Physics 9 Fall 2010

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_c color). 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 an electron-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 original electron very likely had littlenal
velocity. Let's see 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 spiraling mo-

tion for the particles.

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

in the process e???? +
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, where the
positrons are shown in red, and the electrons

in green.

(a) Initially we have an electron and a photon. Since the
photon isn't charged, the initial charge is just that on the electron, ????e. After the reaction, we've produced an electron and
positron pair. The net charge of the pair is ????e+e
= 0, and since we still have an electron the net charge after the reaction is ????e, and so charge is conserved.

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

magnetic _eld. This leads to motion in di_erent directions.

(c) The electron is curving in the counterclockwise direction,
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 to the
left, and so the direction of the magnetic _eld

is out of the page.

(d) The faster the speed of the particle, the bigger the
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 it from is their kinetic energy, and so, they slow down. The
slower they move, the smaller 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.