In the Gaussian units of the paper, the total <magnetic pressure> is $p_B=(B_R^2+B_\phi^2)/(8\pi)$. Define the initial <plasma beta> $\beta_p=8\pi p_0/B_0^2\gg1$, assuming a nonzero initial field. The solution from part (b) gives
$$
p_B(R,t)=\frac{B_0^2}{8\pi}\left(\frac{R_0}{R}\right)^2\left[1+\frac94\frac{GMt^2}{R^3}\right],\qquad
\frac{p_B}{p}=\frac1{\beta_p}\left[1+\frac{9GMt^2}{4R^3}\right].
$$
For any $t>0$ this ratio decreases strictly with $R$, tends to infinity as $R\downarrow0$ and tends to $1/\beta_p<1$ as $R\to\infty$. The <magnetic-pressure equality radius under Keplerian winding> is therefore unique in the formal profile on $R>0$:
$$
\boxed{R_{\rm eq}(t)=\left[\frac{9GM}{4(\beta_p-1)}\right]^{1/3}t^{2/3},\qquad\zeta=\frac23.}
$$
The same expression proves $p_B>p$ for $R<R_{\rm eq}$ and $p_B<p$ for $R>R_{\rm eq}$. At equality, $|B_\phi|/|B_R|=\sqrt{\beta_p-1}\gg1$, justifying the toroidal-dominated approximation
$$
R_{\rm eq}\simeq\left(\frac{9B_0^2GM}{32\pi p_0}\right)^{1/3}t^{2/3}.
$$
The source's existence claim refers to the indefinitely extended radial profile and the stated neglect of back-reaction. For a disk with a finite inner and outer edge, an equality radius lies in the disk only while the displayed value lies between those edges. If the initial field were zero, no magnetic winding or equality radius would occur.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 314 / 4 / c / Solution

{title2=$R_{\rm eq}\propto t^{2/3}$}

For <Keplerian rotation>, initial $B_R\propto R^{-1}$ and thermal pressure $p\propto R^{-2}$, <axisymmetric magnetic winding> gives $B_\phi=-(3/2)\Omega tB_R$. If the initial <plasma beta> is $\beta_p>1$, the formal equality of <magnetic pressure> and gas pressure occurs at
$$
R_{\rm eq}^3=\frac{9GMt^2}{4(\beta_p-1)}.
$$
The magnetic-to-thermal pressure ratio decreases with radius, so magnetic pressure dominates inside this radius. In a disk with a finite radial range, an equality radius exists in the disk only when this formal radius lies within that range.

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 Incoming links: Magnetic-pressure equality radius under Keplerian winding