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Mass Spectrometer and Cyclotron

A magnetic force is always sideways to the motion, so a charge in a magnetic field moves in a circle of radius r = mv/qB. A mass spectrometer first speeds ions up with a voltage V, then bends them in a field B; heavier ions make bigger circles (r = (1/B)√(2mV/q)), so masses can be told apart. A cyclotron keeps a particle spinning between two D-shaped boxes and pushes it each time it crosses the gap. The time for each half-turn is the same, so the particle gains energy and moves to bigger circles (f = qB/2πm).

🎬 Step-by-step story

  1. The floor shows a magnetic field pointing down into the page (the crosses). A charge that is not moving feels no magnetic force at all.
  2. Now the charge moves. The magnetic force pushes it sideways, always at right angles to its path. So it turns in a circle. Change the speed or the field and watch the radius change.
  3. Mass spectrometer, step 1. A voltage speeds up an ion, then the ion enters the field through a slit and bends in a half-circle. It lands on a plate at distance d = 2r.
  4. Now two ions with different masses. The heavy ion (orange) makes a bigger half-circle than the light ion (blue), so they land at different spots. The plate reads the mass.
  5. Cyclotron. A particle starts in the middle of two D-shaped boxes. At every gap it gets a push and speeds up, so its circle grows. But each half-turn takes the same time!
  6. Free play. Switch between circle, spectrometer and cyclotron, and move the sliders. Try to predict what will change before you move one.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

If the magnetic force does no work, how can a cyclotron give a particle energy?

The energy comes from the electric field in the gap, not from the magnet. The magnet only keeps the particle in its circular path and brings it back to the gap.

Why does a faster particle not finish its circle sooner?

It has a bigger circle to travel. The extra speed and the extra distance cancel, so the time is the same: T = 2πm/qB.

Why does the heavy ion land farther in a mass spectrometer?

It is harder to turn, because of its larger mass. Its circle is bigger, r ∝ √m, so d = 2r is bigger.

Why is the ion first speeded up with a voltage?

So that every ion of the same charge starts with a known energy qV. Then the only thing that changes the radius is the mass.

Why is there no electric field inside the dees?

The hollow metal boxes shield the inside. The particle can then move in a pure circle with no extra push. The push is only in the gap.

Why can the magnetic field not push a resting charge?

The force is qvB. With v = 0 the force is zero. A magnet needs the charge to move.

A charge moving in a magnetic field

A magnetic field pushes only on a moving charge. The push (force) is always at right angles to the velocity. A sideways push can turn the particle, but it cannot make it faster or slower. So the speed stays the same and the path is a circle.

The magnetic force qvB is the force that points to the centre of the circle. Set it equal to the needed centre-seeking force mv²/r:

qvB = mv²/r, so r = mv/(qB)

Time for one full circle: T = 2πr/v = 2πm/(qB). The speed v cancels out. A faster particle makes a bigger circle, but it needs the same time to go round it.

Mass spectrometer: sorting ions by mass

A mass spectrometer measures the mass of atoms or molecules. It has three jobs.

  1. Make ions. The sample is given an electric charge q.
  2. Speed them up. An ion falls through a voltage V and gains energy qV. So ½mv² = qV and v = √(2qV/m).
  3. Bend them. The ion enters a magnetic field B through a slit and moves in a half-circle. It hits a plate at distance d = 2r.

Put v into r = mv/qB:

r = (1/B) √(2mV/q), so m = qB²r²/(2V)

For the same q, V and B, the radius depends on mass: r ∝ √m. A heavy ion lands farther from the slit. Two isotopes of the same element (same charge, a little different mass) land in two separate marks. The plate tells you the mass, and the size of each mark tells you how much of each isotope there was.

Cyclotron: speeding up particles in small steps

To give a particle a lot of energy with a small voltage, a cyclotron uses the same voltage again and again.

Each push makes the circle bigger, because r ∝ v. The path is a spiral that grows outwards.

Key point: the time for each half-turn is πm/(qB), the same for small and large circles. So one fixed frequency of the voltage works for the whole journey:

f = qB/(2πm) (the cyclotron frequency)

Energy and limits of a cyclotron

The particle leaves at the edge of the dee, where r = R. Then v = qBR/m, and the final kinetic energy is

KE = q²B²R²/(2m)

More energy needs a bigger B or a bigger R. It does not depend on the gap voltage; a smaller voltage only means more turns.

Limits: (1) At speeds near the speed of light the mass grows, so the timing slips out of step. (2) Electrons are too light, so they reach such speeds quickly; cyclotrons are used for protons and heavier ions. (3) A very big magnet is costly.

Try it: a practical

Try it: In the 3D, choose Circle. Predict what happens to r if you double the speed, then check. Predict what happens to the time for one turn. Then choose Spectrometer and set the heavy mass to 4. Predict how far its mark is compared to the light ion (answer: twice as far, because √4 = 2). At home: tie a ball on a string and whirl it. A longer string or a faster ball is like a bigger r. The string is the force that bends the path, just like B.

Key formulas and definitions

Worked examples

1. A proton (m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C) enters a field of 0.50 T at 3.0 × 10⁶ m/s at right angles to the field. Find the radius of its path.

r = mv/(qB) = (1.67 × 10⁻²⁷ × 3.0 × 10⁶)/(1.6 × 10⁻¹⁹ × 0.50) = 5.01 × 10⁻²¹ / 8.0 × 10⁻²⁰ = 0.063 m = 6.3 cm.

2. If the speed of the particle in the example above is doubled, what happens to the radius and to the time for one circle?

r ∝ v, so r doubles to 12.6 cm. T = 2πm/qB does not contain v, so the time for one circle stays the same.

3. A proton moves in a field of 1.0 T. Find the cyclotron frequency.

f = qB/(2πm) = (1.6 × 10⁻¹⁹ × 1.0)/(2π × 1.67 × 10⁻²⁷) = 1.52 × 10⁷ Hz, about 15 MHz.

4. In a mass spectrometer, singly charged neon-20 ions land 20.0 cm from the slit. Where do neon-22 ions land (same V and B)?

d ∝ √m. d₂ = 20.0 × √(22/20) = 20.0 × 1.0488 = 20.98 cm, about 21.0 cm. The two isotopes are about 1 cm apart.

5. A proton and an alpha particle (mass 4 times, charge 2 times) are accelerated through the same V and enter the same B. Compare their radii.

r = (1/B)√(2mV/q), so r ∝ √(m/q). For the alpha: √(4/2) = √2 times the proton's. rα : rp = 1.41 : 1.

6. A cyclotron has dees of radius 0.50 m and a field of 1.5 T. Find the energy of protons that leave it, in MeV.

v = qBR/m = (1.6 × 10⁻¹⁹ × 1.5 × 0.50)/(1.67 × 10⁻²⁷) = 7.2 × 10⁷ m/s. KE = ½mv² = 4.3 × 10⁻¹² J. Divide by 1.6 × 10⁻¹³ J/MeV: about 27 MeV.

Common mistakes

Practice quiz

1. A charge moves at right angles to a magnetic field. Its path is a:
2. The radius of the circle is:
3. In a mass spectrometer, for the same charge, V and B, a heavier ion lands:
4. The frequency of the alternating voltage in a cyclotron is:
5. As a particle spirals outwards in a cyclotron, the time for each half-turn:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is a mass spectrometer in simple words?

It is a machine that weighs atoms. It gives ions a push with a voltage, bends them with a magnet, and reads the mass from where they land.

Why does a cyclotron work with a fixed frequency?

A faster particle moves on a bigger circle, and the two effects cancel. Every half-turn takes the same time πm/qB, so the voltage can flip at one steady rate.

Why can a magnetic field not speed up a particle?

The magnetic force is always at right angles to the motion, so it does no work. It changes the direction of the velocity, not its size.

Where this is taught

China高二Selective 2 Ch.1 Ampère and Lorentz forces

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