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Mathematics for Physics

233 Energy

A measure of a system’s capacity to produce change that is used throughout physics to keep track of what can happen.

Conservation of energy. The total energy of an isolated system stays constant. An isolated system exchanges neither matter nor energy with its surroundings. This principle is used to balance energy for allowed changes.

Conservation of energy is

\[ \Delta E_{\mathrm{sys}} = 0 \]

where

  • \(E_{\mathrm{sys}}\) is the total energy of the isolated system.

Conversion of forms. Energy exists in many forms that convert into one another. Mechanical energy is often split into kinetic energy of motion and potential energy of configuration. This principle is used to follow a conversion such as falling, heating, or radiation.

The First Law. Energy crosses a system boundary as work and as heat. Work is transfer through organized motion. Heat is transfer driven by a temperature difference. This principle is used to write the First Law.

The First Law is

\[ \Delta U = q + w \]

where

  • \(\Delta U\) is the change in internal energy.
  • \(q\) is the heat absorbed by the system.
  • \(w\) is the work done on the system.

Mass-energy equivalence. Rest energy is equivalent to mass. This principle is used to convert a mass defect into a nuclear energy release.

The mass-energy relation is

\[ E_{0} = mc^{2} \]

where

  • \(E_{0}\) is the rest energy.
  • \(m\) is the rest mass.
  • \(c\) is the speed of light.

Energy quantization. On the quantum scale, bound systems have discrete allowed energies. This principle is used to compute spectral lines from level differences.

Field and radiation energy. Energy can reside in fields and can travel through empty space as radiation. This principle is used to account for sunlight and radio waves.

Note: Mechanical energy is often split into motion energy and stored energy. As entropy rises, energy tends to degrade into less useful thermal forms.

233.1 References

  1. Jaffe, R. L., & Taylor, W. The Physics of Energy. Cambridge University Press, 2018. — conservation; forms; quantization; degradation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — transfer by work and heat.
  3. Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books, 2017. — mass-energy; four-momentum.
  4. Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics. — energy in fields.
  5. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — radiation; field energy.