= Solution
Here $r\gg\lambda_D$, so the thin-double-layer result gives
$$
\boxed{\mu_e=\frac{(8.85\times10^{-12})(78.5)(0.045)}{10^{-3}}=3.13\times10^{-8}\ \mathrm{m^2\,V^{-1}\,s^{-1}},}
$$
and therefore \b[$v=3.13\times10^{-5}$ m s$^{-1}$], along the field for the positive <zeta potential>.
In the opposite limit $r\ll\lambda_D$, the <Hückel electrophoretic mobility> is $\mu_e=2\epsilon\zeta/(3\eta)$, giving \b[$2.08\times10^{-8}$ m$^2$ V$^{-1}$ s$^{-1}$], two-thirds of the thin-layer value at the same $\zeta$. Both asymptotic expressions are independent of radius at fixed <zeta potential>; the crossover depends on $r/\lambda_D$. These classical approximations neglect double-layer relaxation and strong-potential corrections.
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