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. 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.

Barlow undertook his experiments with the aim of determining whether long-distance telegraphy was feasible, and believed he proved that it was not. 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. 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.

In 1827, Georg Ohm published a different law, stating that the current varies inversely with the wire's length, not its square root and varies directly with voltage (V); that is, $I \propto \frac{A}{L}\cdot V$. 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 ciruit" whereas Barlow did not. In particular, Ohm explicitly includes a term for what we would now call internal resistance of the battery. Barlow does not have this term and approximates 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.