Two Gradients and One Dipole Force

A permanent point dipole with fixed moment $\mathbf p$ is centered at $\mathbf R$ in a prescribed external electrostatic field $\mathbf E(\mathbf r)=-\boldsymbol\nabla\phi_{\mathrm{ext}}(\mathbf r)$. The external field is smooth near $\mathbf R$, and the dipole's orientation is held fixed while $\mathbf R$ is varied. Represent the dipole by $+q$ at $\mathbf R+\mathbf d/2$ and $-q$ at $\mathbf R-\mathbf d/2$, where $\mathbf p=q\mathbf d$ points from the negative charge to the positive charge. Take the ideal-dipole limit $d\to0$ and $q\to\infty$ with $\mathbf p$ fixed. (a) Begin with the exact external interaction energy of the two charges and derive the limiting energy $U(\mathbf R)$. Use $\mathbf F=-\boldsymbol\nabla_{\!R}U$ to obtain the force, explicitly stating what is held fixed. (b) Independently expand the exact net force on the two charges and take the same limit. (c) In components, derive the relation between $\boldsymbol\nabla_{\!R}(\mathbf p\cdot\mathbf E)$ and $(\mathbf p\cdot\boldsymbol\nabla_{\!R})\mathbf E$. Explain why the energy and charge-limit answers agree for the stated field and why the two notations are not interchangeable for a general vector field. (d) Apply the result to $\phi_{\mathrm{ext}}(x,y,z)=V_0xy/L^2$, where $V_0$ has units of volts and $L>0$ has units of length. Find all Cartesian components of $\mathbf F$ and verify their SI units.

Write the exact differences $q[\phi_{\mathrm{ext}}(\mathbf R+\mathbf d/2)-\phi_{\mathrm{ext}}(\mathbf R-\mathbf d/2)]$ and $q[\mathbf E(\mathbf R+\mathbf d/2)-\mathbf E(\mathbf R-\mathbf d/2)]$ before expanding.