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Question

An alternating electric field, of frequency ν, is applied across the Dees (radius =R) of a cyclotron that is being used to accelerate protons (mass =m, charge =e). 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=2πmνe and K=2mπ2ν2R2
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B
B=mνe and K=m2πνR2
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C
B=mνe and K=2mπ2ν2R2
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D
B=2πmνe and K=m2πνR2
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Solution

The correct option is A B=2πmνe and K=2mπ2ν2R2
Frequency is given as (ν)=qB2πm

Where,
q= charge on particle =e (for proton)
B= operating magnetic field
m= mass of charged particle

ν=eB2πm B=2πmνe ...(1)


Also, R=mvqB

Where,
v= velocity of charged particle
R= radius of circular path taken by charge

v=qBRm ...(2)

Now, kinetic energy, K=12mv2

From equation (2),

K=12m[q2B2R2m2]

K=e2B2R22m=e2R22m[2πmνe]2 [From (1)]

K=2mπ2ν2R2

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