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Arc lamps, two carbon rods with a bright electric discharge between them, once lit many streets and halls. They often hissed and flickered. Hertha Ayrton set out to explain why, and ended up writing a law for how an arc behaves. This post covers what she found, a calculation with her own equation, and what you can borrow from her method.
Who was Hertha Ayrton?
The American Institute of Physics' Physics Today note on Ayrton records that she was born in Portsea, England, on 28 April 1854, studied mathematics at Girton College, Cambridge, and passed her examinations in 1881, though, like all women at Cambridge then, she did not receive a degree. She married the electrical engineer William Ayrton, who studied electric arcs. Her entry in Who Was Who gives her death as 26 August 1923 and describes her as the only woman member of the Institution of Electrical Engineers.
Why did the electric arc hiss?
In an arc lamp the positive carbon develops a hollow, white-hot "crater". At low currents the arc burns quietly. Turn the current up and, at some point, the arc suddenly starts to hiss, and the voltage across it suddenly drops.
Ayrton tested one explanation after another and wrote up her experiments in The Electric Arc (1902). Her key tests, from the summary of the chapter on hissing:
When the arc was enclosed so that little or no air reached it, the sudden drop in voltage did not happen, even at three times the current that made the open arc hiss.
Puffs of air sent at the crater produced the drop even at small currents. Oxygen gave the same drop as air.
Nitrogen or carbon dioxide sent at the crater gave no drop.
Gas blown through the negative carbon, so that it could not reach the crater directly, gave no drop.
Her conclusion, in her words: the drop "is due to the oxygen in the air getting directly at the crater and combining with the carbon at its surface." She also explained the sound itself. Burning makes a small cloud of gas that briefly shields the crater; the gas clears, air rushes back, and the cycle repeats. These rushes and stops set up an irregular vibration, and the current oscillates with it.
She first presented this work in her paper The hissing of the electric arc in the Journal of the Institution of Electrical Engineers in 1899. Physics Today notes that she became the institution's first female member after presenting it.
What was the Ayrton equation for the arc?
Ayrton wanted a single rule for the voltage V (in volts) needed to keep a quiet arc of length l (in millimetres) carrying a current A (in amperes). In chapter 6 of her book she fitted her measurements on solid carbons with
V = 38.88 + 2.074 l + (11.66 + 10.54 l) / A
She checked it against 81 points from her curves: 58 calculated voltages were within 0.05 V of the measured ones. The constants belong to her carbons; the shape of the law is the lesson.
Worked example: what happens when the current rises?
Take an arc 2 mm long.
Fixed part: 38.88 + 2.074 x 2 = 43.03 V.
Current-dependent part: (11.66 + 10.54 x 2) / A = 32.74 / A.
Now try three currents:
A = 4: V = 43.03 + 8.19 = 51.2 V. Power = 4 x 51.2 = about 205 W.
A = 8: V = 43.03 + 4.09 = 47.1 V. Power = about 377 W.
A = 12: V = 43.03 + 2.73 = 45.8 V. Power = about 549 W.
The voltage falls as the current rises. Compare a resistor obeying Ohm's law, where doubling the current doubles the voltage. The ratio V/A drops from 12.8 to 5.9 ohms between 4 and 8 amperes, so the arc has no fixed resistance at all. NCERT's Class 12 chapter on Current Electricity lists exactly this kind of behaviour, "V ceases to be proportional to I", among the limitations of Ohm's law.
This falling curve is why arc lamps are run with a resistor in series. If the current creeps up, the arc voltage drops, which would let even more current flow; the series resistor takes a larger share of the supply voltage as the current rises and holds it steady.
Common mistakes with arcs and non-ohmic devices
Assuming V/I is constant. For an arc, V/I depends on the current. Always read the value off the actual curve or equation at that current.
Forgetting units in fitted formulas. In Ayrton's equation l is in millimetres and A in amperes. Put l in metres and the answer is nonsense.
Mixing up V/I with the slope. At 8 A, dV/dA = -32.74 / 64 = -0.51 V per ampere: negative, while V/I is positive.
Our post on Ohm's law limitations and non-ohmic devices has more examples for board and JEE revision. If a V-I graph is hard to picture, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read.
What else did she work on?
Her arc papers went to the Royal Society too, including The mechanism of the electric arc in its Philosophical Transactions. Physics Today notes that she was the first woman nominated as a Fellow of the Royal Society, that the Society declined to admit a married woman, and that it awarded her the Hughes Medal in 1906.
What can a student take from how Hertha Ayrton worked?
Look for the pattern, not just the facts. Her preface says her aim was not to add more isolated facts about the arc but to see how they bore on one another.
Design tests that can rule ideas out. Each gas she blew at the crater was a test that could have proved her wrong.
Fit an equation, then check it. She did not trust her formula until she had compared it with 81 measured points.
Practice routine: take a non-ohmic graph from your textbook, read V at three currents, and compute V/I and the slope at each.
Next step: redo the worked example for a 4 mm arc at 5 A and 10 A, and check that the voltage falls as the current rises.
