Measuring energy without getting fooled

James Prescott Joule, in formulating what is now known as Joule’s Law, found that various forms of energy such as mechanical, electrical and heat are essentially identical and can be changed one into the other. His work formed the theoretical basis for the First Law of Thermodynamics. Joule further investigated the phenomenon of magnetostriction. He discovered how current through a resistance related to heat dissipated, which became Joule’s First Law. One result of all of this was disproving the Caloric Theory, which held that heat consists of a self-repellent fluid called caloric that flows from hotter to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in solids and liquids.

The SI unit of energy bears Joule’s name. The initial impetus for Joule’s scientific research was a plan to replace aging steam engines in the family brewery with recently-invented electric motors. He propounded Joule’s First Law in 1841, stating that “the heat which is evolved by the proper action of any voltaic current is proportional to the square of the intensity of the current, multiplied by the resistance to conduction which it experiences.” This is what engineers are referring to when they talk about the I²R dissipation of heat in a resistive circuit.

Joule eventually considered the amount of work that could be derived from various forms of energy. He had found the principle of convertibility of energy. A paper titled On the Mechanical Equivalent of Heat described an experiment in which Joule used a falling weight to spin a paddle in a well-insulated barrel of water. He measured the rise in temperature of the water and found the mechanical equivalent. It differs a little from our current figure, but Joule is rightfully credited with seeing the absolute correspondences among disparate forms of energy.

Unfortunately, it is easy to get confused about units of energy, torque, power, and other related parameters, whether they involve Joules or not. In mechanics, the concept of force (in some direction) is closely analogous to the concept of torque (about some angle); similarly, mass is analogous to moment of inertia while distance is analogous to angle. The SI unit for torque is the newton-meter (N-m), which algebraically has the same dimensions as the Joule (J). But the two units are not interchangeable.

Though energy units have the name Joule, there’s no special name for units of torque, which explains the compound name derived from the constituent parts. There’s also a distinction between the two units in that energy is a scalar quantity – the dot product of a vector force and a vector displacement. But torque is a vector – the cross product of a distance vector and a force vector. Torque and energy are related to one another by the equation E = τθ where E is energy, τ is (the vector magnitude of) torque, and θ is the angle swept (in radians). Radians are dimensionless, so it follows that torque and energy have the same dimensions.

Another potential source of confusion is in the conversion of energy units. Mechanical engineers may be familiar with the fact that 1 J = 9.4782×10⁻⁴ BTU = 0.7376 ft⋅lb. Electrical engineers may have a harder time remembering that Joules convert into units of watt-hours, not watts (1 J = 2.7778×10⁻⁴ W⋅h).

A Watt is a unit of power. A watt-hour (Wh or W-h) is a unit of energy. Watt indicates the rate of using energy in J/sec. The classic analogy invoked to clarify the relationship compares the rate of using energy to how fast water flows out of a water pipe. A light bulb rated at 100 W consumes 100 J/sec. A 100-W light bulb used for an hour will consume 100 W-h of energy or 0.1 kW-h.

One final conversion worth mentioning is that 1 J = 6.24150974×10¹⁸ eV. Thus units of electron-volts are energy units. They aren’t a special kind of voltage or potential difference. An electron volt is the amount of energy gained (or lost) by the charge of a single electron moving across an electric potential difference of one volt. Voltage is energy per unit charge or V = J/C where C = coulombs.

Note that to complicate matters a bit, high-energy and plasma physicists may use the electron volt in units of momentum or temperature. But it is unusual to see this interpretation outside of these rarified disciplines.