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

• A

  $\frac{9 p}{32 \pi \epsilon_{0}} \hat{k}$
• Correct answer
  B

  $\frac{-7 p}{32 \pi \in_{0}} \hat{k}$
• C

  $\frac{7 p}{32 \pi \in_{0}} \hat{k}$
• D

  none of these

Solution

The given point is on axis of $\frac{\vec{p}}{2}$ dipole and at equatorial line of $\vec{p}$ dipole so that field at given point is $\left(\vec{E}_{1}+\vec{E}_{2}\right)$
$\begin{array}{c} \vec{E}_{1}=\frac{2 K(p / 2)}{2^{3}}=\frac{K p}{8}(+\hat{k}) \\ \vec{E}_{2}=\frac{K p}{1}(-\hat{k}) \\ \vec{E}_{1}+\vec{E}_{2}=-\frac{7}{8} \mathrm{~K} p(-\hat{k})=-\frac{7 p}{32 \pi \epsilon_{0}} \hat{k} \\ 250 \times 9.8 \end{array}$

Hence, the correct answer is option: 'B'.