The joule-second (J s, or J∙s) is the mathematical product of an SI Derived Unit, the joule (J), and an SI Base Unit, the second (s). The joule-second describes the amount of action occurring in a physical system through a summation of energy (or heat, or work) over time. In mathematical terms, this summation of energy means that the quantity of energy becomes integrated over time to give a number - an answer to the question.
In SI base units the joule-second becomes kilogram-meters squared-per second or kg∙m2∙ s-1. Dimensional Analysis of the joule-second yields M L2 T-1. Note the denominator of seconds (s) in the base units.
Source of Confusion
The joule-second should not be confused with the physical process of joules per second (J/s). In legitimate physical processes, when the unit of time appears in the denominator of a ratio, the described process occurs at a rate. For example, in discussions about speed, an object like a car travels a known distance of kilometers spread over a known number of seconds, and the car’s rate of speed becomes kilometers per second (km/s). In physics, work per time describes a system’s power; defined by the units of Watts (J/s), or joules per second. A similar analogy does not apply to the expression kg∙m2∙ s-1 because the units kg∙m2 in the numerator have no physical meaning like the numerators of distance and energy shown in these examples.
The joule-second also appears as the unit of measure in classical mechanics for the angular momentum of a rotating object and in quantum mechanics within the definition of Planck's constant. Angular momentum is the product of an object’s moment of inertia, in units of kg∙m2 and its rotational velocity in units of m∙s-1∙m-1, or simply s-1 (i.e., Hz). This product of moment of inertia and rotational velocity yields kg∙m2∙ s-1 or the joule-second. Planck's constant represents the energy of a wave, in units of joule, divided by the frequency of that wave, in units of s-1. This quotient of energy and frequency also yields the joule-second (J∙s).
- BIPM. Le Système international d’unités / The International System of Units (‘The SI Brochure’). Bureau international des poids et mesures, eighth edition, 2006, updated 2014. URL http://www.bipm.org/en/si/si_brochure/, ISBN 92-822-2213-6.
- Schlamminger, S.; Haddad, D.; Seifert, F.; Chao, L. S.; Newell, D. B.; Liu, R.; Steiner, R. L.; Pratt, J. R. (2014). "Determination of the Planck constant using a watt balance with a superconducting magnet system at the National Institute of Standards and Technology." Metrologia. 51 (2): S15. arXiv:1401.8160 . Bibcode:2014Metro..51S..15S. doi:10.1088/0026-1394/51/2/S15. ISSN 0026-1394.