First slide

Force on a charged particle moving in a magnetic field

Question

In a mass spectrometer used for measuring the masses of ions, the ions are initially accelerated by an electric potential V and then made to describe semicircular paths of radius R using a magnetic field B. If V and B are kept constant, the ratio $(\frac{charge on the ion}{mass of the ion})$ will be proportional to

Moderate

A

1/R
B

R
C

R²
D

1/R²

Solution

The radius of the orbit in which ions moving
$R = \frac{mv}{qB}$
The angular frequency of rotation of the ions about the vertical field B is given by
$ω = \frac{v}{R} = \frac{qB}{m} = 2 πv$
where v is frequency.
Energy of ion is given by
$E = \frac{1}{2} {mv}^{2} = \frac{1}{2} m {(Rω)}^{2} = \frac{1}{2} {mR}^{2} B^{2} \frac{q^{2}}{m^{2}}$
or $E = \frac{1}{2} \frac{R^{2} B^{2} q^{2}}{m}$ ... (i)
If ions are accelerated by electric potential V, then energy attained by ions
E = qV ... (ii)
$qV = \frac{1}{2} \frac{R^{2} B^{2} q^{2}}{m}$
or $\frac{q}{m} = \frac{2 V}{R^{2} B^{2}}$
If V and B are kept constant, then $\frac{q}{m} \propto \frac{1}{R^{2}}$

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