A small electric dipole of dipole moment $\vec{P}$ is placed on the x-axis at the point $(1, 0) .$ The dipole moment vector forms an angle of $30^{\circ}$ with the x-axis. A non uniform electric field has been applied in the region given by the vector $\vec{E} = x^{2} \hat{i} + y^{2} \hat{j} .$ The electric force acting on the dipole is most appropriately given by

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$(2 P \cos 30 °) (2 \hat{j})$

$(2 P \cos 30 °) (\hat{i} + \hat{j})$

$(2 P \cos 30 °) (\hat{i} + 2 \hat{j})$

$(2 P \cos 30 °) (\hat{i})$

answer is B.

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Detailed Solution

We will first calculate the $x$ component of force on the dipole. $x$ component of field at a point having $x$ co-ordinate $x$ is

$E_{x} = x^{2}$

Field at $(x + Δ x)$ is $(E_{x} + Δ E_{x})$ where

$Δ E_{x} = (\frac{d E_{x}}{d x}) Δ x = 2 x (d \cos 30 °) = 2 x d \cos 30 °$

$∴ F_{x} = q (E_{x} + Δ E_{x}) - q E_{x} = q Δ E$ $= 2 q x d \cos 30 ° = [2 P \cos 30 °] x$

At $x = 1$ $F_{x} = 2 P \cos 30 °$

Similarly, $F_{y} = [2 P \sin 30 °] y$ due to $E_{y}$

At $y = 0, F_{y} = 0.$ (According to the given function, $E_{x}$ does not give $F_{y}$ and $E_{y}$ does not give $F_{x}$ )

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