Question

The oscillating frequency of a cyclotron is $10 \,MHz$ . If the radius of its Dees is $0.5 \,m$ , the kinetic energy of a proton, which is accelerated by the cyclotron is

• A. 10.2 MeV
• B. 2.55 MeV
• C. 20.4 MeV
• D. 5.1 MeV

Answer 1

Answer: (D)

5.1 MeV

Answer 2

KE of charged possible in a cyclotron,
${{E}_{k}}=\frac{{{q}^{2}}{{B}^{2}}{{r}^{2}}}{2m}$
But frequency $f=\frac{qB}{2\pi m}$
$\therefore$ ${{E}_{k}}=\frac{{{(2\pi mf)}^{2}}{{r}^{2}}}{2m}=2{{\pi }^{2}}m{{f}^{2}}{{r}^{2}}$
Or ${{E}_{k}}=2\times {{(3.14)}^{2}}\times 1.67\times {{10}^{-27}}\times {{(10\times {{10}^{6}})}^{2}}$
$\times {{(0.5)}^{2}}$
$=8.23\times {{10}^{-13}}J$
$\therefore \,\,{{E}_{k}}\,=\frac{8.23\,\times {{10}^{-13}}}{1.6\,\times {{10}^{-19}}\,}\,$
$=5.1\,\times {{10}^{6}}\,eV=5.1\,MeV$