The cyclotron (see figure) is a device used to accelerate elementary particles s

Question

The cyclotron (see figure) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in circular orbits in the magnetic field B.
The particles are accelerated to higher speeds each time they pass through the gap between the metal "dees," where there is an electric field E. (There is no electric field inside the hollow metal dees.)
The electric field changes direction each half-cycle, owing to an AC voltage V = V0 sin(2 ? f t), so that the particles are increased in speed at each passage through the gap.

If the radius of the cyclotron is 2.530

Explanation / Answer

let the maximum velocity of the proton be v.so,

balancing the magnetic and centrifugal forces,

mv^2/r=q(vXB)

or v=qBr/m

where m=mo/(1-v^2/c^2)^0.5

or v^2=(1.6*10^-19*0.262*0.253)^2*(1-v^2/(3*10^8)^2)/(1.67*10^-27)^2

=6.35*10^6 m/s

so kinetic enegy=0.5mv^2

=0.5*1.67*10^-27*(6.35*10^6)^2/(1-(6.35/300)^2)^0.5

=3.367*10^-14 J

=210437.5 eV

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