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A cyclotron's frequency is 8MH z. What s...

A cyclotron's frequency is `8MH z`. What should be the operating magnetic field for accelerating protons ? If the radius of its dees is 50 cm, calculate the k.E ( in ` M`e V) of the proton beam produced by the accelerator .

Text Solution

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The cyclotron's frequency `v = 18 MHz`
` = 8 xx 10^(6) ` Hz
The mass of the proton an ` = 1.67 xx 10^(-27) `kg.
The charge of the proton ` q = 1.6 xx 1010^(-19) ` C
Radius of the dees ` r = 50 cm = 50 xx 10^(-2) ` m
Magnetic field B = ?
k.E of the proton k.E = ?
Magnetic field ` B = (2 pi m v)/q `
`B = (2 xx 3.14 xx 1.67 xx 10^(-27) xx 8 xx 10^(6))/(1.6 xx 10^(-19))`
` = (83.90 xx 10^(-21))/(1.6 xx 10^(-19))`
` = 52.438 xx 10^(-2)`
` B = 0.524 T`
k.E ` = 1/2 mv^(2)`
`v = r omega = r xx 2 pi v`
` = 0.5 xx 2 xx 3.14 xx 8 xx 10^(6)`
` v = 25.12 xx 10^(6)` m/s
k.E ` = 1/2 xx 1.67 xx 10^(-27) xx (25.12 xx 10^(6))^(2)`
` = (41.95 xx 10^(-21) xx 25 .12 xx 10^(6))/2 `
` = 526.89 xx 10^(-15) " J (or) " 5.269 xx 10^(-13) J`
To convert into MeV
`= (526.89 xx 10^(-15))/(1.6 xx 10^(-19)) = 3.29 xx 10^(6) eV`
k.E `= 3.29 xx 10^(6) ` MeV
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