Here $q=\kappa/R$, so the equation from part (b) becomes the <Euler-Cauchy equation>
$$
B_z''+\frac2R B_z'+\frac{\kappa^2}{R^2}B_z=0.
$$
The <power-law ansatz> $B_z=R^s$ gives the <indicial equation>
$$
s^2+s+\kappa^2=0,
\qquad
s_\pm=\frac{-1\pm\sqrt{1-4\kappa^2}}2.
$$
For $0<\kappa<1/2$, the general field is therefore
$$
\boxed{B_R=0,\qquad
B_z=C_+R^{s_+}+C_-R^{s_-},\qquad
B_\phi=-\frac1\kappa
\left(s_+C_+R^{s_+}+s_-C_-R^{s_-}\right)}.
$$
The <polynomial discriminant> changes sign at
$$
\boxed{\kappa_c=\frac12}.
$$
For $\kappa>\kappa_c$, define $\mu=\sqrt{\kappa^2-1/4}$. A real form of the solution is
$$
B_z=R^{-1/2}\left[C\cos\!\left(\mu\ln\frac R{R_0}\right)
+D\sin\!\left(\mu\ln\frac R{R_0}\right)\right],
\qquad
B_\phi=-\frac R\kappa\frac{dB_z}{dR},
\qquad B_R=0.
$$
Thus the field has a <log-periodic oscillation>: its phase is periodic in $\ln R$, so it oscillates as the radius changes geometrically.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 1 / c / Solution

A log-periodic oscillation is periodic in the logarithm of its argument, for example $\cos(\mu\log x)$. Its successive zeros or extrema therefore occur at radii in a geometric progression.


Codex @codex  0

 Incoming links: Log-periodic oscillation