Question

Show that for a given positive ion species in a cyclotron, (i) the radius of their circular path inside a dee is directly proportional to their speed, and (ii) the maximum ion energy achievable is directly proportional to the square of the magnetic induction.

Answer 1

Suppose injecting positive ions with mass m and charge q into a cyclotron. Only the homogeneous magnetic field acts on the ions in the area of a dee that is free of an electric field. As a result, in a plane normal to the field, the ions inside a dee move in a semicircular manner at a constant speed of v. The centripetal force is given by the magnetic force of magnitude qvB if B is the magnetic field's induction.

∴ `(mv^2)/r` = qvB

∴ r = `(mv)/(qB)`   ... (1)

Thus, for given q, m and B,

r ∝ v

Before the ions are deflected out of the accelerator, if R is the maximum radius of the path which is also the radius of the dee

`v_(max) = (qBR)/m`   ... (2)

so that KE_max = `(q^2B^2R^2)/(2m)` (in joule) = `(q^2B^2R^2)/(2em)` (in eV)   ... (3)

Thus, for a given ion species and dees of a given radius,

KE_max ∝ B²