   Chapter 11.11, Problem 33E

Chapter
Section
Textbook Problem

An electric dipole consists of two electric charges of equal magnitude 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 in the figure 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. To determine

To show:

E is approximately proportional to 1D3 when P is far away from dipole

Explanation

1) Concept:

The binomial series:

If k is any real number and x<1, then

1+xk=n=0knxn=1+kx+kk-12!x2+kk-1k-23!x3+

2) Given:

E=qD2-qD+d2

3) Calculation:

The electric field E at a point P given in the figure below is

E=qD2-qD+d2

To expand E as power of  dD rewrite  E as

E=qD2-qD+d2

=qD2-qD21+dD2

=qD21-1+dD-2

Use binomial series to expand 1+dD-2 as,

1+dD-2=1+-2dD+-2-2-12!dD2+-2-2-1-2-2

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