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 $\left(\frac{\text{Charge on the ion}}{\text{Mass of the ion}}\right)$ will be proportional to

• A. $\frac{1}{R}$
• B. $\frac{1}{R^2}$
• C. $R^2$
• D. R

Answer 1

The correct option is B $\frac{1}{R^2}$

Let $q$ be the charge and $m$ be the mass of the particle. Energy is increased only by the electric field doing work.

$\Rightarrow qV=\frac{1}{2}m{v}^2\Rightarrow v=\sqrt{\frac{q}{m}}\sqrt{2V}$

The magnetic field provides centripetal force for motion in circle:

$\Rightarrow \frac{m{v}^2}{R}=qvB$

$\Rightarrow v=\frac{qBR}{m}\Rightarrow \frac{1}{BR}\sqrt{2V}=\sqrt{\frac{q}{m}}$

$\Rightarrow \frac{q}{m}=\frac{2V}{B^2{R}^2}$

$\Rightarrow \frac{q}{m}\propto \frac{1}{R^2}$