The cyclotron frequency is a one-line calculation, and it alone separates the options.

Step 1 — the cyclotron angular frequency. An electron in a magnetic field circles at

\(\omega_{c}=\dfrac{eB_{0}}{m}\)

\(=\dfrac{1.6\times10^{-19}\times0.336}{9.11\times10^{-31}}=\dfrac{5.376\times10^{-20}}{9.11\times10^{-31}}=5.90\times10^{10}\ \text{rad/s}\)

— statement (B). Note that the answer depends only on the flux density; the anode voltage, beam current and radii play no part in it whatever. Statements (A) and (C) are the same figure with the exponent wrong by four and six orders of magnitude, so this one calculation eliminates options 1, 3 and 4 and leaves option 2.

Step 2 — the Hull cut-off voltage. This is the anode voltage below which the magnetic field bends every electron back before it reaches the anode:

\(V_{c}=\dfrac{e}{8m}B_{0}^{2}a^{2}\left(1-\dfrac{b^{2}}{a^{2}}\right)^{2}\)

with a the anode radius and b the cathode radius. The ratio term is the same on any consistent length units:

\(\left(1-\dfrac{b^{2}}{a^{2}}\right)^{2}=\left(1-0.25\right)^{2}=0.5625\)

and \(\dfrac{e}{8m}B_{0}^{2}=2.1954\times10^{10}\times0.112896=2.4785\times10^{9}\).

A note on the arithmetic, stated plainly. Taking the printed 10 cm literally, \(a^{2}=0.01\ \text{m}^{2}\) and the formula gives about 13.9 MV. The printed value of 139.50 kV is reproduced exactly by taking the radii as 10 mm and 5 mm instead, that is \(a^{2}=10^{-4}\):

\(V_{c}=2.4785\times10^{9}\times10^{-4}\times0.5625=1.394\times10^{5}=139.4\ \text{kV}\)

So the question's dimensions are inconsistent by a factor of ten, but (D) is the intended value and (E) matches no reading at all. The cyclotron frequency settles the answer independently, so the ambiguity does not affect the choice.

Why both quantities matter in a magnetron. The device oscillates only when the anode voltage is below cut-off, so that electrons curve back and form the rotating spokes of space charge that exchange energy with the resonant cavities. The cyclotron frequency sets the timescale of that circular motion, and the cavities are dimensioned so their resonance matches the spoke rotation.

Hence, the correct statements are (B) and (D).