Extra Homework Problems and Hints

Due in class Friday February 17^th.

Note that now that the text books are in I will not be typing the problems from the text anymore.I will still put the problem numbers in here so that you have a single reference point for the homework.

1) Grant and Phillips 1.5

2) Grant and Phillips 1.7

3) Grant and Phillips 1.10

4) Grant and Phillips 2.1

5) In Cartesian Coordinates the position of a moving particle is given by the vector
r(t) = x(t)i + y(t)j z(t)k.
The velocity is then given by the vector
v(t) = x'(t)i + y'(t)j z'(t)k.
Life is less simple in spherical polars where the position is given by the three functions
r(t), θ(t), and φ(t)
but the position is written
r(t) = r(t)ê(t)
where ê is a unit vector pointing in the instantaeous radial direction. (Yes, I know that this is not the usual notation but I am hampered by html's poor support for mathematics. Obviously we normally write that as an r with a hat on it but I can't figure out how to do that in html.) Now, the trick is that the unit vector itself changes as time passes.

a) Use the chain rule and careful diagrams to find the velocity vector in spherical polars as a function of time in terms of r(t), θ(t), and φ(t) and their derivatives.
b) Apply this to find the velocity of particle whose position is given by the functions
r(t) = Sqrt(1 + 4 t²), θ(t) = tan^-1(1/4t), and φ(t) = t.

6) Consider the electrostatic potential .

Please deduce and explain everything there is to know about V, E, and the charge distribution that produced them from an examination of V.

a) What do the equipotential surfaces look like? Sketch a cross-section of the equipotentials to show the spacing.

b) Use the relation E = -grad V to find the electric field and thus sketch the field lines.

c) Use the differential form of Gauss's Law to find the charge distribution .

d) This is a somewhat peculiar charge distribution. One expects that for very large r the system should look like a point charge but that would give a somewhat different potential. Can you say anything about why this is so?

NOTE this is a description of the electron density in a Hydrogen atom in its ground state. You may like to think about the effect of including the nucleus.

7) A solid plastic sphere of radius R1 has a charge -Q1 uniformly distributed over its surface. A concentric metal sphere with inner radius R2 and outer radius R3 surrounds the plastic sphere. There is a charge +Q2 on its inner surface and a charge +Q3 on its outer surface. The total charge on the metal, Q2 + Q3, is greater than Q1.

Use Gauss's Law to find the magnitude and direction of the electric field at various radii, r, from the centre of the solid sphere. Consider separately the cases

a) r < R1 (inside the plastic sphere), b) R1 < r < R2 (in the air gap), c) R2 < r < R3 (in the metal), and d) R3 < r (outside the metal).

If -Q1 = -5nC what can you say about Q2? Explain.

8) The potential far from an electric quadrupole is given by .

Find the Electric field that gives rise to this potential.(This is definitely easiest in spherical polars and you should use this as a chance to get practice with Appendix B.)

9) Consider the bisector plane of an electric dipole. For example, in the figure below, where the charges are +Q at +k and -Q at -k, this is the plane z = 0. First find the electric field on that plane. Then find the TOTAL flux that passes across that plane in the +ve z direction. Discuss the relationship between your answer and Gauss's Law.