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Ch3.2 - Multipole expansions The potential due to a point...

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Multipole expansions The potential due to a point charge is given by a simple expression : V( r ) = | | q 0 4 q r r - πε Suppose the charges are distributed over a localized region , then the potential is given by V( r ) = - V 3 0 r d 4 1 ' | | ) ( r' r r' ρ πε , which could be very complicated if the distribution is not regular in shape. However, if we are observing the charge from a very far away distance, then we can make a reasonable approximation V( r ) 2245 - V 3 a 0 r d 1 4 1 ' ) ( | | r' r r ρ πε = | | a total 0 q 4 1 r r - πε , where a r is an average position of the charge distribution. That is, the localized distribution of charge appears to be like a point charge if they are observed from a distance very far away. Suppose we have a neutral distribution of charge, e.g. a neutral atom contains a nucleus and with electrons distributed around it, with total charge = 0. Our approximation becomes V( r ) = 0 . In this case, our approximation is not good enough. We shall learn how to make better approximation systematically. First, we will consider a model system that consists of two equal and opposite point charges that are separated by a fix distance, i.e. an electric dipole system. From a very long distance from the dipole, our first approximation gives V( r ) = 0 since Σ q = 0 .
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Since the distance of the observation point from +q and –q are not equal, the cancellation is not complete and V( r ) = - + - + r r q q 4 1 0 πε 0 (3.1) We expand r + and r under the condition r >> d (i.e. far away) 2 1 2 2 2 1 2 2 r d r 4 d 1 r 2 d r 2 2 d r + = + = ± θ θ cos cos ) ( r + ± 2245 + = - ± ... cos cos θ θ r 2 d 1 r 1 r 4 d r d 1 r 1 1 2 1 2 2 r 2 r d r d r 1 1 1 θ θ cos cos = 2245 - - + r r V( r ) 2 0 r qd 4 1 θ πε cos 2245 (3.2)
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Therefore, the influence of this electric dipole, which is a neutral charge combination, is still felt by an observer, but it falls off with distance much faster than that for the potential of a system with a net charge, i.e. 2 r 1 goes to zero faster than r 1 as r . Eq.(3.2) can be rewritten as V( r ) = 2 0 r 4 1 r p ˆ πε , where p , called the dipole moment , is pointing from the –ve charge to the +ve charge, with a magnitude | p | = qd . The potential due to a dipole is the left behind from the incomplete cancellation of the potentials from two equal and opposite charges that are not exactly at the same position We can play this game of ‘incomplete cancellation’ further by putting two dipoles next to, but not overlapping, one another. Consider the charge system below : For simplicity we have put the observation point P and all the charges on a line. The potential at P is given by
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+ - - + - - + - + - + + + = 2 d 2 L r q 2 d 2 L r q 2 d 2 L r q 2 d 2 L r q 4 1 V 0 P πε - + - + + = - - 1 1 0 r 2 d r 2 L 1 r 2 d r 2 L 1 r q 4 1 πε + - -
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