignored in the following, but the output may or may not be connected to a load resistance $R$ -- depending on whether the switch is closed or open. The transformer input and output voltages are $\nu_{\text{in}}(t)=V_{\text{in}%% }\cos \omega t$ and $\nu_{\text{out}}(t)=V_{\text{out}}\cos (\omega t+\phi _{%% \text{out}})$ with $V_{\text{in}}=(120$ V$)\sqrt{2}$ and $\omega =2\pi \times 60$ sec$^{-1}$.

With the switch open, the behavior of this system is governed by the equations

$\nu_{\text{in}}(t)-L_{1}\frac{di_{1}}{dt}=0$ $\nu_{\text{out}}(t)-M\frac{di_{1}}{dt}=0$

While with the switch closed,

$\nu_{\text{in}}(t)-L_{1}\frac{di_{1}}{dt}-M\frac{di_{2}}{dt}=0$ $\nu_{\text{out}}(t)-L_{2}\frac{di_{2}}{dt}-M\frac{di_{1}}{dt}=0$

$\nu_{\text{out}}(t)=i_2 R$

Consider first the situation of the transformer plugged into the wall socket but no connection made to the transformer output, i.e., the switch is open in the accompanying circuit diagram. Hence, there is no current in the secondary coil, i.e., $i_{2}(t)=0$, and the simpler first set of equations apply. What is the phase angle between the voltage across the primary coil, $v_{\text{in}}(t)$, and the current that flows in the primary coil, $i_{1}(t)$? (Give a numerical value in degrees.) What is the instantaneous power delivered to the primary coil as a function of time? (Express your answer in terms of the parameters given.)

Consider now closing the switch and connecting the transformer output to some device which, for simplicity, is modeled as an $R=100$ $\Omega$ load resistance. What is the time-averaged power dissipated in the secondary circuit?

What is the time-averaged power delivered by $\nu_{\text{in}}$?

QP 15

At the mid-point of a long, thin, straight wire that carries a steady current $I$ is a capacitor made of two parallel, circular plates of radius $%% R$ and separation $D$, with $D\ll R$, as shown. The current charges the capacitor.

a) What is the magnitude of the magnetic field $%% B(r)$ half way between the plates and at a distance $r$ from the axis that passes through their centers? (Give the answer for the whole range $0<r<\infty$.)

b) Sketch $B(r)$ versus $r$ for $0 \leq r\leq 2R$. On the same graph use a dotted line to represent the magnitude of the magnetic field a distance $r$ from the wire at a location along the wire that is far from the capacitor. (Please make the diameter of your dots about double the thickness of your first line so that there is no ambiguity if there are values of $r$ for which the two magnetic fields have the same value. Also, please take some care in making the sketch. If the function in question is linear, don't give it curvature and vice versa. If it is concave down, don't draw it concave up. And if it is differentiable, don't draw it with a kink.)

QP 16

$\parbox{3.0 in}{ A straight wire carries a current $I(t)$ along th... ... the lower, $z<0$ hemisphere? Explain your answer. \bigskip \bigskip \par }$ $\parbox{2.4in}{ \begin{figure}[h] \centerline{{\epsfxsize=1.5in \epsfbox{QP2cI.eps}}} \end{figure}}$

$\parbox{3.0 in}{ d) Evaluate the magnitude of the magnetic field $B$... ...zero for $\theta =0$ and half of the total for $\theta =90^{\text{o}}$.) }$ $\parbox{2.5in}{ \begin{figure}[h] \centerline{{\epsfxsize=1.5in \epsfbox{QP2cII.eps}}} \end{figure}}$

QP 17

For the Earth, the Sun's gravitational attraction dominates by far over the effects of radiation pressure. Comets, however, have tails consisting of small particles of condensed dust and ice.

(a) Assume that the dust grains in a comet's tail are reflecting spheres and that they all have the same density, $\rho$. Show that particles with radius less than some critical value will be blown out of the solar system. You may approximate the spheres as reflecting disks oriented toward the sun in considering the effect of incident radiation.

(b) Estimate the critical size numerically, assuming the density $\rho =1$ gm/cm$^3$ and the power output of the sun $W_{\text{Sun}}=3.8\times 10^{26}$ Watts. You may need the following constants: G $=6.7\times 10^{-11}$ N m$^2$/kg$^2$ and the mass of the sun $M=2\times 10^{30}$ kg.

QP 18

A microwave communications link consists of two identical antennae. Each antenna works symmetrically in send or receive mode.

a) Assume antenna 1 radiates 1 W/m$^2$ into a beam of circular cross section with a diameter equal to the diameter of the parabolic reflector, $D$ = 1 m. Calculate $E_{\sc rms}$ and $B_{\sc rms}$ for the radiated beam.

b) Assume the parabolic reflector falls off antenna 1, which still radiates the same power but now emits this power isotropically, i.e., uniformly in all directions. How much power is received at Station 2?

QP 19

(a) A high-speed ($v=0.99c$) express train is traveling past the Pasadena station. Axel, Brad, and Clyde are standing in the station. Axel and Clyde are at the ends of the station platform; Brad is in the center. (Note that all observers are wearing high-priced, accurate [and precise] Swiss watches that have been synchronized in their own rest frame.) The proper length of the train is 100 m. What is the length of the train if measured in the rest frame of the station?

(b) Axel measures the time it takes for the train to pass him, front to back. What does he measure?

(c) Three observers are riding the train, Adrienne, Beatrice, and Chloe. Adrienne and Chloe are at the ends of the train; Beatrice is at the center. Each of them notices the time on Brad's watch as they pass him, along with the time on their own watch. Beatrice notes that both on her own watch and on Brad's watch $t=t^{\prime }=0$. What time does Adrienne observe on her watch and on Brad's as she passes him?

(d) What time does Brad observe for Beatrice's passage?

QP 20 (The saga of QP15 continues....)

(a) Suddenly, as if the participants lived in a Physics text gone terribly wrong, two huge lightning bolts hit the front and back of the train. Axel and Clyde record the times of the bolts, and they are simultaneous in the station frame. (All participants survive, as this is the Hollywood version.) A little bit later, Brad sees the bolts hitting simultaneously and faints instantly. How much later?

(b) Denise (the ``Fourth Woman'') happens to be standing in the train at such a location as to pass Brad at the instant he faints. Does she also see both bolts of lightning simultaneously? Is she standing in the center of the train? How do you know? What does she conclude about the simultaneity of the bolts hitting the train in her frame?

QP 21

Sir Bevis and Count Rumpkopf were to joust to the death in the lists at Canterbury in the Spring tourney. The wizard Merlin approached Bevis with an offer. In return for rights to all Bevis' lands, Merlin would provide enchanted oats for Bevis' horse that would allow the horse to run at relativistic speed. Merlin explained the advantage as follows. The knights' lances were of equal length when at rest. If they approached each other at relativistic speed, in Bevis' own rest frame, Rumpkopf's lance would be genuinely shorter. Having the longer lance, Bevis could pierce Rumpkopf's chest and then jump out of the way of Rumpkopf's approaching lance. (Of course, unbeknownst to Bevis, Merlin made the same deal with Rumpkopf.)

Let $L_o$ be the rest length of each knight's lance. They charge at each other with equal speed $v'$ relative to the ground. Let $v$ be the speed at which Rumpkopf approaches Bevis as determined in Bevis' rest frame. (Express answers in terms of $L_o$ and $v$.)

a) What is the length of Rumpkopf's lance in Bevis' rest frame?

Consider carefully the following possible events. Event A: Bevis' lance pierces Rumpkopf's chest. The horses continue running at a constant speed, and (event B:) Rumpkopf's lance pierces Bevis' chest.

b) What is the distance between event A and event B in Bevis' rest frame? (Hint: If the answer is not immediately obvious, consider the above sketch, which is meant to represent an instant in Bevis' rest frame.)

c) What is the time between event A and event B in Bevis' rest frame?

d) How does the distance of part b) divided by the time of part c) compare to the speed of light, $c$? (Is it greater than $%% c$, less than $c$, or is it not determined without numerical values for the parameters of the problem?)

e) Does Sir Bevis have time after seeing his lance pierce Count Rumpkopf to jump off his horse and avoid death? (You may assume that his nerves are excellent and his reaction time is essentially zero.)

f) What is the time between event A and event B in Merlin's rest frame (in which the horses have equal [but opposite] speeds)?

QP 22

In an electron-positron collider, the electron is accelerated to an energy of 5.11 MeV. The electron mass is 511 keV/c$^2.$

(a) What is the velocity (expressed as $v/c$), kinetic energy, and momentum of the electron?

A positron is accelerated in the opposite direction to the same energy and collides with the electron.

(b) What is the maximum mass of a new particle $X$ created by the collision [ $e^{+}+e^{-}\rightarrow X$] ? What is the new particle's momentum in this case?

(c) What is the velocity (expressed as $v/c$) of the positron in the electron's rest frame? [Careful! This calculation requires some precision.]

(d) What is the total energy in this frame? Discuss why most accelerators now use colliding beams as opposed to fixed targets.

A new particle, the Techion, is created with a rest mass 3 MeV and total energy 6 MeV in the laboratory frame. In the lab frame, the Techion travels 2 m between where it is created and where it decays.

(e) What is the proper decay time (i.e., the decay time in its own rest frame) for the Techion?