P19-160 – Electric Field Created by an Electric Dipole

E_{net}=E_+-E_-=\frac{kq}{{\left(x-d/2\right)}^2}-\frac{kq}{{\left(x+d/2\right)}^2}

\begin{aligned}
E_{net}&=\frac{kq}{\displaystyle{{\left(x-\frac{d}{2}\right)}^2}}-\frac{kq}{\displaystyle{{\left(x+\frac{d}{2}\right)}^2}} \\
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&=\frac{kq}{\displaystyle{x^2{\left(1-\frac{d}{2x}\right)}^2}}-\frac{kq}{\displaystyle{x^2{\left(1+\frac{d}{2x}\right)}^2}} \\
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&=\frac{kq}{x^2}\left[\frac{1}{\displaystyle{{\left(1-\frac{d}{2x}\right)}^2}}-\frac{1}{\displaystyle{{\left(1+\frac{d}{2x}\right)}^2}}\right]
\end{aligned}

\begin{aligned}
E_{net}&=\frac{kq}{x^2}\left[\frac{1}{\displaystyle{{\left(1-\frac{d}{2x}\right)}^2}}-\frac{1}{\displaystyle{{\left(1+\frac{d}{2x}\right)}^2}}\right] \\
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&=\frac{kq}{x^2}\left[{\left(1-\frac{d}{2x}\right)}^{-2}-{\left(1+\frac{d}{2x}\right)}^{-2}\right] \\
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&\approx \frac{kq}{x^2}\left[1+\frac{2d}{2x}-\left(1-\frac{2d}{2x}\right)\right] \\
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&\approx \frac{kq}{x^2}\cdot \frac{2d}{x} \\
\\
&\approx \frac{2kqd}{x^3}
\end{aligned}

\boxed{{\overrightarrow{E}}_{net}\approx\frac{2kqd}{x^3}\ \ \hat{x}\approx \frac{2kp}{x^3}\ \ \hat{x}}