8.022 at ESG Fall 2004 Problem Set 06 due Friday October 22

Reading:

Purcell Chapter 3

Problems from the text: 3.3: The hint is valid. If your integration is not “simple,” try something else. You might even see what you get if the field lines leave the charge at an arbitrary angle. 3.5: Besides the calculation, make sure you understand the physics involved, in that two unnamed observers are claiming two different things. Whom do you support? 3.6: Should be straightforward. For those interested in a more mathematical treatment of what has become essentially a two-dimensional problem, see the notes “Dipoles in Two Dimensions”, linked from our page. 3.8: Try doing this for arbitrary parameters instead of the given 5 cm, 8 cm and 10 esu/m. 3.17: I get those numbers, so we’ll hope they’re correct. 3.20: The thing here is to decide which parameters involved in the geometry are kept fixed, and which vary. I’ve found two different ways that work, and there may be others. 3.22: In years past, a variation on this was a test question. 3.28: See comments for 3.6 above.

**(A)**

So far, we have constructed a scalar potential field φ_{t}(r, θ) and the associated vector field E_{t}(r, θ), both in spherical coordinates. These fields have the properties that φ_{t} = 0 for r = R_{0}, and you have shown that E_{t} = E_{n}n̂ at r = R_{0}; that is, the vector electric field is normal to the equipotential at r = R_{0}.

(1) Assume, then, that the sphere of radius R_{0} is a conductor, and that e = 0 for r < R_{0}. Find an expression for σ(θ), the surface charge density on the sphere.

(2) Find the dipole moment due to this charge, given by

∫_{sphere} r dq,

where r is, as usual, the vector from the origin to a point on the sphere. You will want to express r in tems of x̂, ŷ, ẑ, the angles θ and ϕ, and R_{0}. Hint: The result should not be surprising.