Question #42304

An electric dipole of dipole moment magnitude $p$ is initially placed at the origin $O$ of a coordinate system, pointing in the $+y$-direction. It is immersed in a non-uniform electrostatic field given by $\vec{E} = \alpha (x \hat{i} - y \hat{j})$, where $\alpha$ is a positive constant. The dipole is moved to the point $A(d, d)$ and rotated to point in the $+x$-direction. Consider two distinct processes for this transition:

Process 1: The dipole is first translated from $O$ to $A$ along the line $y=x$ while maintaining its orientation in the $+y$-direction. Then, at point $A$, it is rotated in the $xy$-plane to align with the $+x$-direction. Process 2: The dipole is first rotated at $O$ in the $xy$-plane to align with the $+x$-direction. Then, it is translated from $O$ to $A$ along the line $y=x$ while maintaining this new orientation.

Let $W_{T1}$ and $W_{R1}$ be the translational and rotational work done by the electric field in Process 1, respectively. Let $W_{T2}$ and $W_{R2}$ be the corresponding works in Process 2. Which of the following options correctly gives these quantities?