9—Vector Calculus 1 2819.41 Compute the divergence and the curl of
yˆx−xˆy
x^2 +y^2, and ofyxˆ−xyˆ
(x^2 +y^2 )^29.42 Translate the preceding vector fields into polar coordinates, then take their divergence and curl.
9.43 As a review of ordinary vector algebra, and perhaps some practice in using index notation, translate the
triple scalar product into index notation and prove first that it is invariant under cyclic permutations of the vectors.
(a)A~.B~×C~=B~.C~×A~=C~.A~×B~. Then that
(b)A~.B~×C~=A~×B~.C~.
(c) What is the result of interchanging any pair of the vectors in the product?
(d) Show why the geometric interpretation of this product is as the volume of a parallelepiped.
9.44 What is the total flux,
∮ ~
E.dA~, out of the cube of sideawith one corner at the origin?(a)E~=αxˆ+βyˆ+γˆz (b)E~=αxˆx+βyyˆ+γzzˆ.
9.45 The electric potential from a single point chargeqiskq/r. Two charges are on thez-axis:−qat position
z=z 0 and+qat positionz 0 +a.
(a) Write the total potential at the point(r,θ,φ)in spherical coordinates.
(b) Assume thatr aandr z 0 , and use the binomial expansion to find the series expansion for the total
potential out to terms of order 1 /r^3.
(c) how does the coefficient of the 1 /r^2 term depend onz 0? The coefficient of the 1 /r^3 term? These tell you
the total electric dipole moment and the total quadrupole moment.
(d) What is the curl of the gradient of each of these two terms?
The polynomials of section4.9will appear here, with argumentcosθ.
9.46 For two point chargesq 1 andq 2 , the electric field very far away will look like that of a single point
chargeq 1 +q 2. Go the next step beyond this and show that the electric field at large distances will approach a
direction such that it points along a line that passes through the “center of charge” (like the center of mass):
(q 1 ~r 1 +q 2 ~r 2 )/(q 1 +q 2 ). What happens to this calculation ifq 2 =−q 1? You may find the results of problem 31
useful. Sketches of various cases of course.