Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 77 / 3 / f

A flagellar bundle and the counterrotating body supply oppositely signed axial torques separated along the swimming direction. Their leading torque moments cancel, and the next term is a rotlet dipole, decaying as $r^{-3}$. For the ideal axisymmetric bacterium the torque vector is parallel or antiparallel to $\mathbf{e}$.

To fix the sign convention, define a signed dipole strength $D$ with units torque times length and setDifferentiating givesMore generally, differentiating $\mathbf{R}\times \mathbf{r}/r^3$ gives $\mathbf{R}\times \mathbf{e}/r^3-3(\mathbf{e}\cdot \mathbf{r})(\mathbf{R}\times \mathbf{r})/r^5$; the first term vanishes when $\mathbf{R}\parallel \mathbf{e}$. A derivative with respect to source position instead of observation position changes the definition of the signed moment. The physical handedness depends on which torque is ahead, so it cannot be fixed without that convention.