Ranking the cyclotron radius

Problem

Four particles with the same kinetic energy in an identical field $B$: (A) proton; (B) electron; (C) alpha particle; (D) deuteron. Rank by orbit radius, using $r \propto \sqrt{m}/q$.

Solution

At equal kinetic energy $K$ and field $B$, from the relation $r = \dfrac{mv}{qB}$ and $K = \tfrac12 mv^2$ one obtains: $r = \frac{\sqrt{2Km}}{qB} \propto \frac{\sqrt m}{q}$

Evaluating $\sqrt m / q$ in units of $m_p$ and $e$:

• proton: $m = 1,\ q = 1 \Rightarrow \sqrt1/1 = 1$;
• electron: $m \approx 1/1836,\ q = 1 \Rightarrow \approx 0{,}023$;
• alpha: $m = 4,\ q = 2 \Rightarrow 2/2 = 1$;
• deuteron: $m = 2,\ q = 1 \Rightarrow \sqrt2/1 \approx 1{,}41$.

So the electron has a very small radius, proton and alpha coincide, the deuteron is the largest. $\ev{B \ll A = C < D}$

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Topics: Magnetismo Concepts: Moto circolare uniforme Skills: Ragionamento di ranking