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The given is the Radio astronomers detect electromagnetic radiation at $45MHz$from an interstellar gas cloud

The objective to find the magnetic field strength inside the gas cloud

The method used here is cyclotron motion to find the magnetic field

The cyclotron motion is defined as a particle travelling in a uniform circular motion perpendicular to the magnetic field at a constant speed. The magnetic field causes the circular motion to produce a circular frequency, which is given by equation $(29.21)$in the form

$f_{cyc}=\frac{qB}{2\mathrm{Ï€³¾}}\left(1\right)$

Where $q$denotes the particle's charge, $B$the magnetic field, and $m$the particle's mass. The frequency is determined by the magnetic field, not the particle's velocity, as indicated by equation $\left(1\right)$. We rewrite equationdata-custom-editor="chemistry" $\left(1\right)$so that $B$is in the form

$B=\frac{2{\mathrm{Ï€³¾f}}_{\mathrm{cyc}}}{q}\left(2\right)$

Now we plug in the electron's $f_{cyc}$, $m$, and $q$values to get $B$.

role="math" localid="1648904991159" $$\begin{gathered} B=\frac{2{\mathrm{Ï€³¾f}}_{\mathrm{cyc}}}{q} \\ =\frac{2\mathrm{Ï€}(9.1×{10}^{-31}\mathrm{Kg})(45×{10}^6\mathrm{Hz})}{1.6×{10}^{-19}C} \\ =1.61×{10}^{-3}T \\ =\boxed{1.61mT} \end{gathered}$$