8.022 at ESG Fall 2004 Problem Set 02 due Friday September 24

Reading

Finish Chapter 1 of Purcell Problems from the text

• 1.21

• 1.27 (optional; you could spend a lot of time on this one if you wanted)

• 1.30

• 1.34

• 1.35 A useful, interesting and amusing calculation; use the text’s hint to do the radial integral first. A picture is of great use. A further hint; put your origin at one of the charges, not at the midpoint of the line segment seperating them. If you check our web page, you’ll see under “Supplemental Class Notes” some links to possibly helpful hints.

More With Dipole Fields

We will rederive in class on Friday September 17 the proper expression for the electric potential due to a dipole p = p ẑ,

φ_{d} = p \frac{cos θ}{r^{2}} , r > 0

in spherical coordinates.

(A) Find the electric field E = −∇ φ_{d}. Of course, you still have your table of vector operators handy. You have your choice of coordinate systems. However, you are asked to express your answer in a “hybrid” form (that’s my own terminology); you’re shooting for

E = \frac{1}{r^{3}} [3 r̂(p · r̂) − p] ;

this form is a common one for the electric field due to an ideal dipole.

(B) Sketch a few field lines in the x-z plane; use a plotting program if you wish. The main thing is to show proper directions for r > 0, θ = 0, r > 0, θ = π/2 and θ = π. Also, from your analytic result, find the value(s) of θ for which E_{z} = 0.

(C) Find the energy density corresponding to this field and the total energy in the region r > R_{0} where R_{0} is some constant radius. We’ll deal with r < R_{0} in due time.