Mathematical Tools for Physics - Department of Physics - University

9—Vector Calculus 1 241

Look for a solution for the magnetic field in the formB~ =ˆzBz(r). What is the total current per

length in this case? That is, for a length∆zhow much current is going around the axis and how is

this related toBzalongr= 0?

Examine also the special case for whichf(r) = 0except in a narrow rangea < r < bwithb−ab

(thin). Compare this result to what you find in introductory texts about the solenoid.

9.50 For a spherical mass distribution such as the Earth, what would the mass density function have

to be so that the gravitational field has constant strength as a function of depth? Ans:ρ∝ 1 /r.

9.51 Use index notation to derive

(~

A×B~

)

.

(~

C×D~

)

=

(~

A.C~

)(~

B.D~

)

−

(~

A.D~

)(~

B.C~

)

9.52 Show that∇.(A~×B~) =B~.∇×A~−A~.∇×B~. Use index notation to derive this.

9.53 Use index notation to compute∇ei~k.~r. Also compute the Laplacian of the same exponential,

∇^2 =div grad.

9.54 Derive the force of one charged ring on another, as shown in equation (2.35).

9.55 A point massmis placed at a distanced > Rfrom the center of a spherical shell of radiusR

and massM. Starting from Newton’s gravitational law for point masses, derive the force onmfrom

M. Placemon thez-axis and use spherical coordinates to express the piece ofdM withindθand

dφ. (This problem slowed Newton down when he first tried to solve it, as he had to stop and invent

integral calculus first.)

9.56 The volume charge density is measured near the surface of a (not quite perfect) conductor to

beρ(x) =ρ 0 ex/aforx < 0. The conductor occupies the regionx < 0 , so the electric field in the

conductor is zero once you’re past the thin surface charge density. Find the electric field everywhere

and graph it. Assume that everything is independent ofyandz(andt).