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ચાર બિંદુવત વિદ્યુતભારો $-q, +q, +q$ અને $-q$ $y$ અક્ષ પર $y = -2d$, $y = -d, y = +d$ અને $y = +2d$ પર છે.$x$ અક્ષ પર $x = D\,\,(D > > d)$ પાસે વિદ્યુતક્ષેત્ર કોના સમપ્રમાણમાં હશે?
$E \propto \frac{1}{D}$
$E \propto \frac{1}{D^3}$
$E \propto \frac{1}{D^2}$
$E \propto \frac{1}{D^4}$
Solution
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 $\because \mathrm{d}<<\mathrm{D}$
$=\frac{2 \mathrm{KqD}}{\mathrm{D}^{3}}\left[1-\frac{3}{2} \frac{\mathrm{d}^{2}}{\mathrm{D}^{2}}-\left(1-\frac{3 \times 4 \mathrm{d}^{2}}{2 \mathrm{D}^{2}}\right)\right]$
$=\frac{2 \mathrm{KqD}}{\mathrm{D}^{3}}\left[\frac{12}{2} \frac{\mathrm{d}^{2}}{\mathrm{D}^{2}}-\frac{3}{2} \frac{\mathrm{d}^{2}}{\mathrm{D}^{2}}\right]$
$=\frac{9 \mathrm{kqd}^{2}}{\mathrm{D}^{4}}$
