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A and B are two points on the axis and the perpendicular bisector, respectively, of an electric dipole. A and B are far away from the dipole and at equal distances from it. The fields at A and B are ${\vec{E}}_{A} a n d {\vec{E}}_{B} .$ Then

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${\vec{E}}_{A} = - 2 {\vec{E}}_{B}$

$| {\vec{E}}_{B} | = \frac{1}{2} | {\vec{E}}_{A} |, a n d {\vec{E}}_{A}$ is perpendicular to ${\vec{E}}_{B}$

${\vec{E}}_{A} = {\vec{E}}_{B}$

${\vec{E}}_{A} = 2 {\vec{E}}_{B}$

answer is C.

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Detailed Solution

Electric field due to dipole at axial position is

$E_{a x i a l} = \frac{1}{4 π ε_{0}} \frac{2 p}{r^{3}} (a l o n g t h e d i p o l e m o m e n t)$

Electric field due to dipole at equatorial position

$E_{e q} = \frac{1}{4 π ε_{0}} \frac{p}{r^{3}} (o p p o s i t e d i p o l e m o m e n t)$

Hence ${\vec{E}}_{A} = - 2 {\vec{E}}_{B}$

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