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  4. Three short electric dipoles, each of dipole moment P, are placed at the vertices of an equilateral triangle of side length L. Each dipole has its moment, oriented parallel to the opposite side of the triangle as shown in the fig. The electric field strength at the centroid of the triangle is √a(K P/L3). Then value of a is: <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/p-jb7bqvtkjlpnwy9o.jpg />

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Q. Three short electric dipoles, each of dipole moment $P,$ are placed at the vertices of an equilateral triangle of side length $L.$ Each dipole has its moment, oriented parallel to the opposite side of the triangle as shown in the fig. The electric field strength at the centroid of the triangle is $\sqrt{a}\frac{K P}{L^{3}}.$ Then value of $a$ is :
Question

NTA AbhyasNTA Abhyas 2022

Solution:

Solution
From superposition
$E=\left(\frac{2 K P}{r^{3}}\right)$ opposite to $2\overset{ \rightarrow }{P}$
$=\frac{2 K P}{\left(\frac{L}{\sqrt{3}}\right)^{3}}$ $r=\frac{2}{3}\times \frac{\sqrt{3} L}{2}=\frac{L}{\sqrt{3}}$
$=6\sqrt{3}\left(\frac{K P}{L^{3}}\right)$
$\therefore a=108$