OHM'S LAW

Nowadays high school and university students use voltmeters and ammeters to demonstrate Ohm's law. These devices were not available in Ohm's time! (Moreover, these devices rely on laws (including Ohm's law) not known at the time.) How did Ohm do it? Suggest experiments (using only equipment available in 1826, or similar present-day equipment) showing that:
(a) "voltage" is proportional to "current";
(b) for fixed "voltage", the "current" is proportional to the cross-sectional area of the wire and inversely proportional to its length.

The solution:

As a voltage source we can use batteries. Ohm first used "chemical batteries" but those had very short life-time, i.e their electromotive force (EMF) varied during the experiment. Eventually, he used a thermocouple as a voltage source. The voltage, or more correctly the EMF, was therefore N*E where N is the number of batteries connected in series and E is EMF of a single battery. If the resistance of wire was X, and internal resistance of the battery was r, then the current I was given by

I=N*E/(N*r+X)

Current can be measured by placing a magnetized needle hanging on a string at certain fixed distance from the wire. Since we are not supposed to know exactly the angular dependence of the magnetic force, we will simply for each current twist the string on which the needle hangs until the needle returns in its position before the current began flowing. The angle by which the string was twisted is proportional to the force momentum, and thus measures the strength of the current. (Of course we assume that forces are indeed proportional to the current.)

We will begin the experiment by keeping the same wire (the same X) and changing N. By plotting 1/I versus 1/N, we will establish r/E, and will establish the "voltage-current" relation of the Ohm's law. Now, by lengthening the wire of making its cross section larger we can investigate dependence of resistance on geometry of the wire. (Of course we cannot measure the actual absolute values of the resistance, but all we need is the dependence on length and cross section area...) Ohm published his results on geometry dependence of X in Journal fur Chemie und Physik, 46, p. 160 (1826).

Nice short description of Ohm's work in its historic context can be found in the book History of Physics (Storia della Fisica) by Mario Gliozzi.

Comment: Groshaus suggested to measure the voltage by attaching one of the leads of the battery to electroscope. Can it be done with sufficient accuracy?