This is the Final Exam for College Physics II. It is comprehensive and covers every topic dealt with in the course. There are 6 questions and your score will consist of your best 5 answers. Each question is worth 6 points. The first three questions deal with the electromagnetic half of the course, the second three questions deal with the optics and modern physics parts of the course. There will be no retest for this test, so feel free to draw on the test itself and hand it in with your answers.

Imagine that you have been shipwrecked on a desert island, with nothing but a cargo container full of simple DC circuit elements. MacGyver like, you decide to build, using only resistors, capacitors, batteries and wires, a microwave transmitter, with which you intend to communicate with a satellite overhead in order to request a rescue.

Q. 1 You have an RC circuit like the one illustrated, in which there are two wires in parallel with the capacitor which do not quite meet each other. The endpoints of these two wires form a potential spark gap, where, if the electrical voltage gets high enough, a spark of current might arc across the air in the gap. This happens when the voltage between the two wire endpoints reaches the breakdown voltage of the air in the gap. When this happens the resistance of the air becomes very low (whereas normally it is so high that we can treat it as infinite), and a lot of current flows through, discharging the capacitor.

a) Draw a graph of voltage versus time showing how the
voltage across the capacitor changes with time once the battery is switched on.
On your graph label the voltage V and the time constant of the circuit t, but don’t worry about what their numerical
value is yet. Assume that V > V_{b}, but not much greater, where V
is the battery voltage and V_{b} is the breakdown voltage of the air in
the spark gap, and that the time it takes the capacitor to discharge through
the gap is close to zero seconds. Draw the graph out to at least 4 time
constants of the circuit (4t). Note
that once the air in the gap does not have current flowing through it, it will
return to normal, and will need a new voltage of V_{b} across it before
it will breakdown again. (4 points)

b) On the circuit diagram, and assuming the circuit is in a state where the capacitor is charged up, use + signs and – signs to label the positive and negative poles of the battery, the positively and negatively charged plates of the capacitor, and also which sides of the spark gap have a positive or a negative electric potential (voltage). Indicate also the direction of conventional current in the circuit as drawn. (2 points)

Q. 2 a) We now focus on the electric field in the capacitor and in the spark gap in the circuit illustrated in Q. 1. In two different diagrams, sketch the electric field lines and the equipotential lines in the capacitor and the spark gap when the capacitor is charged up almost to the breakdown voltage. (4 points)

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Resistor side of capacitor | | Battery side

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Resistor side of spark gap Battery side

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b) At all times the voltage across the capacitor and the
voltage across the spark gap are equal. Since this is the case, can you
identify, from the diagram you drew in a, why the air in the spark gap
undergoes electrical breakdown when the voltage reaches V_{b}, but the
air in the capacitor does not? (2 points)

Q. 3 a) We wish to use the device illustrated in Q. 1 to
generate microwaves with wavelength of 10 cm. If the only resistor you have is
one with R = 1 kW, explain (with
calculations) why the capacitance you should choose is about .33 x 10^{-12}
F. (3 points)

b) The only kind of capacitor you have available is a
standard C = 1 pF = 1 picoFarad = 1 x 10^{-12} F size, but you can have
any number of these you like. To get close to the capacitance you need from a)
should you wire these in series or in parallel? How many capacitors will you
need? (3 points)

Q. 4 We are trying to use our little microwave antenna circuit to transmit signals to a satellite which is almost directly overhead. As the microwave passes upwards through the atmosphere it will move through increasingly rarified air and the refractive index of the air will decrease accordingly. This means the microwave beam will be refracted as it travels upwards, and therefore the satellite will think the atmosphere is a different thickness from what is actually is. Suppose there are 5 layers in the atmosphere, with the satellite being in space, with a refractive index of n=1, the next highest layer being the Thermosphere (n=1.0005), then the Mesosphere (n=1.001), then the Stratrosphere (n=1.002) and finally we are at sea level in the Troposphere (n=1.003). If the height of the satellite is 42,000 km, what height does the satellite think it is above us, based on refraction of your signal? (6 points) Hint: There is short cut to do this problem, and several of the numbers given turn out not to be needed. For a bonus point, which numbers given are not needed?

Q. 5 a) The satellite passing overhead is not moving toward or away from your little microwave transmitter, so there is no Doppler shift involved. But the satellite is moving, so there is a relativistic time dilation effect. To the satellite, the antenna is moving, and therefore the period of the electrical circuit is altered, just as any clock would be. If the frequency of your transmitter is f, as you see it, what is the frequency f’ of the microwaves actually received by the satellite, if the satellite is moving across the sky with speed v? Do not use numbers, simply express your answer (and your calculation) algebraically, relating f’ to f in terms of v the speed of the satellite and c, the speed of light in a vacuum. (3 points)

b) Obviously if the frequency of the microwaves is different as the satellite sees it, then the wavelength l’ of the waves received by the satellite must be different from the wavelength l of the waves as you see them emitted by your circuit. Calculate the relationship between l’ and l, in terms of the speed v of the satellite, using your result from part a) and one of the two axioms of special relativity (state which axiom you are using). As in part a) express your calculation and result in algebraic form. (3 points)

Q. 6 a) If the satellite overhead is a GPS satellite then we can use the timing of signals between it and our circuit to establish our position. For these purposes it is critical that the uncertainty in the time Dt that the photons are emitted and received is small. If quantum effects turn out to be significant, and in order for this uncertainty to be small, do you think it would be better to use higher frequencies or lower frequencies of photons, as this will certainly affect our construction of the circuit? (3 points)

b) The usual rule of thumb is that quantum effects will be important only when we are dealing with a small number of particles, in this case photons. Let’s suppose we choose the frequency and wavelength used in Q. 3 (from out point of view, not the satellite’s, which is to say a wavelength of 10 cm), and we construct our circuit so that the total microwave power transmitted is 10 W. How many photons are being emitted by our antenna every second? (3 points)