Physics 1c Practical – QP Problems – 2012

QP 1

V

V

B

L’

Ex

L

D

d

(not to scale)

x

x

x

x

x

x

x

x

B

l

v

Oscilloscope display tubes use voltages applied to parallel plates both to accelerate a beam ofelectrons up to high speed and to steer the beam vertically and horizontally to produce the desireddisplay. 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 1mm wide), which glow red, green, or blue (depending on the phosphor) when struck. The Earth’smagnetic field will also contribute to the deflection of the beam — but by a direction and an amountthat depends on the location and orientation of the tube relative to the Earth.

Assuming an accelerating high voltage of V = 20,000 V and a tube length or electron path lengthof L′ = 50 cm from high voltage plates to the screen, estimate the maximum possible total deflectiond of the beam at the screen due to the Earth’s field.

(From the magnitude of your answer relative to the pixel size you will learn whether the effect ofthe Earth’s field is negligible, minor, or something that must be actively compensated. By the way,computer monitors have much smaller pixels, e.g. 0.2 mm, to allow clear display of text.)

QP 2

A Los Angeles County civil engineer, looking over plans for the light rail system (before it wasfinally built) was concerned about a particular aspect of one of the design proposals. The highvoltage line that was to power the electric locomotives lay between the tracks, which were groundedand served as the circuit return for the locomotive power. But buried just beneath the high voltageline were sensing and signal control devices. Remembering her freshman E&M, she was concernedabout induced EMF ’s due to the AC power line effecting the sensors and controls. So she proposedthe 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) = Io cosωt

1

and let the device of interest be modeled by a square loop of wire of area A located a distance Rfrom the power line. R A1/2 so that the magnetic field due do I(t) can be approximated as aconstant at any given time over the whole area A. Assume that the power line lies in the plane of Ato maximize its effect.

I(t)

R

A

What is the emf 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 is1.5 MW (i.e., 1.5× 106 watts). In the design, R was about 30 cm, and A is typically no bigger than10 cm × 10 cm (i.e., 10−2 m2).

What is the numerical estimate of the emf E induced in a typical device due to the power line, forthe values given?

QP 3

At the Caltech Athletics Department 1978 awards dinner, Mary Mack (B.S. ’78), three-time NCAAWestern Division jump rope champ, was presented a deluxe jumping rope whose outer layer was abraid of copper and brass threads. It sparkled, but it also conducted electricity from one end to theother. Hence, when jumping in the Earth’s magnetic field (B ∼ 0.5× 10−4T), Miss Mack generateda voltage V between the ends of the rope. (Also, Miss Mack could easily do 5 jumps per second.)

To determine whether it was safe to jump without insulating gloves, first consider the inducedvoltage V . In particular, write

V (ϕ, t) = Vmax · f(ϕ) · g(t)

where Vmax is the maximum value of V (ϕ, t), ϕ is the direction she faces with ϕ = 0 being due(magnetic) north, ϕ = π/2 being due west, etc., and t is the time.

a) What is Vmax? Express your answer first algebraically in terms of B (the field of the Earth), l(the length of the rope), and ω, ν, or T (the angular frequency, frequency, or period of the motion,respectively [ω = 2πν = 2π/T ]) and assuming that the motion is uniform and the rope forms asemicircle. Write down reasonable numerical values for the relevant parameters to find a numericalestimate, in Volts, of the potential generated.

b) What is g(t) in terms of the parameters of the problem?

c) What is f(ϕ)?

2

Held tightly in sweaty hands, with full hand contact, the rope-to-two-hands (in series) resistancewas only 500Ω in total. Compared to that contact resistance, the rope and her internal resistanceswere negligible.

d) Estimate the largest possible peak current induced by the Earth’s field that passed through herarms 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 wellas a horizontal component, was there an orientation for which Miss Mack would have generatedessentially 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?

3

QP 4

θ

x

v

B

figure 1

A conducting bar is sliding at velocity v to the right on a V-shaped conducting rail, as shownin figure 1. There is a uniform magnetic field B out of the page. The rails are frictionless andresistanceless. The bar has a resistance λ per unit length. The half-angle of the V is θ.

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

4

QP 5

In a standard design of an AC voltage transformer, two wire coils are wound around the centralpost of an iron “core” (which is really more of a frame) as shown in the sketches below. Iron is chosenfor its outstanding magnetic permeability. Note that in the sketches, the dashed lines denote edgesthat 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 Ohmicheating and consequent power loss and because, by Lenz’s Law, they reduce the desired magneticfield.

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

5

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, whichone 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 tobe less than one lamination thickness. Would there be eddy currents in D? Where? How big? Whatlimits their magnitude?

6

QP 6

figure 2

r

r

d

1

2

N turns

An iron toroid of rectangular cross section has an inner radius r1, an outer radius r2, and a thicknessd, as shown in figure 2. A large number of turns, N , of wire are wound uniformly around the ironcore, which has magnetic permeability µ. (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. Derivean expression for B within the iron toroid as a function of the radial distance from the symmetryaxis (r1 < r < r2) in terms of the quantities given.

(b) Using the result of part (a), obtain an expression for the self-inductance L of this coil. Youshould obtain a result of the form L = N2 · L0, where L0 is the single-turn inductance. What is L0

in terms of the properties of the toroid given above (i.e., µ, r1, r2, and d)?

QP 7

Consider a long air-core solenoidal coil with N turns and total inductance L0. A constant currentI is flowing in the coil. Answer the following in terms of the quantities L0, N , and I.

(a) What is the magnitude of the magnetic flux ΦB through each turn?

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

Now a rod of soft iron with magnetic permeability µm is inserted into the solenoid, completelyfilling its interior volume. The current through the coil is held fixed at the value I. Answer thefollowing in terms of the quantities L0, N , and I (as above) and µm.

(c) What is the new magnetic flux Φ′B 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 ofthe increase or decrease in terms of the principle of conservation of energy.

7

QP 8

The formula derived in ZAP! and used in Experiment 10 for the op-amp amplifier circuit shownbelow is correct in the limit that the “open loop” gain, G, of the op-amp itself is enormous.

V V

R

R

-

in out

2

1

+

Open loop gain G is defined by

Vpin 6 − Vground = G(V(+)pin 3 − V

(−)pin 2)

where the four voltages are defined in the next figure.

(In actuality, the op-amp Vground is slightly ambiguous until some sort of feedback is established.Another one of the op-amp’s undrawn inputs can be used to adjust that value to agree exactly withsome other reference voltage. This would allow, for example, adjusting the output voltage of your101× amplifier to zero when the amplifier input voltage is zero.)

Vpin 2(-) -

+

V

VV

ground

pin 6

pin 3(+)

Your op-amp has a value G ' 5 × 105 for DC applications. For slowly varying sinusoidal inputsthis value is maintained, but at some high frequency it begins to drop — eventually reaching valuesthat aren’t big at all.

(The frequency at which a significant decline in G sets in is different for different models of op-amp.However, there are always engineering trade-offs, and better high frequency response is not always thesole desideratum; in some cases it is actually undesirable.

Hence, it may be of interest to know how the amplifier circuit above behaves for G 6=∞.)

8

a) Find the formula for the amplification factor Vout/Vin, for finite G, in terms of R1, R2, and G.You should assume that no current flows in or out of the + or − op-amp inputs even though voltagesare applied.

This last idealization concerning op-amp input currents is not precisely valid either. The currentthat 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 Rin to ground which precedes an idealop-amp that allows no input current flow. This is represented in the following figure.

R in

R in

-

+

-

+

real ideal

b) Find the formula for the amplification factor Vout/Vin of the original amplifier circuit in termsof R1, R2, and Rin, assuming G =∞.

(In contrast to all the previous real-world modifications to ideal behavior explored in this quiz,modern designs have made this one pretty irrelevant. In particular, your op-amp has Rin ∼ 1012Ω.)

c) Describe in words (a few is OK, not more than four sentences) what happens to Vout in theamplifier circuit with a real op-amp when the + and − inputs are inadvertently reversed, i.e., asshown below.

Vin

V

-

out

+

(no electricalconnection

here)

9

QP9

R L

C = 1 µ f

figure 4

X

Y

Z

In your laboratory kit you find an unlabeled inductor. Let R be its internal resistance and L beits inductance. You quickly determine R to be 35 Ohms using your ohmmeter. Curiosity drives youto go to the help lab where you find a 1000 Hz signal generator and a 1 microfarad capacitor. Youraffinity 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.

10

QP 10

Consider a battery connected to an inductor through a switch, as shown. The point of this problemis 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 ' 100pF = 10−10 F. (It is, after all, two pieces of metal, separated a small distance by an insulator.) Thelargest resistance in the circuit is Rint, the internal resistance of the battery, and Rint ' 0.6 Ω, whilethe unloaded voltage of the battery is Vo, with Vo ' 3 V. The inductor has an inductance L, withL ' 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 = 0serves as initial conditions on the t ≥ 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) = Vo/Rint.

[In your answers for parts a) and b), please use the symbols Vo, Rint, C, and L rather than theirnumerical values.]

a) What is the differential equation that governs the t-dependence of Q(t) for t ≥ 0? (Hint: drawthe effective circuit diagram and follow the voltage drops and rises all the way around a la Kirchhoff’sloop rule.)

b) The answer to part a) can be cast into the form of the equation for the RLC circuit (i.e., withoutany Vo) by a change in variables that involves shifting Q(t) by a time-independent constant. Whatis 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., Rint ≡ 0), it would oscillateindefinitely as t→∞. (Imagine, however, that I(0) were still some finite initial value.) What wouldbe the period of those oscillations (in seconds, using the numerical values as provided)?

d) Taking into account the actual value of Rint, estimate the decay time of the current, i.e., thetime it takes to drop roughly to 1/e of its t = 0 value (in seconds, using the numerical values asprovided).

11

QP 11

C

R

R

+

-

1

2

v vIN OUT

i

figure 5

L

Recall that a well-designed circuit with an op-amp can be analyzed using two properties of idealop-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 vIN . This circuit forms afrequency-dependent amplifier. It has the property that the input impedance is very large. In otherwords, no matter how large a voltage is applied at vIN , very little current is drawn through the input.

(a) Draw a phasor diagram for the two resistors, the capacitor, and the inductor, indicating i, vR1,

vR2, vC , vIN , and vOUT on the diagram.

(b) What is the ratio of voltage amplitudes VOUT /VIN for this circuit?

(c) In the region of frequency where ωL > 1/(ωC), does the input voltage vIN lead or lag theoutput voltage vOUT ?

(d) What is the ratio of voltage amplitudes VOUT /VIN for values of the frequency ω which are verylarge? What is the ratio for ω very small?

(e) At what frequency ω0 do you expect the signal to be a maximin? (Note: it is not necessary todifferentiate to write down this answer!) What is the value of VOUT /VIN at ω = ω0?

12

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 toadjust the volume and tone of the output with knobs on the face of the guitar. The initial electricalsignal 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 witha variety of variable R filters.

a) If we consider this as a system with a specified vin(t) and a desired vout(t), sketch a circuitthat could serve as a low-pass filter using a single capacitor C and a single variable resistor R, i.e., itwould pass from “in” to “out” all frequencies well below some adjustable point and seriously attenuatesignals 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.

v (t)in ? v (t)

out

For historical reasons, the industry standard for such variable resistors is a 250 kΩ pot, i.e., 0 ≤R ≤ 250 kΩ.

b) What is the minimum value of C that ensures that the roll-over frequency, i.e., the point wherethe filter gives Vout/Vin = 1/2 for the respective amplitudes of sinusoidal voltages, can be varied toinclude the range of 1000 to 10,000 Hz when combined with a 250 kΩ pot? (Don’t forget to distinguishbetween the angular frequency ω in radians per second and the oscillation cycle frequency υ in Hz.)

While the considerations above give a reasonable estimate of the appropriate C, real guitars arenot wired this way. A relevant consideration is that the pick-up is not an ideal voltage source. Whileit is true that the vibrating magnetized strings induce voltages in the pick-up coil, the coil’s voltageoutput is influenced by the actual current and its time derivative. Just like the battery and powersupply internal resistances that you measured, a magnetic pick-up coil has an internal resistance thatcan be as high as 10 kΩ — 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 onthe circuit at audio frequencies.

Hence a better model of the pick-up coil is a voltage source vin (due to the action of the strings)in series with the coil’s self-inductance L and its internal resistance Rint. A typical hook up isthen described by the diagram below, where Rpot is the 0 to 250 kΩ variable resistor and C is theaccompanying capacitor.

v v

L R

R

C

int

potoutin

c) Assuming an input voltage of angular frequency ω, find the formula for the voltage amplituderatio Vout/Vin for the circuit above in terms of L, Rint, Rpot, C, and ω.

13

QP 13

A variable inductor (inductance Ladj) can be used as a dimmer for lights in an AC circuit, and it’sfar more efficient than using a variable resistor. Consider a 100 W bulb plugged into a wall socket(which provides 60 Hz 120 VRMS

AC ).

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

R

R

V

bulb

AC

adj

If Radj is adjusted so that the bulb runs at 25 W, what is the total (time-averaged) power beingdrawn from the wall socket? (You should assume that the bulb is approximately “Ohmic” over therange 25 to 100 W, i.e., has a fixed resistance Rbulb such that it runs at 100 W when connecteddirectly into the wall socket.)

And compare to using an adjustable Ladj, wired in series with the bulb, as shown below.

R

L

V

bulb

AC

adj

For what value of Ladj will the bulb run at 25 W?

For that value of Ladj, what is the total (time-averaged) power being drawn from the wall socket?

14

QP 14

Consider a 10:1 step-down transformer that plugs into a 60 Hz, 120 V (rms) wall socket andprovides 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. Therelevant inductances are L2 = L = 0.010 H, L1 = 100L = 1.0 H, and (mutual) M = 10L = 0.10H so that L1/M = 10, L2/M = 1/10, and L1L2 = M2. The resistances of the coils themselves arenegligible and are to be ignored in the following, but the output may or may not be connected to aload resistance R — depending on whether the switch is closed or open. The transformer input andoutput voltages are νin(t) = Vin cosωt and νout(t) = Vout cos(ωt + φout) with Vin = (120 V)

√2 and

ω = 2π × 60 sec−1.

i (t)2

outv (t)v (t) L L

M

i (t)

R

1 2in

1

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

νin(t)− L1di1dt = 0

νout(t)−M di1dt = 0

While with the switch closed,

νin(t)− L1di1dt −M

di2dt = 0

νout(t)− L2di2dt −M

di1dt = 0

νout(t) = i2R

Consider first the situation of the transformer plugged into the wall socket but no connection madeto the transformer output, i.e., the switch is open in the accompanying circuit diagram. Hence, thereis no current in the secondary coil, i.e., i2(t) = 0, and the simpler first set of equations apply.

What is the phase angle between the voltage across the primary coil, vin(t), and the current thatflows in the primary coil, i1(t)? (Give a numerical value in degrees.)

What is the instantaneous power delivered to the primary coil as a function of time? (Express youranswer in terms of the parameters given.)

Consider now closing the switch and connecting the transformer output to some device which, forsimplicity, is modeled as an R = 100 Ω load resistance.

What is the time-averaged power dissipated in the secondary circuit?

What is the time-averaged power delivered by νin?

15

QP 15

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

I

R

r

D

I

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

b) Sketch B(r) versus r for 0 ≤ r ≤ 2R. On the same graph use a dotted line to represent themagnitude of the magnetic field a distance r from the wire at a location along the wire that is farfrom the capacitor. (Please make the diameter of your dots about double the thickness of your firstline so that there is no ambiguity if there are values of r for which the two magnetic fields have thesame 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 isdifferentiable, don’t draw it with a kink.)

16

QP 16

A straight wire carries a current I(t) along the negative z-axis from -∞ to the origin. There thewire ends, and a charge q(t) builds up on a small metal sphere centered at the origin. The goal ofthis problem is to compute the resulting magnetic field B at points P and P ′ using the fact that thissituation has rotational symmetry about the z-axis and using the Ampere-Maxwell law, i.e., includingthe displacement current.

Both P and P ′ are located a distance R in the y-direction from the z-axis. P is at z = 0, and theline from the origin to P ′ makes an angle θ with the z-axis as shown.

a) What is the electric flux ΦE(t) that passes through the upper (z > 0) hemisphere of radius R“centered” at the origin?

b) Use the z > 0 hemisphere to deduce the magnitude B of the magnetic field at P in terms ofI(t) and R, using the Ampere-Maxwell law.

c) Would you get the same answer if you used the lower, z < 0 hemisphere? Explain your answer.

z>0 hemisphere

y

z

P

P’

I(t)

θ

R

θ

r

polar cap

d) Evaluate the magnitude of the magnetic field B at P ′. The following geometrical fact shouldprove useful: The part of the surface area of a sphere of radius r (i.e., a sphere with 4πr2 total area)that forms the “polar cap” within an angle θ of the polar axis is 2πr2(1− cos θ). (Note that this caparea is zero for θ = 0 and half of the total for θ = 90o.)

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QP 17

For the Earth, the Sun’s gravitational attraction dominates by far over the effects of radiationpressure. 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 thesame density, ρ. Show that particles with radius less than some critical value will be blown out ofthe solar system. You may approximate the spheres as reflecting disks oriented toward the sun inconsidering the effect of incident radiation.

(b) Estimate the critical size numerically, assuming the density ρ = 1 gm/cm3 and the poweroutput of the sun WSun = 3.8× 1026 Watts. You may need the following constants: G = 6.7× 10−11

N m2/kg2 and the mass of the sun M = 2× 1030 kg.

QP 18

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

a) Assume antenna 1 radiates 1 W/m2 into a beam of circular cross section with a diameter equalto the diameter of the parabolic reflector, D = 1 m. Calculate Erms and Brms for the radiated beam.

L = 10 km

D = 1 m

b) Assume the parabolic reflector falls off antenna 1, which still radiates the same power but nowemits this power isotropically, i.e., uniformly in all directions. How much power is received at Station2?

18

QP 19

ChloeBeatriceAdrienne

Axel Brad Clyde

Photograph taken by camera at the stationfigure 7

v=0.99c

(a) A high-speed (v = 0.99c) express train is traveling past the Pasadena station. Axel, Brad, andClyde are standing in the station. Axel and Clyde are at the ends of the station platform; Brad isin the center. (Note that all observers are wearing high-priced, accurate [and precise] Swiss watchesthat have been synchronized in their own rest frame.) The proper length of the train is 100 m. Whatis 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 areat the ends of the train; Beatrice is at the center. Each of them notices the time on Brad’s watch asthey pass him, along with the time on their own watch. Beatrice notes that both on her own watchand on Brad’s watch t = t′ = 0. What time does Adrienne observe on her watch and on Brad’s asshe 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 lightningbolts hit the front and back of the train. Axel and Clyde record the times of the bolts, and they aresimultaneous in the station frame. (All participants survive, as this is the Hollywood version.) Alittle 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 topass Brad at the instant he faints. Does she also see both bolts of lightning simultaneously? Isshe standing in the center of the train? How do you know? What does she conclude about thesimultaneity of the bolts hitting the train in her frame?

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QP 21 (Following long established tradition, we escape to the realm of Fantasy & Science Fictionto elucidate issues in Special Relativity.)

Sir Bevis and Count Rumpkopf were to joust to the death in the lists at Canterbury in the Springtourney. 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 relativisticspeed. Merlin explained the advantage as follows. The knights’ lances were of equal length when atrest. If they approached each other at relativistic speed, in Bevis’ own rest frame, Rumpkopf’s lancewould be genuinely shorter. Having the longer lance, Bevis could pierce Rumpkopf’s chest and thenjump out of the way of Rumpkopf’s approaching lance. (Of course, unbeknownst to Bevis, Merlinmade the same deal with Rumpkopf.)

Let Lo 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 inBevis’ rest frame.

(In Bevis’ rest frame, Merlin and the ground itself approach Bevis at speed v′ wherev′ = (c2/v)1− [1− (v/c)2]1/2.)

(Express answers in terms of Lo and v.)

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

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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 isnot 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 parametersof the problem?)

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

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

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QP 22

In an electron-positron collider, the electron is accelerated to an energy of 5.11 MeV. The electronmass is 511 keV/c2.

(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− → X] ? Whatis 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 beamsas opposed to fixed targets.

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

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

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