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

Using the relationship for equilibrium of forces:

$F_E = F_M$

Where:

• Electric force, $F_E = qE$
• Magnetic force, $F_M = qvB$

Setting them equal gives: $qE = qvB$

Solving for E, we have:

$E = vB$

Substituting the known values:

• $v = 1.5 × 10^5$ m s$^{-1}$
• $B = 0.12$ T

Thus, $E = (1.5 × 10^5) × (0.12) = 1.8 × 10^4 \, \text{V/m}$

In a uniform magnetic field, the centripetal force needed for circular motion is provided by the magnetic force:

$F_c = F_M$

Where:

• Centripetal force, $F_c = \frac{mv^2}{r}$
• Magnetic force, $F_M = qvB$

Setting these equal gives: $\frac{mv^2}{r} = qvB$

Rearranging for r:

$r = \frac{mv}{qB}$

This shows that the radius r is directly proportional to mass m and inversely proportional to charge q and magnetic field B.

Using the formula for radius of the path:

$r = \frac{mv}{qB}$

Substituting known values:

• $v = 1.5 \times 10^5$ m s$^{-1}$
• $B = 0.12$ T
• Mass of $^{6}Li^+$ ion = $1.0 × 10^{-26}$ kg
• Charge of $^{6}Li^+$ ion, $q = 1.6 × 10^{-19}$ C

Calculating r:

$r = \frac{(1.0 \times 10^{-26}) (1.5 \times 10^5)}{(1.6 \times 10^{-19})(0.12)}$

After calculations, find the distance from small aperture to detector surface.

Using energy conservation, the work done by the electric field equals the gain in kinetic energy:

$qEd = \frac{1}{2} mv^2$

Where:

• $d$ = distance between electrodes = 3000 V
• Charge of ion, $q = 1.2 × 10^{-19}$ C
• Mass of ion = $1.2 × 10^{-26}$ kg

To find v, solve for speed:

$v = \sqrt{\frac{2qEd}{m}}$

Substituting the values:

• $E = 6000 \text{V}$
• $d = 3000 \text{V}$

Thus, v is determined after performing calculations.

The time taken for ions to travel from aperture P to the detector relates closely to their mass and charge due to their varying accelerations. The ions of different masses will have different speeds and consequently take different times.

The relationship can be described by:

$t = \frac{l}{v}$

Where t is the time interval, l is distance traveled, and v is speed derived from mass and charge. Therefore, by measuring these time intervals, we can deduce the mass of the ions as the time intervals will differ according to the equation relating mass, charge, and speed.