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sity distribution (r), therefore these expressions must be identical.
Problem j: Use the equivalence of these expressions to derive that for r0 < r the following
identity holds:
1l 4 Y ( ')Y ( 0 '0 ) r0l :
XX
1
r0 = l=0 m=;l (2l + 1) lm (17.39)
lm rl+1
jr ; j
The derivation of this section could also have been made using (17.39) as a starting
point because this expression can be derived by using the generating function Legendre
polynomials and by using the addition theorem for obtaining the msummation. However,
these concepts are not needed in the treatment of this section that is based only on the
expansion of functions in spherical harmonics and on the use of Green's functions.
As a last exercise let us consider the special case of a spherically symmetric mass
distribution: = (r). For such a mass distribution the potential is given by
V (r) = GM (17.40)
r
;
17.5. DIPOLES, QUADRUPOLES AND GENERAL RELATIVITY 251
where M is the total mass of the body. The gradient of this potential is indeed equal to
the gravitational acceleration given in expression (6.5) for a spherically symmetric mass
M.
Problem i: Derive the potential (17.40) from (17.38) by considering the special case that
the massdensity depends only on the radius.
17.5 Dipoles, quadrupoles and general relativity
We have seen in the section 6.2 that a spherically symmetric mass leads to a gravitational
eld g(r) = ^=r2, which corresponds to a gravitational potential V (r) =
r =r.
;GM ;GM
Similarly, the electric potential due to a spherically symmetric charge distribution is given
by V (r) = q=4 "0 r, where q is the total charge. In this section we investigate what
happens if we place a positive charge and a negative charge close together. (Since there
is no negative mass, we treat for the moment the electrical potential, but we will see in
section 17.6 that the results also have a bearing on the gravitational potential.)
+q + r =+ a /2
 r=  a/2
q
Figure 17.3: Two opposite charges that constitute an electric dipole.
Consider the case that a positive charge +q is placed at position a=2 and a negative
charge is placed at position
;q ;a=2.
Problem a: The total charge of this system is zero. What would you expect the electric
potential to be at positions that are very far from the charges compared to potential
for a single point charge?
Problem b: The potential follows by adding the potentials for the two point charges.
Show that the electric potential generated by these two charges is given by
q q
4 "0 V (r) = + a=2j : (17.41)
a=2j ;
jr ; jr
Problem c: Ultimately we will place the charges very close to the origin by taking the
limit a 0. We can therefore restrict our attention to the special case that a r.
!
Use a rst order Taylor expansion to show that up to order a:
1 = 1 1 (r a) : (17.42)
a=2j r 2r3 ;
jr ;
CHAPTER 17. POTENTIAL THEORY
252
Problem d: Insert this in (17.41) and derive that the electric potential is given by:
4 " V (r) = q (r a) (17.43)
0 r3
;
Now suppose we bring the charges in gure 17.3 closer and closer together, and suppose
we let the charge q increase so that the product p = qa is constant, then the electric
potential is given by:
4 "0 V (r) = (^r2p)
r (17.44)
;
where we used that r = r^. The vector p is called the dipole vector. We will see in the next
r
section how the dipole vector can be de ned for arbitrary charge or mass distributions.
In problem a you might have guessed that the electric potential would go to zero
at great distance. Of course the potential due to the combined charges goes to zero
much faster than the potential due to a single charge only, but note that the electric
potential vanishes as 1=r2 compared to the 1=r decay of the potential for a single charge.
Many physical systems, such as neutral atoms, consists of neutral combinations of positive
and negative charges. The lesson we learn from equation (17.43) is that such a neutral
combination of charges does generate a nonzero electric eld and that such a system will
in general interact with other electromagnetic systems. For example, atoms interact to
leading order with the radiation (light) eld through their dipole moment 52]. In chemistry,
the dipole moment of molecules plays a crucial role in the distinction between polar and
apolar substances. Water would not have its many wonderful properties if it would not
have a dipole moment.
Let us now consider the electric eld generated by an electric dipole.
Problem e: Take the gradient of (17.44) to show that this eld is given by
1
E(r) = 4 " r3 (p 3^ (^ p)) :
rr (17.45)
;
0
Hint, either use the expression of the gradient in spherical coordinates or take the
gradient in Cartesian coordinates and use (4.7).
The electric eld generated by an electric dipole has the same form as the magnetic
eld generated by a magnetic dipole as shown in expression (4.3) of section (4.1). The
mathematical reason for this is that the magnetic eld satis es equation (4.13) which
states that (r B) = 0 while the electric eld in free space satis es according to equation
(4.12) the same eld equation: (r E) = 0. However, there is an important di erence, the
electric eld is generated by electric charges. In general, this eld satis es the equation
(r E) = (r)="0 . In the example of this section we created a dipole eld by taking two
opposite charges and putting them closer and closer together. However, the magnetic eld
satis es (r B) = 0 everywhere. The reason for this is that the magnetic equivalent of
electric charge, the magnetic monopole, has not been discovered in nature.
Problem f: The fact that magnetic monopoles have not been observed in nature seems
puzzling, because we have seen that the magnetic dipole eld has the same form as
the electric dipole eld that was constructed by putting two opposite electric charges
17.5. DIPOLES, QUADRUPOLES AND GENERAL RELATIVITY 253
close together. Can you think of a physical system or device that does generate the
magnetic dipole eld but that does not consist of two magnetic monopoles placed
closely together?
Monopole Dipole Quadrupole

+ +
+
  +
.
Figure 17.4: The de nition of the monopole, dipole and quadrupole in terms of electric
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