Set 1  Due Friday December 1, 2006

• 5.39, p. 247 Review of Hall Effect
• 5.48 p. 250 Use Biot-Savart law to calculate B components from a circular loop
• 6.1 p. 259 torque on one loop from another
• Derive Eq. 5.86 from 5.85 (Take the curl and evaluate)

• a) 5.23 { Find B, then use curl B to find J }  b) 5.25 b) { Make sure A is continuous at r = R }
• 6.25 a).  p. 283 { 6.25 b) is a bonus question } {I did this one using potential energy.}
• 6.26, p.283
• Calculate the attractive force between two identical loops of wire,  carrying currents of 2500 A parallel to each other. Each loop radius is 9.0 mm, and their centers are a distance of 3.0 mm apart. {I did a 200-point integration in Excel to calculate a component of B acting on one loop from the other. You could do the same thing in Maple.}
• 5.55. p. 252 {This one is simple only if you get a bright idea. }

• Clearly  record your data (both trials) for the toy magnet and angle vs distance. Put down your partners' names. Explain how you did the analysis to find the magnetic moment of this magnet and give its value with units.
• Carefully explain your data (write English phrases) for the strong magnets suspended by a string. With equal care, explain your analysis, and say what the maagnetic moment was for this magnet.
• Calculate the magnetization of your small strong magnet from its magnetic moment, explaining how you did it. Include the units. Calculate the effective circumferential current in a single magnet of thickness 3 mm.
• a) Work out the transmission coefficient T for a plane electromagnetic wave from index n1 to a layer of thickness d and index n2, then back out again to index n1 on the far side. Take an initial angle of theta to the normal going in. Have the E fields all lying in the plane of incidence. (This is a re-hash of what we did in class) b) Find T for n1 = 1.5, n2 = 1, n3 = n1 = 1.5, and an incident angle theta of 50 degrees to the normal, with d = 2 lambda. (A numerical value is required.)
• Same as part a) of the previous problem, but now the E fields are all perpendicular to the plane of incidence. Leave your answer symbolic.

• Turn in the worksheet where you obtain the equation for laplacian E =  mu sigma ...  etc.  This also requires you set up the coupled equations for the real and imaginary parts of E inside the wire.
• Turn in a numerical integration for the magnitude of E inside a copper wire (sigma = 6.7 E7 S/m) wire of 2 mm diameter at a frequency of 1500 kHz (1.5 MHz). This requires a plot of |E| vs r, and should have E = 1 on the axis (r=0). You'll need to take the first derivatives =0 on the axis.. Give the value of E at the outer edge of the wire, and its value at a radius of 0.90 mm (90% of the way to the edge) (Counts as two problems.) This can be done in Excel using an RK2 scheme, or in Maple, or maybe something else.

• Griffiths' 9.28, p. 411
• For the next three problems, see class notes on waveguides and cavities (on the web)
• Numerically integrate the equation R'' = -R  -1/x R' and find the spot (away from x=0) where its slope is zero (use only 100-200 steps if you do it RK2 in Excel). Email me your output.
• Do the same but for m=1 as a separation constant in cylindrical coordinates. You should find the slope vanishes at a much smaller value of x than for m=0! Send me this one via email also.
• Find the lowest resonant freq of a cylindrical cavity of length 20.6 mm and radius 14.5 mm. This is the cavity which resonates on the old microwave gear when the wavelength is around 3.25 cm. Check your wavelength (for m = 1!) and see that it is fairly close.

• Show that when u/R = 1 we are at the critical angle from n1 to n2.
• Work out the condition for odd TE modes (analogous to cos u = + / - u/R , odd quadrants).
• For a slab whose thickness is 10 microns, at an omega of 1 x 10^15 rad/s, and n1 = 1.5 and n2 = 1.46, determine the valid even TE solutions.  Determine the launch angle for each of these (theta in n1).  {This is readily done in Maple. Could also be done on a spreadsheet. }
• Use the 1957 table from the Radio Amateur's Handbook (p. 325) to calculate the characteristic impedance for RG-58 and RG-59 coaxial cable, and for twinlead 214-056, 214-076, and 214-022. (Most, but not all will check out ok.) Use Good's Eq. 17.17.

• An electric dipole antenna of length 25 cm radiates 30 mW at 28 MHz. Find the magnitude of the current magnitude.
• A magnetic dipole antenna of diameter 15 cm carries a current of amplitude of 0.20 A at 57 MHz. Find the radiated power. {See Griffiths sect  11.1.3}
• Griffiths prob 11.3, p. 450
• Griffiths prob 11.4, p. 450
• Find the radiation resistance of a 40-cm rod operating at 75 MHz. Explain why your method probably won't work for750 MHz.

• In the 8-layer filter I have emailed you, find a relation between the widths a and b, and the indices of refraction, in the parameters as given. I suggest you save the spreadsheet, then work on it under another name, so you keep the original values. Change the parameters so that you have a center frequency at 490 nm. Aside from this peak, there should be no wavelength in the visible region with more than 20% transmission.
• Antenna problem 1, Rad Notes p. 7, after Eq (47), work out an integral
• Antenna problem 2, Rad Notes p. 7, after Eq. (52), find R_rad of 1.5 wave antenna (make sure you get the right results for the half and full wave antennas first).

• Griffiths' 12.45 p. 534 (counts as 2 problems; check how the forces should be related)
•     a) show that Emax/Emin for a relativistic particle is gamma^3
•     b) integrate E dot dA for a relativistic particle over a sphere of radius R
•         i) what do you expect to get?
•        ii) how will this calculation depend on beta and gamma?
• Worksheet on E and B transmformations (counts as 2 problems)