Physics 212 – Final Exam NAME:__Albert Einstein________

Summer 2002 STUDENT ID#:______007______

Friday 15 August 1110 - 1310

Prof. Milos Steinhart

INSTRUCTIONS: This is a closed book exam. Calculators are allowed. You may use your own formula sheet but it may contain only formulas. You must show all of your work in the space provided for each question and you must include appropriate units and significant figures. If you need more space, use the other side of the paper but mark clearly to which problem and which question is your work related. Try to make your work as clear and transparent as you can. Do not use other papers!

You are bound by the University’s Honor Code. Any evidence of cheating must be reported to the College of Arts and Sciences.

PLEASE CIRCLE ALL ANSWERS

All problems are worth 20 points.

SCORE

Problem 1 ___20________________

Problem 2 ___20________________

Problem 3 ___25________________

Problem 4 ___20________________

Problem 5 __ 20________________

TOTAL __105________________

Problem 1: A solid nonconducting sphere of radius r₀ has a total charge Q. The charge density is directly proportional to the radius, so: r_e(r) = b r. Calculate

a) (6) the exact behavior of the charge density in terms of Q:

Total charge must be integrated through thin spherical shells in which r_e can be considered constant.

Q = so b = Q/pr₀⁴ and r_e(r) = Qr/pr₀⁴

b) (6) the electric field at points inside the sphere (r < r₀):

From Gauss’ law E(r) 4pr² = 1/e₀ = pbr⁴/e₀ = Qr⁴/e₀r₀⁴ Þ

E(r) = Qr²/4pe₀r₀⁴

c) (4) the electric field at points on the sphere (r = r₀):

The electric field on the surface will be continuous, so we can substitute into b) r = r₀ :

E(r) = Q/4pe₀r₀²

d) (4) the electric field at points outside the sphere (r > r₀):

From the Gauss’ law but now all the charge is enclosed:

E(r) 4pr² = Q/e₀ Þ E(r) = Q/4pe₀r²

Problem 2: Suppose you are looking at two current loops in the plane of the page as shown in the figure. When the left circuit is closed at some instant by the switch:

(To get any credit here, mark clearly all the directions and explain your choice!)

a) (4) what is the direction of induced current in the right loop?

The current in the left loop will be clockwise due to the polarity of the power source. After closing the loop the magnetic field, which goes out of the paper in the second loop will grow and so will the magnetic flux. Induced current will flow in clockwise direction to oppose this growth.

b) (4) does the situation change after some time?

After a short time an equilibrium field will be reached and magnetic flux will be constant and there will be no induced current.

c) (4) what is the direction of the induced current in the right loop if it is pulled to the right?

If we pull the right loop to the right, it will get in weaker field and the magnetic flux will decrease and a counterclockwise current will be induced to support the flux.

d) (4) what happens if we rotate the right loop in the plane around the left one at a constant distance?

Due to the symmetry we would move in constant field and there will be no change of the flux and no induced field.

e) (4) what happens when we open the left circuit?

If we open the left circuit the magnetic field will disappear and a counterclockwise current will be induced in the right loop to support the decreasing flux.

Problem 3: Our favorite local radio station broadcasts on a frequency 1040 kHz. If we find at home a coil with the inductance L = 4 mH,

a) (4) what capacitance do we need to build a simple one-circuit radio?

We use a LC resonance circuit, usually parallel (but serial is also possible) and from the Thompson’s formula:

b) (4) Make drawing of the (main parts of the) circuit.?

c) (4) How long would the l/2 dipole antenna be?

l = c / f = 3 10⁸ /1.04 10⁶ = 288 m Þ l/2 = 144 m

d) (4) If we listen for 8 minutes, How many waves will pass the antenna during this time?

We calculate how many periods have past during the time t = 480 s.

T = 1/f = 1/(1.04 10⁶) = 9.61 10^-7 s

t/T = 480 / 9.61 10^-7 = 5 10⁸ waves

e) (4) How far will the first wave get by then?

In 480 s the light as well as other EMW gets to d = ct = 1.44 10¹¹ m = 144 million km

f) [optional +5 points] Compare the previous distance to some distance important in Nature.

This is roughly the distance from the Earth to the Sun.
Problem 4: An aquarium filled with water (n_w = 1.33) has flat glass sides whose index of refraction is n_g = 1.58. A beam of light from outside the aquarium strikes it at a 43.5° angle of incidence. What is the angle of this light when it

a) (5) enters the glass?

From Snell’s law n₁ sinj₁ = n₂ sinj₂ = n₃ sinj₃

sinj₂ = 1/n₂ sinj₁ Þ j₂ = 25.8°

b) (5) when it after enters the water?

Since the walls of the glass are parallel, the angle of incidence is also j₂ :

sinj₃ = n₂/n₃ sinj₂ Þ j₃ = 31.2°

c) (5) What would be the angle if the same beam entered water directly from air?

sinj₃ = n₁/n₃ sinj₁ Þ j₃ = 31.2° , which is the same!

d) (5) Make some physically reasonable conclusion from b) and c) considering also the conditions of validity of your statement.

Refraction from one medium into another one doesn’t depend on third medium, which would separate these two under the condition the separating medium has parallel sides. There can be even more parallel layers present.

Problem 5: Multiple choice. Pick the single best answer or answers to each conceptual question below. Please include a brief explanation for partial credit. Each answer is worth 4 credits.

1. A capacitor is connected to an tunable oscillating power source. As the frequency of the power source increases, the impedance or capacitive reactance

a) increases.

b) decreases.

c) remains the same.

d) depends on the direction of the current.

2. In planar harmonic electromagnetic wave, the magnetic field achieves its maximum when the electric field

a) is also at its maximum.

b) is at zero.

c) is at its minimum.

d) is at some intermediate value.

3. Which is (are) true? The energy carried by an electromagnetic wave in a vacuum

a) propagates at the speed of the light.

b) consists of equal contributions from the electric and magnetic fields.

c) propagates along the direction of the electric field.

d) all of the above.

4. The phenomenon of dispersion occurs when

a) there is total internal reflection.

b) the index of refraction depends on the wavelength.

c) there is a virtual image.

d) the incident beam is completely reflected.

5. For a thin lens made of two spherical surfaces, the focal length depends on

a) the index of refraction of the lens.

b) the radii of the two spherical surfaces.

c) the assumption of the incident rays near the optical axis.

d) the lateral size of the lens.