Physics 273 – Fall 2005
Optional Review Problem Set
Optional: Hand in neat, well-organized solutions for any 5 of these problems for up to 10%
increase of one exam score
Due Friday Dec. 16
Recommended: Work all the problems to help prepare for final exam. Also – rework
problems on exams 1 and 2, and look through class notes/assignments for other topics (e.g.
Doppler, intensity vs. distance, polarization, multi-slit interference…) that aren’t included
in these problems.
1. Wave summation: Review the material in H&L 2.7, 11.4-5, 13.1-4, as well as class notes on
amplitude modulation. Look at web links on Fourier synthesis and Fourier series.
a) Estimate the periods present in the sum wave shown, and use the estimates to find the
frequencies of the two cosine waves that were summed to produce the wave.
b) The wave shown is the sum of a series cosine waves of frequency fo, fo+f, fo+2f, fo+3f, …
fo+(N-1)f. Estimate the three periods that define this sum wave and use them to determine
the average frequency, the frequency difference f, and the number N.
c) The wave shown has a carrier frequency fc, and is amplitude modulated by a cosine wave of
lower frequency fm. Find the two frequencies. Write the equation for an amplitude
modulated wave, then use trig identities to prove that there are three frequencies present in
the wave form, and then find their values.
d) Suppose you are given an analytical formula f(t) for a waveform of period T. Write an
expression for the sum of sine and cosine functions that equals f(t). Write the integral
expressions that would allow you to determine the magnitude of each term in the sum.
2. AC circuits: review Tipler Ch. 29, and H&L 1.5 and 9.2-3. Also Kirchoff’s rules, Tipler 255, and class notes.
Problem: Consider an RL circuit with a resistor R, inductor L, DC source of voltage Vo, and a
switch S, all in series. The switch is initially open and then at t = 0 the switch is closed.
a) Draw the circuit and write the equation for the voltage drop across each component. What is
the differential equation describing I(t), the current? Find the solution for I(t).
b) From part a obtain a formula for the voltage across the inductor VL(t) and the energy stored
in the inductor EL(t). Sketch VL(t) and EL(t) vs. t on the same graph.
c) Let L = 0.50 H, Vo = 12 V, and assume that at t=3.0 ms, I(t) = 0.45IF, where IF denotes the
current after a long time. Determine IF and R.
d) After the current has reached IF, we set the source voltage Vo to zero. At what time after that
has the magnetic energy stored in the inductor decayed to 25% of the value it had just before
the voltage was set to zero?
e) (extra: not required for credit) Consider replacing the DC voltage source with an ac source,
V = Vocos(t). Define imaginary voltage and current functions, Iˆ and Vˆ, such that V =
Re(Vˆ) and I = Re(Iˆ), and Vˆ = Voeit, Iˆ = Ioˆeit. Solve for Ioˆ in the exponential form for a
complex number. Take the real parts to find V and I. What is the value of the phase
difference between the current and voltage functions?
3. Generic wave motion: Review H&L 2.1-4, 4.1-4, 4.6, 6.4. and web animation of wave
reflections, and wave impedance worksheet.
Problem: A transverse sinusoidal wave on a taut wire has an amplitude of 2 mm and a
wavelength of 0.72m. The wave speed is 200 m/s. The wave is traveling in the positive x
direction, and the displacement is in the y direction.
a) Write the one-dimensional wave equation describing the motion of the wave. Find the value
of the tension in the wire given the linear density of 2.45 x10-2 kg/m.
b) Write the wave equation in real form and in imaginary form, assuming that the displacement
= 0 at x=0, t=0. Demonstrate that both functions satisfy the wave equation. Include real
numbers with units for all the parameters possible.
c) Find the maximum vertical speed of the wire. Consider the position x = 1.2 m and find the
first time at which the vertical speed of the wire is zero.
d) The wire has a smooth junction to a thicker wire of linear density .2205 kg/m. Find the
speed, frequency and wavelength of the wave in the second wire.
e) Find the power incident on the junction from the smaller wire, and the power transmitted into
the larger wire.
4. Standing waves, longitudinal waves: Review H&L Ch. 6, longitudinal waves H&L Ch 5
and 4.2-3. Look at web page animations of reflections and standing waves.
Problem: A standing sound wave in a pipe with length L = 0.80 m is described by:
 x,t   Asin 2x /Lsin 2t   /4
where the axis of the pipe is parallel to the x direction,  is the displacement, and  is the
frequency. The speed of sound is 331.5 m/s.

a) What are the wavelength and frequency of the wave?
b) Fill in the x coordinates of the nodes (zeros) and anti-nodes (local max/min in values) in ,
Ý  , and P (the pressure), in the table. How is the pipe terminated?

t
parameter

nodes
anti-nodes

Ý 

t
P
c) 
Carefully draw a graph showing P(x) at times t = 0, 1/8, 1/4, 3/8, 1/2, 5/8The axes
of the graph should be labeled quantitatively and each curve indicated clearly.
d) Assume there is no energy loss or input. What is the energy stored in the pipe by the standing
wave.
5. Electromagnetic waves: Review Tipler Chs. 30 & 31, H&L 9.4-5, 12.1-3.
Problem: Consider an electromagnetic wave traveling in vacuum in the positive z direction with
frequency 100 MHz. The wave is polarized in the +x direction (electric field). Be sure to
specify vector directions where appropriate in the following.
a) What spatial variable do the electric and magnetic field depend on? Write the partial
differential equation defining E.
b) What is the wave vector of the motion? Write the solution for the electric field E and
specify the values of any quantities in the solution that can be determined.
c) What is the magnetic field B of the wave? What is the Poynting vector?

d) The refractive index of vacuum is n = 1. What is the impedance? Suppose the plane wave is
incident on an interface to a material of refractive index n = 1.523. What is the impedance of
 is the magnitude of the wave vector of the transmitted wave? What
the material? What
fraction of the incident energy is transmitted?
For an angle of incidence is  = 15o, draw carefully the incoming plane wave and the
transmitted wave (using the wave-front presentation, as well as indicating the wavectors).
Show that the component of the wave vector parallel to the interface is conserved (this is a
short derivation – if it’s taking more than a few lines, think more simply).
6. Transmission lines: Review H&L 9.1-4, class notes and worksheet on wave impedances.
Problem: A coaxial cable has a ratio of inner to outer radii of 15, and is filled with dielectric
material of  = 2.00o.
a) Find the impedance Zc of the cable and the speed of an EM wave in the cable.
b) The cable is terminated with resistors of i) 55, ii) 115, and iii) 400. An EM wave of
power Po is transmitted into the cable. Find the power dissipated in the resistor for each of the
three cases.
c) The cable is to be used to provide power to an electronic circuit of impedance Z = 9Zc.
Explain what you can do to optimize the power transmission to the circuit.
7. Displacement current/Maxwell’s relationships: Tipler Ch. 30, H&L 9.4-7.
Problem: A long, uniform, straight wire of radius a, and resistance per unit length R (units /m)
is carrying a steady current I.
a) Find the magnetic field (magnitude and direction) at the surface of the wire.
b) On the basis of symmetry, what is the direction of the electric field at the surface of the wire?
Do you expect that the magnitude E of the electric field will be different at different positions
parallel to the wire? What is the Poynting vector? What does this mean?
c) Assume that Ohm’s law holds for this wire, and the voltage drop across a wire length x is
Ex. Show that the integral of the Poynting vector over the surface of the wire is equal to the
resistive power dissipation.
8. Optics – refraction/imaging principles: Review Tipler 32-1,2 and H&L 12.3-4, 6-7.
Problem: Light is incident from a medium with n = 1, onto a spherical interface to a material of
refractive index 1.520 and radius of curvature R1 = +12 cm, as shown in figure a.
a) Consider a light ray that is initially parallel to the optical axis and at height h = 1 cm. Find the
angles , ’ and . Use the value of  to find the distance C where the light ray crosses the
optical axis.
b) Assume that there is an object at a distance s from the surface. Use the small-angle focusing
equation to find the values of s’ for s = 40 cm, 100 cm and infinity. Compare these values with
the result of part a. What is the significance of the comparison?
c) Now consider a second spherical interface of R2 = -16 cm, that terminates the material as
shown in figure b. The separation between the
two interfaces is t. For an object at s = infinity,
find the position of the object s2’ created by the
two surfaces as a function of t. (Treat the
image of the first interface as the object for the
2d interface.) What is the significance of your
result for the case that t becomes very small?
9. Optics – practical instruments and diffraction limit: Review Tipler Ch 32, H&L 11.7,
12.9-10
Problem:
In normal operation, a telescope creates a virtual image at negative infinity by matching the focal
points of the two lenses as shown. Thus the distance between the lenses t = f1 + f2. In this case
the angular magnification  = tane/tan is given by  = -f1/f2.
a) It is possible to change the relative position of the objective and ocular so that the virtual
image is at the near-point of the eye. What distance t between the two lenses is required for
this case? Assume the eye-ocular distance is negligible and use the partmeter dnp for the
near-point distance of the eye.
b) Find the expression for the angular magnification for this case in terms of f1, f2, and dnp.
c) If the diameter of the objective lens is 35 cm, and the ocular lens poses no diffraction limit,
what is the smallest separation of two objects at a distance of 150 km (use light wavelength of
550 nm) that can be resolved?
10. Diffraction/interference: Review Tipler 31-7, 33, H&L 11.1-6. Worksheet on wave
impedances and boundary conditions.
Problem: Consider a thin film of refractive index n1 on a material of refractive index n2.
Consider the cases i) n1 > n2, and ii) n1 <n2.
a) Tabulate the phase shifts of the electric field of an normally incident electromagnetic wave
for reflection and transmission at each interface. (if your table isn’t trivially easy for me to
interpret, I won’t grade it).
b) Find the conditions needed to create an antireflection coating, or an anti-transmission coating
for each case. (You must justify your result in a form short and clear enough for me to be
willing to read for credit.)
11. Interference: this is an old exam problem from the last time I taught optics.
Problem: (Principle of the Rayleigh refractometer) Light of wavelength  is incident normally
on a flat surface with two slits of separation d. The width of the slits is not important for this
problem. When a block of material of thickness t is placed in the light path past one of the slits,
the central maximum in the intensity is moved from  = 0 to ’.
a) Assuming very small angles, find the expression for the phase shift of each light ray between
the slit and a detector at a distance L that is positioned at an angle .
b) Find the expression for the angle ’ where the difference in phases between the two rays = 0.
c) Very small angles can be calibrated by using the interference pattern formed without the
block present. Find the angle, 1, of the first maximum in intensity in terms of d, if the
wavelength is 550.000 nm. If ’ = 20.50001, find the refractive index n’ of the material in
the block. (note values given to 6 figures)