Question

An alternating electric field, of frequency v, is applied across the dees (radius $=$ R) of a cyclotron that is being used to accelerate protons (mass $=$ m). The operating magnetic field (B) used in the cyclotron and the kinetic energy (K) of the proton beam, produced by it, are given by

• A. $B=\frac{mv}{e}$ and $K=2m{\pi }^2{v}^2{R}^2$
• B. $B=\frac{2\pi mv}{e}$ and $K=m^2\pi v^2{R}^2$
• C. $B=\frac{2\pi mv}{e}$ and $K=2m{\pi }^2{v}^2{R}^2$
• D. $B=\frac{mv}{e}$ and $K=m^2\pi v{R}^2$

Answer 1

Answer: (C)

$B=\frac{2\pi mv}{e}$ and $K=2m{\pi }^2{v}^2{R}^2$

Time period of cyclotron is

$T=\frac{1}{v}=\frac{2\pi m}{eB};B=\frac{2\pi m}{e}v;R=\frac{mv}{eB}=\frac{P}{eB}$

$\Rightarrow P=eBR=e\times \frac{2\pi mv}{e}R=2\pi mvR$

$K.E.=\frac{P^2}{2m}=\frac{(2\pi mvR{)}^2}{2m}=2{\pi }^{2}m{v}^2{R}^2$