Virtual Cloud Chamber — Particles vs Antiparticles

Concept: A curved track on its own cannot give the sign of a charge; you need the direction of travel first.
Serves: Reading a cloud-chamber photograph the way a lead plate lets you read it.
Inquiry: What photograph would an e⁺ going up and an e⁻ going down each leave, and what does a lead plate add?
Model: r = p/(|q|B) is integrated step by step, and the lead takes energy along the path actually travelled inside it.
Idealisation: e± lose energy in lead mainly by bremsstrahlung, dE/dx ≈ E/X₀ plus ≈1.3 GeV/m of ionisation; the sim uses an effective X₀ of 8 mm (true value 5.6 mm) so that one typical track reproduces the 63 → 23 MeV Anderson measured across 6 mm. Heavier particles use a collision term ∝ 1/β², so a slow proton stops inside the plate. Multiple scattering in the lead is not modelled.

Principle	Where it appears
Predict first, never blocked	The question strip sits above the stage and the scene starts unfired and paused; no control is ever disabled, and answers can be changed.
Misconception targeted structurally	Answering card 1 draws the mirror-image guess as a dashed arc that visibly misses, while the true reversed e⁻ track slides onto the photograph.
Invariant vs varying	r = p/(\|q\|B): B and p vary on sliders while \|q\|, m and the relation stay fixed; old tracks stay on screen for comparison.
Operational quantities	Every claim is a labelled number: p and r before and after the plate, the energy the lead actually took, and a 5 cm scale bar.
Formal layer on demand	The kinematic chain and the stopping-power estimate live in the ⚛︎ The Physics revision sheet.

Virtual Cloud Chamber

Teach me Hide Text

Speed

Cloud chamber — side viewB out of the page

Guided Inquiry

1 · Dirac’s prediction

Let’s go back to 1928. Dirac’s equation for the electron insists on solutions with negative energy — and they can’t be discarded. His reading: they describe a real particle, an electron’s twin with opposite charge. This chamber is where that claim goes on trial.

2 · One track, no arrowheads

Here’s your first photograph, freshly developed: a single curved track at B = 1.50 T, and nothing else. No head, no tail, no clue which end came first. Do you think an e⁺ travelling one way and an e⁻ travelling the other could both have left this exact track?

3 · A stronger field

Now let’s turn up the field. This track was exposed at B = 1.50 T. The particle and its momentum stay the same, and the field is doubled to 3.00 T. What do you think the radius will do?

4 · Heavy versus light

Now let’s swap in something heavier. An e⁺ at 63 MeV leaves an arc of r ≈ 14 cm. A proton is given the same 63 MeV of kinetic energy. What do you think the proton’s arc will look like?

5 · A plate across the chamber

Now let’s add an obstacle. A 6 mm lead plate crosses the chamber (B = 0.70 T), and an e⁺ is about to cross it, losing ≈ 40 MeV inside. Which side of the plate do you think will carry the tighter arc?

6 · Anderson’s photograph, 1932

And now for the real thing. This is the signature Carl Anderson caught in 1932 — one track, a plate crossing, the tighter arc above. What do you think he could settle from a single photograph like this?

Listen

Beam

“? unknown” fires an unlabelled particle: read its momentum and its bending, then press it again to reveal what it was.

Energy & field

Kinetic energy 63 MeV

Field B (out of the page) 1.50 T

Chamber

Lead plate (6 mm)

Track readout

Slide the lead plate in to compare the arc before and after the lead.

Particle: —
p before plate: —
r before plate: —
ΔE in plate: —
p after plate: —
r after plate: —

r = p/(|q|B)

Read the radius off the photograph, know B, and you have the momentum.

E = Ek + mc², pc = √(E² − (mc²)²)

The chain the kinetic-energy slider uses: this is where “p before plate” comes from.

dE/dx (e±) ≈ E/X₀ + 1.3 GeV/m, X₀,eff = 8 mm ⇒ ΔE(63 MeV, 6 mm) ≈ 40 MeV

Idealised: bremsstrahlung ∝ E (true X₀ in lead = 5.6 mm; 8 mm here stands in for a typical rather than mean single-track loss) plus ionisation, integrated along the path actually taken inside the plate. Anderson's own 1932 track lost 63 → 23 MeV. Heavier particles use a collision term ∝ 1/β², so a slow proton stops inside the lead instead of crossing it.

The Physics

The cloud chamber · what this sim shows, the physics behind it, and every control

At a glance

A charged particle crossing a cloud chamber leaves a trail of droplets, and a magnetic field bends that trail into an arc. This sim fires different particles through the field, with or without a lead plate, so you can see what one photograph does and does not tell you.

What you'll learn

How the radius of a track depends on the particle's momentum and on the field.
How kinetic energy, mass and momentum are linked, and what that does to the radius.
What a lead plate does to a track, and what the change tells you.
What one photograph can and cannot say about the sign of a particle's charge.

The apparatus. The sim fires one particle at a time from the middle of the floor.

What's on screen

Track: droplets along the path, in the species' colour (e⁻ blue, e⁺ orange, μ⁻ violet, μ⁺ pink, p yellow, antiproton green). A label gives species, kinetic energy and B.
Faint old tracks: earlier shots stay as dim ghosts until you press Clear old tracks.
⊙ dots and the B chip: the field points out of the page; the chip (top left) shows its strength.
Status chip (top right): no plate · direction unknown, lead plate in or stopped inside the lead.
Track readout (sidebar): momentum and radius before the plate, energy lost in it, then momentum and radius after.

See itFire an e⁺ at 63 MeV in 1.50 T, plate out. Watch for: droplets building up along a curved track, with a label giving the species, energy and field.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Beam chips	Which particle is fired. Picking one only arms the shot: press Fire to launch it. Every species has \|q\| = 1 e.	e⁻ e⁺ μ⁻ μ⁺ p p̄ opens on e⁺
? unknown	Fires a random species at a random energy, both hidden. Press it again to reveal them.	25–160 MeV
▶ Fire / ↻ Play again	Launches the selected particle. When a shot has finished it becomes Play again, which repeats the same shot.	button
Kinetic energy	The particle's kinetic energy. Moving it redraws the track at once and leaves the previous one as a ghost.	10–200 MeV opens at 63 MeV
Field B	The magnetic field strength, pointing out of the page. Redraws like the energy slider.	0.20–3.00 T opens at 1.50 T
Lead plate (6 mm)	Puts a lead plate across the middle of the chamber. With the plate in, the track is drawn as a moving head.	on / off opens off
Clear old tracks	Removes the faint ghosts of earlier shots.	button
Track readout	Opens the list of numbers for the current track: p and r before the plate, energy lost, p and r after.	panel
▶ Play / ⏸ Pause	Pauses or resumes the drawing. When a shot has finished it shows Play and repeats the shot.	top bar
Speed	How fast the track is drawn. The track and its numbers stay the same.	0.25×–4×
↻ Reset	Restores the starting state of the mode you are in. In free exploration that is an e⁺ at 63 MeV in 1.50 T, plate out, old tracks cleared.	top bar

Fixed: chamber 32 cm × 20 cm; lead 6 mm thick, across the middle; every particle has charge ±1 e.

Why the track is an arc

A charged particle ionises the vapour, and droplets condense on the ions, so the track marks its path. The magnetic force F = q v × B is always perpendicular to the velocity. It turns the particle but does no work, so the speed stays constant and the path is a circle.

The radius is set by the momentum p and the field B. Every species here has |q| = 1 e, so nothing else enters.
The sign of q sets the direction of the bend. With B out of the page, an e⁺ fired upward bends right and an e⁻ bends left.

The practical form: 0.2998 is c written in GeV per tesla-metre.

See itFire an e⁻ and an e⁺ from the same spot at 63 MeV in 1.50 T. Watch for: the two arcs curving opposite ways, mirror images of each other.

What sets the radius

At a given B the radius is proportional to p: double p and r doubles. At a given p, double B and r halves. The energy slider sets kinetic energy, so the sim first converts it to momentum:

E_k B beamset with the Beam chips in the sidebar

These sliders move the sim too.

at 63 MeV kinetic energy	p (MeV/c)	β = v/c	r at 1.50 T
e⁺	63.5	1.00	14.1 cm
μ⁺	131	0.78	29.2 cm
p	350	0.35	77.7 cm

The radius follows momentum, not kinetic energy. For an e⁺, E ≈ pc, so doubling Eₖ to 126 MeV doubles r (14.1 → 28.1 cm). For a proton the same step raises p only 1.44 times (r: 77.7 → 111.7 cm).
Slow tracks are fatter. Ionisation per unit length grows as 1/β², and the sim draws it: the proton's track is thicker than the e⁺'s.

See itFire an e⁺ at 1.50 T, then double B to 3.00 T. Watch for: the new arc with half the radius of the first, which stays as a faint ghost.

See itFire an e⁺ at 63 MeV, then a proton at the same energy. Watch for: the proton's fat, nearly straight track beside the faint e⁺ ghost.

One track, two stories

Nothing on a developed photograph says which way the particle was going. With the plate out, the sim draws the track as a finished photograph, with no head and no direction mark.

Reverse both the charge and the direction of travel and the force is unchanged. An e⁺ going up and an e⁻ coming down leave one and the same arc.
The tempting guess, an e⁻ bending the other way from the same start, is the mirror image. That is a different curve.

One arc, read two ways. The mirror guess misses it.

Key idea. A curved track gives the momentum. The sign of the charge also needs the direction of travel.

See itAdd the reversed e⁻ to the e⁺ photograph. Watch for: the blue reversed e⁻ sliding onto the e⁺ track exactly, while the grey dashed mirror guess misses it.

What the lead plate does

Lead only ever takes energy from a particle. So p falls in the plate, and r = p/(|q|B) shrinks: the arc after the plate is the tighter one.

e⁺ and e⁻ lose energy mainly by bremsstrahlung, which grows with their energy. A 63 MeV positron loses most of it: 41 MeV at 0.70 T.
μ, p and antiprotons lose energy by ionisation, which grows as 1/β². The slower they get, the faster they lose it, so in the sim a 63 MeV proton stops inside the plate.
A track that meets the plate at an angle travels more than 6 mm in it: 6.4 mm for the e⁺ at 0.70 T, 8.7 mm at 1.50 T. At 1.50 T the e⁺ meets the plate 43° off the normal and leaves on a 3 cm curl that falls back into the lead, so the See-its here use 0.70 T.

The crossing at B = 0.70 T, from the sim's own integrator.

The sim does not model multiple scattering: the track leaves the plate heading the way it entered, only on a smaller radius.

See itSlide the plate in for an e⁺ at 63 MeV in 0.70 T. Watch for: the arc above the plate tighter than the arc below it, with r after plate smaller than r before.

See itFire a 63 MeV proton at the plate. Watch for: the track ending inside the lead, and the status chip reading “stopped inside the lead”.

How the sign is read

The side with the tighter arc is the side the particle left by. That fixes the direction of travel.
With the direction known, the bend gives the sign. In this sim, B out of the page, an upward track bending right is positive.
In 1932 Carl Anderson photographed a cosmic-ray track crossing 6 mm of lead in a field of about 1.5 T. It had about 63 MeV before the plate and 23 MeV after. The tighter arc was above, so the particle was moving upward, and its bend meant a positive charge.
The track was too thin for a slow, heavy proton. What remained was a positive electron: Dirac's antiparticle, the positron.

See itSend an e⁻ through the same plate at 0.70 T. Watch for: the exit arc again the tighter one, now bending the other way, to the left.

Where it matters

Collider detectors. ATLAS and CMS measure momentum from track curvature in solenoid fields of 2 T and 3.8 T. In 3.8 T a 1 GeV/c particle bends on a radius of 0.88 m. Tracks are taken to come outward from the collision, so the direction is known and the bend gives the sign.
The positron. Anderson's photograph was the first sighting of antimatter. He shared the 1936 Nobel Prize in Physics for it.
Choosing B. A weak field gives a long radius, and a long radius is hard to measure. At 0.20 T a 63 MeV e⁺ has r = 1.06 m, so the track drifts under 2 cm sideways across the 20 cm chamber.

Go deeper: how the sim models the lead

Straight through 6 mm the formula gives 24.5 MeV out; the sim follows the real, longer path.

The e± law uses X₀,eff = 8 mm, not lead's true 5.6 mm, so that one typical track reproduces Anderson's 63 → 23 MeV. At 0.70 T the sim gives 63 → 22 MeV.
The heavy-particle law is a simple 1/β² form. An antiproton is treated as a negative proton: annihilation in the lead is not modelled.
The track is computed exactly. Only the droplet positions are random, and they change nothing but the look.

Summary

In one line. A track's radius measures momentum, r = p/(|q|B). The sign of the charge also needs the direction of travel, which the lead plate supplies.

Key points

At equal kinetic energy a heavier particle has more momentum, so a wider arc.
One arc fits (q, one direction) and (−q, the opposite direction).
Lead removes energy, so the tighter side is the exit side. That gives the direction, and then the bend gives the sign.

Common mix-ups

"The way a track curls tells you the charge." Only if you also know the direction of travel.
"A bigger radius means more energy." The radius follows momentum. At 63 MeV the proton's arc is 5.5 times the e⁺'s.
"The lead bends the track." The lead only removes energy. The field does all the bending, and the arc tightens because p fell.

Things to try on your own

Fire a 63 MeV e⁺ at 0.20 T, then at 3.00 T, and compare the arcs.
With the plate in and Eₖ = 63 MeV, raise B in 0.05 T steps until the e⁺ no longer comes out of the plate (above 0.90 T).
Fire an e⁺ at 200 MeV, then adjust a proton's energy until its arc has the same radius. The two then have the same momentum.