Question

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${\mathit{E}}_{\mathbf{n}\mathbf{e}\mathbf{x}\mathbf{t}}$ in the expression$\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{o}}}\frac{\mathbf{q}\mathbf{d}}{{\mathbf{z}}^{\mathbf{3}}}\mathbf{+}{\mathit{E}}_{\mathbf{n}\mathbf{e}\mathbf{x}\mathbf{t}}$ ?

Short answer

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

Step-by-step solution

The given data

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

$E=\frac{1}{2\pi {\epsilon }_o}\frac{q}{z^3}\left(\frac{d}{{\left(1-{\left(\frac{d}{2z}\right)}^2\right)}^2}\right)$

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,

$E=\frac{1}{2\pi {\epsilon }_o}\frac{qd}{z^3}$ (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-

$E=\frac{1}{2\pi {\epsilon }_o}\frac{q}{z^3}\left(\frac{d}{{\left(1-{\left(\frac{d}{2z}\right)}^2\right)}^2}\right)$

Expanding using binomial theorem,

$\begin{array}{rcl}E& =& \frac{1}{2\pi {\epsilon }_o}\frac{q}{z^3}\left(\frac{d}{{\left(1-{\left(\frac{d}{2z}\right)}^2\right)}^2}\right)\\ & =& \frac{1}{4\pi {\epsilon }_o}\frac{q}{z^2}\left(\frac{1}{{\left(1-\frac{d}{2z}\right)}^2}-\frac{1}{{\left(1+\frac{d}{2z}\right)}^2}\right)\\ & =& \frac{1}{4\pi {\epsilon }_o}\frac{q}{z^2}\left[\left(1+\frac{d}{z}+\frac{3}{4}\frac{d^2}{z^2}+\frac{1}{2}\frac{d^3}{z^3}+\dots .\right)-\left(1-\frac{d}{z}+\frac{3}{4}\frac{d^2}{z^2}-\frac{1}{2}\frac{d^3}{z^3}+\dots .\right)\right]\\ & =& \frac{1}{2\pi {\epsilon }_o}\frac{q}{z^3}+\frac{1}{4\pi {\epsilon }_o}\frac{q{d}^3}{z^5}+\dots \dots \\ & & \end{array}$

Therefore, the term required in the problem is $\frac{1}{4\pi {\epsilon }_o}\frac{q{d}^3}{z^5}$.