Homework 2
Due in class on Friday, January 26th.
Please note that html is sadly lacking in facilities for representing mathematics. I do the best I can but it is sometimes a little wonky. In particular, denominators that line up in one web browser don't always line up in another so you have to be a little generous in how you interpret multi-line expressions. This is particularly true of problems 1 and 2 this week.
1) Math 5.69. Prove that
d (A x B) = d A x B
+ A x d B
du du
du
where A and B are differentiable functions of u.
2) Math 5.71. If r = a cos ωt + b
sin ωt, where a and b are any constant,
non-collinear vectors and ω is a constant scalar, prove that
a) r x dr = ω(axb)
and b) d2r + ω2r
= 0.
dt dt2
3) The figure below shows three equally spaced charges (the small circles).
a) Find the Electric Field at the four points marked with an x. They are all
on a square grid of side l.
b) Sketch the Electric Field near the sharges (out to about 2l to 3l).
c) Sketch the Electric Field a long way from the charges, out to at least
20l away.
4) The figure below shows four charges placed on the corners of a square
of side l.
a) Sketch the Electric Field near the charges (out to 2-3l).
b) Sketch the Electric Field far from the charges (to at least 20l away).
c) Discuss the relationship between the field in part b and field in part
c of Q2.
5) 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. Find the electric potential at all points in the bisector plane.
6) G & P1.3. The maximum electric field which can be supported by dry air at atmospheric pressure is about 106 Volts/m. What is the maximum potential difference to earth for a conducting sphere of radius 10 cm in air? (Take the distance from the sphere to earth to be infinite. Provided that the actual distance is >> 10 cm the error made will be very small.)
7) G & P1.5. Two concentric conducting spherical shells have radii a and b (a < b). Calculate their capacitance. Show that as their separation becomes small compared to a the formula approaches that for two parallel plates of the same area.
8) G & P 1.7. The uranium nucleus contains 92 protons and has a radius of about 10-14m. Assuming that the positive charge of the protons is uniformly distributed throughout the nucleus, calculate its electrostatic energy in MeV. (Imagine that the nucleus is built up layer by layer. When a sphere of radius r has alreay been built up, how much additional work is required to add an additional spherical shell of thickness dr?) In the uranium nucleus splits into two equal fragments, each of radius 8 x 10-15m, how much electrostatic energy is released?
