= Solution
Choose $\mathbf B=B\hat{\mathbf z}$ and a signed tangential <velocity> $\mathbf v=u\hat{\boldsymbol\phi}$, with $u>0$ counterclockwise viewed from $+z$. The magnetic part of the <Lorentz force> is $euB\hat{\mathbf r}$, while the electric part is $-eV\hat{\mathbf r}/(\eta r)$. The radial <acceleration> for a <circular orbit> is $-u^2/r$, regardless of the sign of $u$. Consequently
$$
-\frac{mu^2}{r}=e\left(-\frac{V}{\eta r}+uB\right),\qquad
\boxed{mu^2+eBr\,u-\frac{eV}{\eta}=0.}
$$
One valid signed-velocity convention therefore has $a=m$, $b=eBr$, $c=-eV/\ln(r_2/r_1)$. If $v$ is required to be a nonnegative <speed>, the two senses of rotation instead satisfy
$$
\boxed{mv^2\pm eBr\,v-\frac{eV}{\eta}=0,}
$$
with the plus sign for counterclockwise motion. For positive $V$, the electric <force> is inward, so it can supply <circular motion> in either direction; in one direction the magnetic <force> opposes it, while in the other it helps. A signed negative root must not be discarded as an unphysical <speed>; it denotes the opposite orbit direction.
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