Question

An alternating electric field of frequency 'v' 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 on proton)

Options

• `(2pi"mv")/"e"`, 2π²mv²R²

• `(2pi^2"mv")/"e"^2`, 4π²mv²R²

• `(pi"mv")/"e"^2`, π²mv²R²

• `(2pi^2"m"^2"v"^2)/"e"`, 2π²mv²R²

Answer 1

`bb((2pi"mv")/"e")`, 2π²mv²R²

Explanation:

Time period, T = 1/V = `(2pi"m")/"eB"`

⇒ B = `(2pi"m")/"e"`v

R = `"mv"/"eB"="P"/"eB"`       [∵ F = `"mv"^2/"R"` = eB]

P = eBR = `"e"xx(2pi"mv"^2)/(2"m")` 

= 2πmvR

K.E = `"P"^2/(2"m")`

= `((2pi"mvR")^2)/(2"m")`

 = 2π²mv²R²