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(C) The electric field due to an electric dipole at a point on its axis (axial position) at a distance $r$ is given by $\vec{E}_{axial} = \frac{1}{4 \

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$\vec{E}_A = -2 \vec{E}_B$

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(C) The electric field due to an electric dipole at a point on its axis (axial position) at a distance $r$ is given by $\vec{E}_{axial} = \frac{1}{4 \pi \varepsilon_0} \frac{2\vec{p}}{r^3}$.The electric field due to an electric dipole at a point on its perpendicular bisector (equatorial position) at the same distance $r$ is given by $\vec{E}_{equatorial} = -\frac{1}{4 \pi \varepsilon_0} \frac{\vec{p}}{r^3}$.Given that point $A$ is on the axis and point $B$ is on the perpendicular bisector,we have $\vec{E}_A = \frac{1}{4 \pi \varepsilon_0} \frac{2\vec{p}}{r^3}$ and $\vec{E}_B = -\frac{1}{4 \pi \varepsilon_0} \frac{\vec{p}}{r^3}$.Comparing these two expressions,we get $\vec{E}_A = -2 \vec{E}_B$.

$A$ 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 distance from it. The electric fields at $A$ and $B$ are $\vec{E}_A$ and $\vec{E}_B$.

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