1. Light in air is incident on the surface
of a transparent substance at an angle of 58_ with

the normal. The reected and refracted rays
are observed to be mutually perpendicular.

(a) What is the index of refraction of the
transparent substance? Hint: Note that

sin (90_???? _) = cos _.

(b) What is the critical angle for the
total internal reection in this substance?

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2. A stationary naval destroyer is equipped
with sonar that sends out 40 MHz pulses of

sound. The destroyer receives reected
pulses back from a submarine directly below

with a time delay of 80 ms at a frequency
of 39.959 MHz.

(a) What is the depth of the submarine?

(b) Based on the frequency shift, is the
submarine ascending or descending?

(c) If the speed of sound in seawater is
1.54 km/s, then what is the vertical speed

of the submarine (remember that, since the
pulse is reected, it's going to be

Doppler shifted twice!).

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3. When electrons transition from one orbit
to lower one, they emit a photon of a speci_c

frequency (and so a speci_ccolor). In one
of these cases the electron jumps from the

third to second orbit, emitting a photon of
frequency f = 4:57 _ 1014 Hz, which is in

the red region. In another case the
electron jumps from the _fth orbit down to the

second, and emits a photon of frequency
6:91 _ 1014 Hz, which is in the blue region of

the spectrum. This light is then sent
through a di_raction grating with 4500 lines/cm.

(a) What are the wavelengths of the two
spectral lines?

(b) What is the distance between the slits
in the grating?

(c) What is the angular separation, __, (in
degrees) between the two m = 1 spectral

lines?

(d) What about for the m = 2 spectral
lines?

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4. Suppose you have two in_nite straight
line charges, of charge per unit length _, a

distance d apart, moving along at a
constant speed v. The moving charge densities

constitute a current I = _v, and so the
charges behave like current-carrying wires.

Because the wires carry the same charge
density, _, they repel each other electrically.

But, because they are current-carrying
wires, with currents moving in the same di-

rection, they attract each other
magnetically. How great would v have to be in order

for the magnetic attraction to balance the
electrical repulsion? Work out the actual

number... Is this a reasonable sort of
speed? (Hint - use Gauss's law to get the electric

_eld, and Ampere's law to get the magnetic
_eld of the wire. Then determine the force

per unit length from each _eld, Felec=L =
_E, and Fmag=L = IB.)

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5. The galactic magnetic _eld in some
region of interstellar space has a magnitude of

1:00_10????9 T. A particle of dust has mass
10:0 _g and a total charge of 0:300 nC. How

many years does it take for the particle to
complete a revolution of the circular orbit

caused by its interaction with the magnetic
_eld? You can assume that the orbit is

perpendicular to the magnetic _eld.

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6. Inchy, an inchworm, is inching along a
cotton clothesline. The 5 meter long clothesline

has a mass of 0.2 kilograms, and is pulled
tight under 100 N of tension. Vivian is

hanging up her swimsuit 0 meters from one
end when she sees Inchy 2.54 cm (one inch)

from the opposite end. She plucks the
clothesline sending a terrifying 3 centimeter high

wave pulse toward Inchy. If Inchy crawls at
1 inch per second, will he get to the end

of the clothesline before the pulse reaches
him? If so, how much time does he have to

spare? If not, how far does he make it
before the pulse kicks poor Inchy o_ the line?

(Note: don't forget that Inchy is running
away from the pulse, so the time for it to

reach him is not just his distance divided
by the wave velocity!)

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Extra Credit Question!!

The following is worth 15 extra credit
points!The _gure to the right shows the actual produc-

tion of the antimatter partner to the
electron,the positron, in a bubble chamber immersed in

a uniform magnetic _eld. The positron has
ex-actly the same mass as an electron, but opposite

charge. The di_erent tracks show the
trajecto-ries of the di_erent particles. In the reaction at

the top, a high-energy gamma ray (just a
high-frequency light wave) is absorbed by an electron,

which scatters away to the right, producing
anelectron-positron pair though the reaction

+ e
! e???? + e+ +
e????;where is the incident gamma ray. A similar ef-fect
occurs in the bottom

reaction, but the originalelectron very
likely had littlenal velocity. Let'ssee what we can tell about

this reaction.

(a) Is charged conserved in this reaction?
Explain.

(b) Why do the electrons and positrons
curve in di_erent directions?

(c) In what direction (into or out of the
page) is the magnetic _eld? How can you

tell?

(d) How does the velocity of the spiraling
electron compare to that of the spiraling

positron? How can you tell?

(e) Accelerating charges radiate energy.
Explain why this leads to the spiralingmo-

tion for the particles.

(f) Positrons and electrons can come
together and annihilate, releasing two photons

in the processe???? + e+ ! + :

Explain why we get two photons, and not
one. Hint: consider the the two particles

initially at rest right next to each other.

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A more colorful picture is seen to the
right, wherethe positrons are shown in red, and the electrons

in green.

(a) Initially we have an electron and a
photon. Sincethe photon isn't charged, the initial charge is justthat on the
electron,????e. After the reaction, we'veproduced an electron and positron pair.
The netcharge of the pair is????e+e = 0, and since we stillhave an electron the net charge after the
reactionis????e, and so charge is conserved.

(b) The electrons and positrons are
oppositelycharged, and so experience opposite forces in the

magnetic _eld. This leads to motion in
di_erentdirections.

(c) The electron is curving in the
counterclockwisedirection, while the positron is curving in the

clockwise direction. From the magnetic
force law,~F = q~v _ ~B , for the positively charged positron

the velocity is down, and the force is
initially tothe left, and so the direction of the magnetic _eld

is out of the page.

(d) The faster the speed of the particle,
the biggerthe radius of the curve. Since the positron has a

bigger spiral, it has a bigger speed.

(e) As the charges radiate, they lose
energy. The only place that they can lose itfrom is their kinetic energy, and
so, they slow down. The slower they move, thesmaller the radius of their
curvature. So, the curve tightens as they slow down,leading to the spiral.

(f) Suppose we had an electron and a
positron at rest and just brought them together.

In this case the initial momentum is zero,
but the photon carries momentum. So,

a single photon would violate momentum
conservation. Two photons, emitted

back-to-back, would conserve momentum. This
means we need two photons.