= Solution
\b[The inferred dipole moment is approximately $0.626\ \mathrm D$.] With the static field parallel to the radiation electric field, the microwave selection rule is $\Delta M_J=0$. A fully resolved three-component first-order <rotational Stark effect> pattern naturally identifies the $J=1\to2$, $|K|=1$ transition: the three initial $M_J=-1,0,1$ values give the three components. $K=0$ would have no first-order shift.
Subtract the lower-state field shift from the upper-state field shift, rather than equating a transition shift to one level's shift:
$$
\delta\nu=\frac{\mu E_{\rm field}KM_J}{h}
\left[\frac1{J(J+1)}-\frac1{(J+1)(J+2)}\right].
$$
For $J=1$ and $|K|=1$, neighbouring components are separated by $\mu E_{\rm field}/(3h)$. Hence
$$
\boxed{\mu=\frac{3h(21\times10^6\ \mathrm{s}^{-1})}{2.0\times10^4\ \mathrm{V\,m}^{-1}}
=2.09\times10^{-30}\ \mathrm{C\,m}=0.626\ \mathrm D.}
$$
This assignment assumes the stated splitting resolves all allowed first-order components, rather than three accidentally unresolved groups of a higher-$J$ transition.
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