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The electric field due to a short electric dipole at a large distance $$(r)$$ from center of dipole on the equatorial plane varies with distance as :
$$E_{\text{equatorial}} = \frac{1}{4\pi\epsilon_0} \frac{p}{(r^2 + a^2)^{3/2}}$$
For a short dipole, $$a \ll r$$, so $$a^2$$ is negligible.
$$E_{\text{equatorial}} \approx \frac{1}{4\pi\epsilon_0} \frac{p}{(r^2)^{3/2}}$$
$$E_{\text{equatorial}} = \frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$$
$$E_{\text{equatorial}} \propto \frac{1}{r^3}$$
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