Barlow's law

Barlow's law was an incorrect physical law proposed by Peter Barlow in 1825 to describe the ability of wires to conduct electricity.^[1][2] It said that the strength of the effect of electricity passing through a wire varies inversely with the square root of its length and directly with the square root of its cross-sectional area, or, in modern terminology:

$I\propto {\sqrt {\frac {A}{L}}}$

where I is electric current, A is the cross-sectional area of the wire, and L is the length of the wire. Barlow formulated his law in terms of the diameter d of a cylindrical wire. Since A is proportional to the square of d the law becomes $I\propto {\frac {d}{{\sqrt {L}}}}$ for cylindrical wires.^[2]

Barlow undertook his experiments with the aim of determining whether long-distance telegraphy was feasible, and believed he proved that it was not.^[1] The publication of Barlow's law delayed research into telegraphy for several years, until 1831 when Joseph Henry and Philip Ten Eyck constructed a circuit 1,060 feet long, which used a large battery to activate an electromagnet.^[3] Importantly, Barlow did not investigate the dependence of the current strength on electric tension (that is, voltage). He endeavoured to keep this constant, so neglected the possibility of solutions such as a high-intensity battery or step-up voltage converters to allow long-distance telegraphy.

In 1827, Georg Ohm published a different law, stating that the current varies directly with voltage (V) and inversely with the wire's length, not its square root; that is, ${\displaystyle I\propto V\cdot {\frac {A}{L}}}$. Ohm's law is now considered the correct law and Barlow's false.

The law Barlow proposed was not in error due to poor measurement; in fact, it fits Barlow's careful measurements quite well. Heinrich Lenz pointed out that Ohm took into account "all the conducting resistances…of the circuit" whereas Barlow did not.^[4] Ohm explicitly included a term for what we would now call the internal resistance of the battery. Barlow did not have this term and approximated the results with a power law instead. Ohm's law in modern usage is rarely stated with this explicit term but nevertheless an awareness of it is necessary for a full understanding of the current in a circuit.^[5]

References

1. Barlow, Peter (1825). "On the Laws of Electro-Magnetic Action as depending on the Length and Dimensions of the conducting Wire and on the question Whether Electrical Phenomena are due to the transmission of a single or of a compound fluid?". Edinburgh Philosophical Journal. 12: 105–113.
2. Aligny, Henry Ferdinand Quarré; Alfred Huet; F. Geyler; C. Lepainteur (1870). Report on Mining and the Mechanical Preparation of Ores. United States of America Government Printing Office. pp. 9–10. Barlow's law was, 'that the conductibility was inversely proportionate to the square root of the lengths and directly as the diameters of the wires or as the square roots of their sections.'
3. Schiffer, Michael (2008). Power Struggles: Scientific Authority and the Creation of Practical Electricity Before Edison. MIT Press. pp. 43–45. ISBN 0-262-19582-8. Barlow's law had devastating implications for anyone who might have considered building an electromagnetic telegraph. Transmitted over a long distance, the current would be undetectable. Indeed, Barlow reported, 'I found such a sensible diminution with only 200 feet of wire, as at once to convince me of the impracticability of the scheme.'
4. E. Lenz, "On the laws of the conducting powers of wires of different lengths and diameters for electricity", Scientific Memoirs vol. 1, pp. 311-324, 1837, originally read to the Academy of St. Petersburgh, 28 November 1834.
5. Nahum Kipnis, A law of physics in the classroom: the case of Ohm's law", Science & Education, vol. 18, iss. 3-4, pp. 349-382, April 2009.