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Physics Question on Electric Dipole

Two point dipoles pk^p \hat{k} and p2k^\frac{p}{2} \hat{k} are located at (0,0,0)(0,0,0) and (1 m,0,2 m)(1\, m , 0,2\, m ) respectively. The resultant electric field due to the two dipoles at the point (1 m,0,0)(1 \,m , 0,0) is

A

9p32πε0k^\frac{9 p}{32 \pi \varepsilon_{0}} \hat{k}

B

−7p32πε0k^\frac{-7 p}{32 \pi \varepsilon_{0}} \hat{k}

C

7p32πε0k^\frac{7 p }{32 \pi \varepsilon_{0}} \hat{ k }

D

none of these

Answer

−7p32πε0k^\frac{-7 p}{32 \pi \varepsilon_{0}} \hat{k}

Explanation

Solution

The given point is on axis of p^2\frac{\hat{p}}{2} dipole and at equatorial line of p^\hat{p} dipole so that field at given point is (E^1+E^2)\left(\hat{E}_{1}+\hat{E}_{2}\right)
E1⃗=2K(p2)23=Kp8(+k^)\vec{E_{1}}=\frac{2 K(p 2)}{2^{3}}=\frac{K p}{8}(+\hat{k})
E2⃗=Kp1(−k^)\vec{E_{2}}=\frac{K p}{1}(-\hat{k})
E1⃗+E2⃗=−78Kp(−k^)=−7p32πε0k^\vec{E_{1}}+\vec{E_{2}}=-\frac{7}{8} K p(-\hat{k})=-\frac{7 p}{32 \pi \varepsilon_{0}} \hat{k}