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Unformatted text preview: MATH 153 Note 1: Solution to Problem 33 in 11.11 Exercise 33 : An electric dipole consists of two electric charges of equal mag- nitude and opposite sign. If the charges are q and- q and are located at a distance d from each other, then the electric field E at the point P at a distance D away from the charge q is E = q D 2- q ( D + d ) 2 . By expanding this expression for E as a series in powers of d/D , show that E is approximately proportional to 1 /D 3 when P is far away from the dipole. Solution : Note how this question differs from most questions of approximat- ing functions with power series; while we ordinarily want to use a partial sum to approximate values close to the center (i.e., small values), here we want to approximate values very far away from the center (i.e., large values). This is the motivation for changing the variable from D to x = d/D . In order to make this variable change, we rearrange the equation by making the substitution D = d/x , as follows: E = q D 2- q...
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- Spring '08