The Half-Strength Potential of a Field-Seeking Particle

A small neutral particle has positive, position-independent isotropic polarizability $\alpha$ and is placed in a prescribed nonuniform electrostatic field $\mathbf E_{\mathrm{ext}}(\mathbf r)$. Its induced dipole moment is $\mathbf p=\alpha\mathbf E_{\mathrm{ext}}$. The external sources are held fixed, and the particle is sufficiently small that field variation across it and higher multipoles may be neglected. (a) Treat $\mathbf p$ temporarily as an independent generalized coordinate. The internal restoring field conjugate to the dipole moment is $\mathbf p/\alpha$. Calculate the work required to polarize the particle from $\mathbf 0$ to $\mathbf p$, add its coupling to the external field, and minimize over $\mathbf p$. Hence derive the effective mechanical potential $U_{\mathrm{eff}}(\mathbf r)$. Explain explicitly why using only $-\mathbf p\mathbin{\cdot}\mathbf E_{\mathrm{ext}}$ after substituting $\mathbf p=\alpha\mathbf E_{\mathrm{ext}}$ gives the wrong factor. (b) Obtain the force on the equilibrated particle in terms of $\nabla |\mathbf E_{\mathrm{ext}}|^2$. State whether a particle with $\alpha>0$ moves toward stronger or weaker field magnitude, and verify the SI units of both energy and force. (c) The particle is constrained to the $z$ axis in a region with prescribed potential $$ \Phi_{\mathrm{ext}}(z)=-E_0L\arctan\!\left(\frac{z}{L}\right), $$ where $E_0>0$ and $L>0$. Find $E_z(z)$, $U_{\mathrm{eff}}(z)$, and $F_z(z)$. Evaluate $U_{\mathrm{eff}}(L)$ and $F_z(L)$, classify the equilibrium at $z=0$, and give the leading forms of the energy and force both near $z=0$ and as $|z|\to\infty$.

Integrate the internal restoring field along a straight path in dipole space: $U_{\mathrm{int}}=\int_{\mathbf 0}^{\mathbf p}(\mathbf p'/\alpha)\mathbin{\cdot}d\mathbf p'$.