d
Electric field at \(\mathrm{p}=2 \mathrm{E}_{1} \cos \theta_{1}-2 \mathrm{E}_{2} \cos \theta_{2}\)

\(=\frac{2 \mathrm{Kq}}{\left(\mathrm{d}^{2}+\mathrm{D}^{2}\right)} \times \frac{\mathrm{D}}{\left(\mathrm{d}^{2}+\mathrm{D}^{2}\right)^{1 / 2}}-\frac{2 \mathrm{Kq}}{\left[(2 \mathrm{d})^{2}+\mathrm{D}^{2}\right]} \times \frac{\mathrm{D}}{\left[(2 \mathrm{d})^{2}+\mathrm{D}^{2}\right]^{1 / 2}}\)

\(=2 \mathrm{KqD}\left[\left(\mathrm{d}^{2}+\mathrm{D}^{2}\right)^{-3 / 2}-\left(4 \mathrm{d}^{2}+\mathrm{D}^{2}\right)^{-3 / 2}\right]\)

\(=\frac{2 \mathrm{KqD}}{\mathrm{D}^{3}}\left[\left(1+\frac{\mathrm{d}^{2}}{\mathrm{D}^{2}}\right)^{-3 / 2}-\left(1+\frac{4 \mathrm{d}^{2}}{\mathrm{D}^{2}}\right)^{-3 / 2}\right]\)

Applying binomial approximation \(\because \mathrm{d}<<\mathrm{D}\) 

\(=\frac{2 \mathrm{KqD}}{\mathrm{D}^{3}}\left[1-\frac{3}{2} \frac{\mathrm{d}^{2}}{\mathrm{D}^{2}}-\left(1-\frac{3 \times 4 \mathrm{d}^{2}}{2 \mathrm{D}^{2}}\right)\right]\)

\(=\frac{2 \mathrm{KqD}}{\mathrm{D}^{3}}\left[\frac{12}{2} \frac{\mathrm{d}^{2}}{\mathrm{D}^{2}}-\frac{3}{2} \frac{\mathrm{d}^{2}}{\mathrm{D}^{2}}\right]\)

\(=\frac{9 \mathrm{kqd}^{2}}{\mathrm{D}^{4}}\)