Chapter 22: Q18P (page 654)

The electric field of an electric dipole along the dipole axis is approximated by equations. 22-8 and 22-9. If a binomial expansion is made of Eq. 22-7, what is the next term in the expression for the dipole’s electric field along the dipole axis, that is, what is $E_{n e x t}$ in the expression $E = \frac{1}{2 π ε_{o}} \frac{q d}{z^{3}} + E_{n e x t}$ ?

Short Answer

Expert verified

The next term in the expression for the dipole’s electric field along the dipole axis is $\frac{1}{4 π ε_{o}} \frac{q d^{3}}{z^{5}}$

Step by step solution

The given data

Expression of electric field due to an electric dipole is given as:

$E = \frac{1}{2 π ε_{o}} \frac{q}{z^{3}} (\frac{d}{{(1 - {(\frac{d}{2 z})}^{2})}^{2}})$

Understanding the concept of electric field

Region in the vicinity of a charge in which if any other charge is placed, it experiences a force, either attractive or repulsive. This region is known as electric field.

The electric field of an electric dipole along the dipole axis,

$E = \frac{1}{2 π ε_{o}} \frac{q d}{z^{3}}$ (i)

Calculation of the next term in the expression for the electric field of a dipole

The equation 22-7 mentioned in the question, is-

$E = \frac{1}{2 π ε_{o}} \frac{q}{z^{3}} (\frac{d}{{(1 - {(\frac{d}{2 z})}^{2})}^{2}})$

Expanding using binomial theorem,

$\begin{array}{rcl} E & = & \frac{1}{2 π ε_{o}} \frac{q}{z^{3}} (\frac{d}{{(1 - {(\frac{d}{2 z})}^{2})}^{2}}) \\ = & \frac{1}{4 π ε_{o}} \frac{q}{z^{2}} (\frac{1}{{(1 - \frac{d}{2 z})}^{2}} - \frac{1}{{(1 + \frac{d}{2 z})}^{2}}) \\ = & \frac{1}{4 π ε_{o}} \frac{q}{z^{2}} [(1 + \frac{d}{z} + \frac{3}{4} \frac{d^{2}}{z^{2}} + \frac{1}{2} \frac{d^{3}}{z^{3}} + \dots .) - (1 - \frac{d}{z} + \frac{3}{4} \frac{d^{2}}{z^{2}} - \frac{1}{2} \frac{d^{3}}{z^{3}} + \dots .)] \\ = & \frac{1}{2 π ε_{o}} \frac{q}{z^{3}} + \frac{1}{4 π ε_{o}} \frac{q d^{3}}{z^{5}} + \dots \dots \end{array}$