A mass spectrometer accelerates doubly ionized atoms of charge 2e over a potential difference V before they enter a uniform magnetic field B which is perpendicular to the direction of motion of the ions. If d is the radius of the ions' path in the magnetic field, what is the mass M of one ion? Express your answer ONLY in terms of V, B, e and d. The potential V is low enough that no relativistic corrections are needed

(B² x e x d²) / V = M

Explanation:

Under electric field by potential gradient if K is the kinetic energy imparted to

the charged particle

K = 1/2 M u² = V x 2e

When charged particle enters the magnetic field with velocity u, it travels on circular path with radius d. So

centripetal force = magnetic force

M u² / d = B x 2e x u

u = (B x 2e x d) / M

Putting this value of u in the relation above

m (B² x 4e² x d²) / 2M² = V x 2e

(B² x 4e² x d²) / V x 2e = 2 M

(B² x e x d²) / V = M