Question

An alternating electric field of frequency 'y' is applied across the dees (radius R) of a cyclotron to accelerate protons (mass m). The operating magnetic field 'B' used and K.E. of the proton beam produced by it are respectively (e = charge of proton).

• A. $\frac{2{\pi }^{2}my}{e^2},4{\pi }^{2}m{y}^2{R}^2$
• B. $\frac{2\pi my}{e},2{\pi }^{2}m{y}^2{R}^2$
• C. $\frac{\pi my}{e},{\pi }^{2}m{y}^2{R}^2$
• D. $\frac{2{\pi }^2{m}^2{y}^2}{e},2{\pi }^2{m}^2{y}^2{R}^2$

Answer 1

Answer: (B)

$\frac{2\pi my}{e},2{\pi }^{2}m{y}^2{R}^2$

$T=\frac{1}{y}$

where T is time

speed of proton, $v=\frac{2\pi R}{T}=2\pi Ry$

Now, for circular motion in the dees-

$\frac{m{v}^2}{R}=evB$

$⟹B=\frac{mv}{eR}$

$⟹B=\frac{2\pi my}{e}$

$K=\frac{1}{2}m{v}^2$

$=\frac{1}{2}m(2\pi Ry{)}^2$

$⟹K=2{\pi }^{2}m{y}^2{R}^2$