Mathematics of Physics and Engineering

Dipoles 171

segment connecting q and —q. In what follows, we will find the approximate
value of U at the points that are far away from both q and —q, that is, when
\r — ro|| is much bigger than the distance ||ri — T2|| between the point
charges. In Problem 4.3, page 429, we discuss the lines of force for the
dipole, that is, the solutions of the differential equation r(t) = —Vf/(r).
To simplify the computations, define x = r — ro and 112 = r\ — TI.
Then

U{r) = i (||t-(l/2)r 12 || ~ ||t + (l/2)r 12 ||) " (3- (^3) - 25)
EXERCISE 3.3.13.C Draw a picture and verify (3.3.25).
For every two vectors u,v, we have ||u — v\^2 = (u — v) • (u — v) =
\u\^2 + \v\^2 — 2u • v. If ||u|| >• ||v|| (that is, ||u|| is much larger than
jj«||), then ||« - v\^2 « ||u||^2 (l - (2u • •u/||w||^2 )). Applying the linear
approximation to the function f(x) = (1 - x)~^1 /^2 at the point x = 0, we
find that f(x) « 1 + (x/2) and
1 1

• T-^\ HI»NI- (
  3

3

26

\\u — v\\ \\u )

For the exact expansion of l/||tt—v\\ in the powers of ||u||/||u||, see Problem

4.4, page 430. Applying this approximation to (3.3.25) with u = x = r — r 0

and v = ±ri2/2, we find

I/(r)«-g-(r~ro)^^12 , ||r-r 0 ||»||r 12 ||. (3.3.27)

ATTEQ \\r - ro\\^6

EXERCISE 3.3.14. B (a) Verify (3.3.27) and show that the approximation

error is of order (||T"I — T,2||/||r — T"o||)^2. (b) Using the relation E = —VC/,

verify that

(i) by computing the gradient of the approximate value U from (3.3.27);

(ii) by computing the gradient of the exact value of U from (3.3.25) and

then approximating the result.

E(r) * -Z- ( ^r^ - S ) • IkH » lkia(l, (3-3.28)

The vector d — qr 12, pointing from the negative charge to the positive

charge, is called the dipole moment of the electric dipole. Taking the limit

as T"i2 —> 0 and q —• 00 so that the vector d stays constant, we get the

point electric dipole. For the point electric dipole at the origin, we