Energy is a quantity that is conserved as a consequence of the time translation invariance of the laws of physics. Below are some formulas calculating energy of different forms.

Kinetic energy is that associated with motion and is defined as for a particle with mass , velocity and momentum . If the mass is a fluid in motion (e.g. wind) with density and volume through cross-sectional area , then .

Work is the result of a force applied over a displacement and is given by the line integral

This has the simple form when force is constant and displacement is linear where is the angle between the force and displacement vectors.

Using Newton’s 2nd law and the relation this can be written as

This is the work-energy theorem which says that work is the change in kinetic energy by a net force. It can also be written as where is momentum.

The above has the rotational analogue where is moment of inertia and is angular velocity and the equation for work becomes

,

where is a torque vector.

This has the simple form in the special case of a constant magnitude tangential force where is the torque resulting from force applied at distance from the centre of rotation.

Note that the time derivative of work is defined as power, so work can also be expressed as the time integral of power:

If the work done by a force field depends only on a particle’s end points and not on its trajectory (i.e. conservative forces), one may define a potential function of position, known as potential energy satisfying . By convention positive work is a reduction in potential, hence the minus sign. It then follows that in such force fields the sum of kinetic and potential energy is conserved.

Some types of potential energy:

- due to a gravitational field: , where are the masses of two bodies, the distance between their centre of masses and is Newton’s gravitation constant.
- due to earth’s gravity at the surface: where and is the object’s height above ground (small compared with the size of the earth).
- due to a spring obeying Hooke’s law: where is the spring constant and the displacement from an equilibrium position.
- due to an electrostatic field: where is Coulomb’s constant and are charges. This can be written as where is a potential function measured in volts.
- for a system of point charges: .
- for a system of conductors: where the charge on conductor is and its potential is .
- for a charged dielectric: the above may be generalised to the volume integral where is charge density and is the potential corresponding to the electric field.
- for an electric dipole in an electric field: where is directed from the negative to positive charge and has magnitude equal to the product of the positive charge and charge separation distance.
- for a current loop in a magnetic field: where is directed normal to the loop and has magnitude equal to the product of the current through the loop and its area.

In electric circuits the voltage drop across an inductance is and the current though a capacitance is . These inserted into the relationship lead to the formulas and for the energy stored in a capacitor and inductor respectively.

Also in electromagnetism the energy flux (flow per unit area per unit time) is the Poynting vector , the cross product of the electric and magnetising field vectors. The electromagnetic energy in a volume is given by ([1])

,

where is the electric displacement field and is the magnetic field. This is more commonly written as when the relationships hold.

In special relativity energy is the time component of the momentum 4-vector. That is, energy and momentum are mixed in a similar way to how space and time are mixed at high velocities. Computing the norm of the momentum four-vector gives the energy-momentum relation

.

This leads to for massless particles (such as photons) and more generally , the mass-energy equivalence relation (here and is rest mass).

In quantum mechanics the energy of a photon is also written as (Planck-Einstein relation) where is Planck’s constant and are frequency and wavelength respectively. Energies of quantum systems are based on the eigenstates of the Hamiltonian operator, an example of which is .

Force is also equal to pressure times area, so another formula for work (e.g. done by an expanding gas) is the volume integral . In thermodynamics heat is energy transferred through the random motion of particles. The fundamental equation of thermodynamics quantifies the internal energy which disregards kinetic or potential energy of a system as a whole (only considering microscopic kinetic and potential energy):

where is temperature, is entropy, is the number of particles and the chemical potential of species . Similar formulas exist for other thermodynamic potentials such as Gibbs energy, enthalpy and Helmholtz energy.

The mean translational kinetic energy of a bulk substance is related to its temperature by where is Boltzmann’s constant.

In thermal transfer the change in internal energy is given by where is mass and is the heat capacity which may apply to constant volume or constant pressure.

The power per unit area emitted by a body is given by the Stefan-Boltzmann law where is the emissivity (=1 for black body radiation) and is the Stefan–Boltzmann constant. This equation may be used to determine the energy emitted by stars using their emission spectrum.

The latent heat (thermal energy change during a phase transition) of mass of a substance with specific latent heat constant is given by .

Finally, the energy of a single wavelength of a mechanical wave is where the mass of a wavelength, the amplitude and the angular frequency [2]. This can be applied to finding the energy density of ocean waves for example [3].

#### References

[1] Poynting Vector. *Brilliant.org*. Retrieved 22:24, December 28, 2018, from https://brilliant.org/wiki/poynting-vector/

[2] Power of a Wave. *Brilliant.org*. Retrieved 21:23, December 30, 2018, from https://brilliant.org/wiki/power-of-a-wave/

[3] Wikipedia contributors, “Wave power,” *Wikipedia, The Free Encyclopedia,* https://en.wikipedia.org/w/index.php?title=Wave_power&oldid=875183814 (accessed December 30, 2018).

[4] Wikipedia contributors, “Work (physics),” *Wikipedia, The Free Encyclopedia,* https://en.wikipedia.org/w/index.php?title=Work_(physics)&oldid=874162163 (accessed December 30, 2018).

[5] Wikipedia contributors, “Potential energy,” *Wikipedia, The Free Encyclopedia,* https://en.wikipedia.org/w/index.php?title=Potential_energy&oldid=873393028 (accessed December 30, 2018).

[6] Wikipedia contributors, “Electric potential energy,” *Wikipedia, The Free Encyclopedia,* https://en.wikipedia.org/w/index.php?title=Electric_potential_energy&oldid=868852409 (accessed December 30, 2018).

[7] Wikipedia contributors, “Thermodynamic equations,” *Wikipedia, The Free Encyclopedia,* https://en.wikipedia.org/w/index.php?title=Thermodynamic_equations&oldid=865388931 (accessed December 30, 2018).

[8] H. Ohanian, *Physics*, 2nd edition, Norton & Company, 1989.

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