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A cyclotron oscillator frequency is 10MH...

A cyclotron oscillator frequency is `10MHz`. What should be the operating magnetic field for accelerating `alpha`-particle? If the radius of the dees is `50cm`, what is the kinetic energy in `MeV` of the `alpha`-particle beam produced by the accelerator?
`(e=1*6xx10^(-19)C, m_alpha=4*0028 a.m.u., 1a.m.u.=1*66xx10^(-27)kg)`

Text Solution

Verified by Experts

Here, `v=10MHz=10^7Hz`,
`r_0=50cm=0*50cm, B=?`
`m_alpha=4*0028xx1*66xx10^(-27)kg`
`=6*645xx10^(-27)kg`,
`q=2e=2xx1*6xx10^(-19)=3*2xx10^(-19)C`.
As, `v=(Bq)/(2pim_alpha)` or `B=(2pim_alphav)/(q)`
`=2xx22/7xx(6*645xx10^(-27)xx10^7)/(3*2xx10^(-19))=1*305T`
Maximum kinetic energy is
`K_(max).=(B^2q^2r^2)/(2m_alpha)`
`=((1*305)^2xx(3*2xx10^(-19))xx(0*50)^2)/(2xx(6*645xx10^(-27)))J`
`=((1*305)^2xx(3*2)^2xx10^(-38)xx0*25)/(2xx6*645xx10^(-27)xx1*6xx10^(-13))MeV`
`=20*5MeV`
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