For Friday September 1, 2006

• 1.11 a) and b) [ gradient ] p. 15
• Approximate using the binomial series a) (101)^(1/2)    b) (3.1)^2    c) (28)^(-1/3)                                        (Use only the leading term and compare to answer from calculator)
• Approximate using taylor's series    a) sin(Pi/3 + 0.02)    b) cos^2(r + x) where r>>x
• Problem 1.5 p. 8 (triple cross product)

• 1.21 a) and b)     p.22    (answer in terms of x,y,z and rectangular unit vectors)
• 1.24 a)                p. 22
• 1.43 [ 1D delta function, p. 49 ]

• Find the electric field on the axis of a hollow cylindrical shell a distance D from the center of the shell along the shell's axis. The shell has a radius R and length L. There is a uniform charge per unit area on the shell's curved surface.
• Ditto except that the cylinder of radius R and length L has a uniform charge per unit volume.
• 1.48 [ 3D delta function, p. 52; be sure to read example 1.16, and then do 1.48]

• 2.9, p. 69
• 2.14, p. 75
• 2.15, p. 75
• 2.17, p 75
• 2.20, p. 79. You can get V by using partial integrations (-dV_x /dx = E_x, etc.)
• 2.23, p. 82

• 2.21, p. 82                                    line integral, gauss's law
• 2.28, p. 87                                    volume integral
• a) 2.29, p. 87   b) 2.31, p. 93       easy
• 2.32 a) and b), p. 95
• 2.35 p. 101
• 2.40 p. 106                                   easy

• 3.8, p. 126
• A +1uC charge is a distance 0.4 m from the center of a grounded conducting sphere of radius 0.3 m. Find the sphere's surface charge density closest to the +1 uC charge, and also the surface charge density farthest from the 1 uC charge, in C/m^2
• Consider a setup like Fig. 3.17. p. 128. Let the V=0 boundaries be at y=0 and +a, with a = 0.20 m.  Vo(y)  is a function which is zero at y=0 and is like a tent, going up to 1000 v at 0.1m in the middle and then back down t zero at y = 0.2 m.   Solve this problem on a spreadsheet, using dx = dy = 0.01 m, so that y has 20 cells across the top, and x has 100 cells down. Then do the iterative average till it settles down. Note the voltage at x=6 cm, and y = 6 cm
• Do this same problem using a series solution of 20 terms in Maple. I'm sending you an email and will post a Maple solution to a different problem, which you can if you wish radically alter to suit this problem. But it does remind you how to do series and summations in Maple. The 'tent' function should be g:=1000*piecewise(y<0.1, 10*y,y>0.1, 10*(.2-y)); .
• For the problem of Example 3.3, p 127 ff, let Vo = 1000 v, and a = 0.20 m. Determine charge/area at x = 0, y=0.10 m (in the middle of the plate). Explain how you did this, and give your result to three significant figures.

• 3.26, p. 149
• 3.27, p. 151
• 3.28, p. 151
• 3.31, p. 154
• 3.33, p. 155

• 4.5, p. 165
• 4.8, p. 165
• 4.9, p. 165
• 4.15. p. 177
• 4.18, p. 184
• 4.26, p. 193

• A 40-uC charge in free space is 0.25 m from a plane boundary with a dielectric material of dielectric constant e = 3 e_o.   Find the force exerted on the 40-uC charge.  (Requires the use of images.)
• A loop of wire has a radius R and carries a current I. a) Find the magnetic field in the plane of the loop at a distance alpha R from the center by integration in Maple. {Hint: express dl in terms of x and y coordinates, and do the same for r. Then the cross product will simplify and be expressible in terms of theta as an integration variable. Theta is the angle between the line to the observer from the center to the line from the center to dl.} b) Evaluate for alpha = 0.5, I = 2 A and R = 1 m {if you do a plot, you should see B increasing as you go away from the center of the loop }
• Make a spreadsheet showing the magnetic field of two coaxial loops of wire, both of radius R, using two sliders (scroll bars) to adjust the positions of each loop. Naturally, you must specify R in a cell in the spreadsheet. The claim is that when the centers of the loops are separated by R, there is an extremely uniform B field along the axis in the center between the loops [ 'Helmholtz coils' ].
• Prob 5.9, p. 219
• Prob 5.11, p. 220 (Add the axial B fields from a bunch of loops.)

For Thursday November 2, 2006

• A cordless toothbrush base has a coil with 100 turns and a diameter of 34 mm. The toothbrush has a coil of 100 turns and a diameter of 18 mm. Assume a  peak current of 100 mA at a frequency of 31 kHz in the base unit and calculate the peak emf generated in the coil. a) Assume (unrealistically) that the B field in the loop of the toothbrush is constant and equal to its value in the center of the loop. b) Work out the flux in the loop of the toothbrush by integrating the magnetic field off-axis, using the B field you calculated in Maple, then complete the calculation. This answer should be larger than in part a). [The wire diameter is about 1/4 mm, and the number of 100 turns in each was an educated guess. The peak current estimate was based on the maximum steady current listed for that gauge wire.]
• Prob 7.20, p. 315
• Prob 7.29, p. 320
• Prob 9.11, p. 382