Mass spectrometers are used to measure the masses of ions - AQA - A-Level Physics - Question 3 - 2020 - Paper 2

Using the equation for the balance of forces, where the electric force equals the magnetic force:

$F_E = F_M$

This gives:

$qE = qvB$

Simplifying, we can find $E$:

$E = vB$

Substituting the known values:

$E = (1.5 imes 10^5 ext{ ms}^{-1})(0.12 ext{ T}) = 1.8 imes 10^4 ext{ V m}^{-1}$.

The centripetal force required to keep an ion moving in a circular path in the magnetic field is provided by the magnetic force:

1. We set the centripetal force equal to the magnetic force:

$\frac{mv^2}{r} = Bqv$

2. Re-arranging this equation gives:

$r = \frac{mv}{Bq}$.

Using the formula we derived for the radius of the path:

1. Substitute in the known values. The mass of the ion $m = 1.0 imes 10^{-26} ext{ kg}$, $v = 1.5 imes 10^5 ext{ ms}^{-1}$, $B = 0.12 ext{ T}$, and $q = e = 1.6 imes 10^{-19} ext{ C}$ (charge of Lithium ion).

Calculate $r$:

$r = \frac{mv}{Bq} = \frac{(1.0 imes 10^{-26} ext{ kg})(1.5 imes 10^5 ext{ ms}^{-1})}{(0.12 ext{ T})(1.6 imes 10^{-19} ext{ C})}$

Calculating yields: $r = 0.0781 ext{ m} = 0.0781 ext{ m} = 7.81 ext{ cm}$.

Using the principle of energy conservation, the kinetic energy (KE) gained by the ion is equal to the work done on it by the electric field:

$KE = qV$

Where:

• $V = 6000 ext{ V}$ is the potential difference.

Using kinetic energy: $\frac{1}{2} mv^2 = qV$

From which we derive: $v = \sqrt{\frac{2qV}{m}}$

Substituting in the values of charge ($q = 1.2 imes 10^{-19} ext{ C}$) and mass ($m = 1.2 imes 10^{-26} ext{ kg}$) gives us the speed of the ion emerging through aperture P.

The time taken for each ion to travel from aperture P to the detector can be related to their respective speeds, which are dependent on their masses:

1. Ions with smaller masses will accelerate more and thus travel faster, leading to shorter time intervals to the detector.
2. Conversely, heavier ions will have greater inertia, resulting in longer travel times.
3. By measuring these time intervals and knowing the acceleration due to fields, the relative masses of the ions can be inferred based on the differences in their speeds.