 End-charged ideal polymer in an electric field

An ideal chain with endpoint charges $+q$ and $-q$ has coupling energy $-q\mathbf{E}\cdot \mathbf{R}$ in a uniform electric field, when $\mathbf{R}$ points from negative to positive charge. Completing the square in its Gaussian end-to-end distribution gives the displayed mean extension. The force conjugate to end separation is $qE$, even though the two endpoints experience opposite forces; adding their magnitudes would double-count the work. The zero-field rms length is $a\sqrt{N}$, and the rms of a single Cartesian component is $a\sqrt{N/3}$. Direct endpoint interactions and charge-induced changes of solvent quality are excluded from this model.