Guided exercises · 3 of 6
More Voltage, Same Energy
Goal. The energy of an ion that reaches the collector is set by the magnet and the machine's radius. The dee voltage buys fewer turns, not more energy.
Who it is for. Readers of How a Cyclotron Works; no prior exercise is needed. One formula with a square in it. About fifteen minutes.
Predict first
Double the push on every gap crossing: does the beam arrive at the collector with double the energy? Most readers say yes; an accelerator's energy comes from its voltage, after all. Write your answer down, with the energy you expect at the collector, before you open the laboratory.
An ion of charge q and mass m, moving well below the speed of light, whose orbit at the collector has radius R in the field B found there, carries momentum qBR, and so an energy
E = (qBR)² / 2m.
The laboratory's reference setting shows 0.596 T and a collector radius of 96.5 mm. For a proton:
qBR = 1.602 × 10⁻¹⁹ × 0.596 × 0.0965 = 9.21 × 10⁻²¹ kg·m/s;
E = (9.21 × 10⁻²¹)² / (2 × 1.673 × 10⁻²⁷) = 2.54 × 10⁻¹⁴ J,
about 158 keV.
That is an ideal estimate: it takes the displayed field, a setting at the centre, as the field at the collector. The dee voltage does not appear in it. What the voltage does set is the number of turns. Taking the displayed dee voltage as the full push an ion receives at each of two gap crossings a turn, another ideal figure, an ion at 5 kV gains 10 keV a turn and needs about 16 turns to reach that energy; at 2.5 kV about 32; at 10 kV about 8. Real pushes are smaller and the turns more.
The shape to expect
Open the laboratory
The link opens the laboratory's prototype machine (named on the page's machine selector) at resonance, with the dee voltage at the reference and two preloaded moves: half the reference voltage and double it. The one control in play is dee voltage. Watch two plots, the radial turn profile and the collected energy distribution; in the Full workspace the chamber top view draws the turns to scale.
Open the Virtual Cyclotron Laboratory (hosted; illustrative, not a calibrated replica)
What you should see
On the laboratory's prototype machine, at half voltage the turn profile shows about twice as many turns, packed closer together; at double voltage about half as many, spaced wider. The energy distribution's peak barely moves between the three settings, and the collected current changes little (an observation about this illustrative model; on a real machine the voltage moves transmission and energy spread as well). The ions were pushed twice as often or half as often, and they stopped at the same place with the same energy — because the collector radius and the field decide where the pushing stops.
To see the whole law at once, a second prepared link sweeps the dee voltage from half to double with everything else fixed and plots collected energy against voltage: a flat line. Open the laboratory with the voltage sweep prepared (hosted; illustrative, not a calibrated replica).
If you see something different
The collected energy may sit a little below the worked 158 keV: the displayed field is the setting at the centre, and the field under the outer turns need not equal it, which the last exercise, Weigh the ion, turns into a measurement. At extreme settings, or on the laboratory's resolved-field machine, the beam itself starts to die as the voltage falls: that is the previous exercise's effect returning, more turns and more accumulated phase slip (Watch an ion fall out of step), and the reason this exercise pins the prototype machine. The exercises interlock. Every reading comes from an ensemble of simulated ions and scatters by a few percent from run to run, a figure that belongs to the simulation alone; the laboratory is an illustrative machine shaped like the reference machine, not a calibrated replica of any real machine.
The physics, cited
A cyclotron textbook, on what limits the energy: it "depends only on field and exit radius" (dg-1376). The rule treats the field-radius product as the magnet's design variable and the energy ceiling it sets; the dee voltage sets the turn count and whether the ceiling is reachable; the ceiling it leaves alone.
The same relation, written the way a designer starts: "The maximum kinetic energy T determines magnetic rigidity: B*rho = sqrt(T^2+2T*E0)/(300*Z)" (dg-075), with B in tesla, ρ in metres, and T and the rest energy E₀ in MeV: the relativistic form of the worked expectation above, fixing the field-radius product before any geometry is drawn.
What the voltage buys instead: "Increasing the D voltage requires fewer turns for acceleration to maximum energy and will compensate for a larger phase shift. However, D voltage is usually limited by … power and spark breakdown" (dg-246). A thesis on a small machine says the same: "higher potential on the dees results in fewer orbits and a shorter time of acceleration" (dg-309).
Background: the field is the specification in the magnet-design page.
Closing question
In one sentence: what did you change, what happened, and why? There is no form; the sentence is for you.