Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 314 / 1 / b / ii / Solution

In the midplane, the vacuum magnetic dipole field has scale $B_0\sim \mu /r^3$, while the result of part i gives $j\sim {\Omega }^2\mu /(c{r}^2)$. Ampère's law then gives the induced-field estimatewhere numerical factors are immaterial in this dimensional analysis. Thus $B_i\sim B_0$ atThis is the light cylinder radius: rigid corotation would have $u=\Omega R_e\sim c$. The nonrelativistic approximation and the assumed unmodified vacuum field therefore fail at the same scale, and a self-consistent pulsar magnetosphere must replace them.