Set 1For Monday September 8, 2008

• 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 first term and the one after that (2 terms altogether)  and compare to answer from calculator.
• Approximate using taylor's series    a) sin(Pi/3 + 0.02)    b) (r + x)^(-3)  where r>>x
• Start with a right circular cylinder of radius 0.030 m and height 0.10 m. It's axis coincides with the z-axis. Now cut away 3/4 of the cylinder along the zx and zy planes. This leaves curved a quarter-section of the cylinder. In a top view it is a quarter-circle from the x to the y axis. a)  Find the flux of a vector V =3  x^ through this section of cylinder, by writing out the integral in cylindrical coordinates, putting in the limits and integrating. b) Use the same method, showing details, to find the flux through this cylindrical section of a vector V = x^ + 1/3 z^ .
  

• 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 rho unit volume.
• 1.48 [ 3D delta function, p. 52; be sure to read example 1.16, and then do 1.48]

• An electric field has only a radial component in spherical coordinates:  E(r) = (15 V/m^3)  r^2, where r is in meters. Find the charge density rho as a function of r in this region of space. Find rho(1.5 m) in C/m^3.
  
• A sphere of radius R = 0.35 m has a charge density rho(r) = (1.7 x 10^-7 C/m^5) r^2. Find the electric field at r = 0.20 m and at r = 0.50 m.
  
• 2.15, p. 75
• There is slab of material of thickness d which extends throughout the xy plane (it's infinite in x and y). This slab extends from z = 0 to z = d. The charge density is zero at z = 0, and increases linearly with z (rho = kz), until it reaches a maximum of rho(+d) = kd. Find the electric field as a function of z, for both z<0 and z >0, and z >d. Keep in mind that the magnitude of E is the same outside the slab on either side of an infinite slab like this. [This resembles Griffiths 2.17, which we will have discussed and done in class.]
  
• 2.20, p. 79. You can get V by using partial integrations (-dV /dx = E_x, etc.)
• In a certain region of space, there is an electric field E = (i^ y  + j^ x) (30 V/m^2) created by stationary electric charges. a) Carry out a test to show that this is a valid electrostatic field  b) Find the potential difference in volts between the points (0,1,0) and 2,1,0)

• There is an infinite sheet of charge lying in the x-z plane. Below the plane (y<0) there is an electric field (40,10,0) V/m. Just below the origin the electric potential is V(0, 1e-8,0) = 25 v. Above the x-z plane there is an electric field (E_x,E_y,0), and at the point (0.5, 1.5,0) m the electric potential is V(0.5 m,1.5,m,0) = 125 v. [ These electric fields and potentials arise from the sheet and from other charges away from the sheet.] Determine the charge per unit area on the plane, and the components E_x and E_y above the plane
• a) Show that the Electric field INSIDE a uniformly charged non-conducting sphere of radius R, and total charge Q  is proportional to r. (We did this in class using Gauss's law.) b) Utilizing this and the fact that outside the sphere is kQ/r^2, determine the electric potential V inside and outside the sphere by doing an integration. Assume V is zero at infinity.
• a) From the last problem, you have both E and V for the sphere, so determine the total energy W of this charge distribution, using Eq. 2.43. b) Now build up the sphere by bringing in little spherical shells of charge dq, to a sphere which already has a radius r, and charge q, and the same charge/volume as did the uniformly charged sphere. Since dW is V dq, this will give you the increment of work needed to do this job. When you have brought it all in, you should get the same answer as for the previous problem.
• Find the total energy W of a uniformly-charged sphere of radius R and charge +Q, which is surrounded by a spherical shell of radius b>R, of charge -Q. Be clear about what method you are using.

• The accompanying figure shows the center of  a 50x 50 grid, which is 0.02 m on a side, and infinitely long coming out of the page. Find the largest and smallest E field magnitudes adjacent to the 100 v equipotential. Identify the cell number(s) of each. Determine the largest and smallest charge densities in C/m^2.
  
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• Find the potential V(x,y) for a long rectangular surface (infinite in the z-direction), where there is a constant 50v along the line x=a, and y = 0 to b, V = 0 along y = 0, x = 0 to a, and V = 0 y = b, and x=0 to a, while dV/dx is 0 along the line x= 0 from y = 0 to b. In other words, the top and bottom have zero volts, the righthand side has 50v, and the lefthand side has zero slope. Create a series solution with 100 terms for this situation, and plot it. In Maple, the plot3D should of course show 50v on the right and zero slope on the left. Let a = 0.20 m and b = 0.15 m.  Turn in the plot (email it to me but please don't show all 100 coefficients). Find the value of V(a/2,b/2) in volts.

• You have a configuration which has symmetry about the z-axis, and you are able to measure the electric field along the z-axis in a region where there is no charge/volume. This turns out to be

              E_z (z-axis) = E_z (0,0,z) = (1.75 V-m^2)/z^3, where z is in meters.

  a) Write down the equation relating derivatives of electric field in this situation (a famous equation!)   It’s simplest to do this in x, y, z.  b) Simplify this equation by observing that behavior in x must be the same in y, because the situation is symmetric about the z-axis. 
  c) You are to find E_x (0.02, 0, 0.25). This is near the z-axis but not right on it. The idea is to do a series expansion about the point (0, 0, 0.25). Done right, this will give you an approximate value of E_x at point (0.02, 0, 0.25)m. Based on what you have in a) and b), which series do you want?  A) Taylor    B) binomial       Go ahead and find an approximate numerical value for E_x(0.02,0,0.25) m .
• You have a conducting sphere of radius R = 0.025 m. There is a charge of +2.4 x 10^-8 C at a distance of 0.050 m from the center of the sphere. The potential of the sphere is + 600 v.  a) Locate the image charge(s) (position and magnitude) which will satisfy the conditions of this problem. b) Determine the maximum charge per unit area (sigma) on the sphere  c) Ditto for the minimum charge/area on the surface of the sphere.

• A spherical surface of radius a has a potential V(a, theta, phi) = P_3(cos theta). A larger spherical surface of radius b>a has a potential V(b, theta, phi) = P_5(cos theta). P_3 and P_5 are two of the 'legendre polynomials. a) Find V(r, theta, phi) for r between a and b, b) Find V(r, theta, phi) for r >b. These solutions will be in the form of a power series, and you are to explicitly show the coefficients.
• We will do prob 3.38 in class. You are to do a similar problem, with charge/length = Qz/a^2. This is positive for positive z, and negative for negative z. As in 3.38 the charge runs from z = -a to z = a. Work out the first two non-vanishing coefficients in the series  V(r>a) = 1/r sum on n C_n (a/r)^n P_n(cos theta).
• For this same (previous problem) charge distribution, calcuate the first four multipole 'moments' ( or those parts with a z-component only) monopole, dipole, quadrupole, hexadecapole). See p. 146 in the text.
• An electric field in air is 100 V/m and makes an angle of 60 degrees to a flat infinite boundary, with silicon on the other side. a) Find the components (x direction, parallel to the surface, y-direction, perpendicular to the surface) of the displacement vector D in air. Include the SI units of D.  b) Find the components of D in silicon, and the angle D makes to the surface. c) Find the components of E inside the silicon, and the angle that E makes to the surface in silicon