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December 30, 2018

A collection of energy formulas

Filed under: science — ckrao @ 10:58 am

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 K = \frac{1}{2} mv^2 = \frac{p^2}{2m} for a particle with mass m, velocity v and momentum p. If the mass is a fluid in motion (e.g. wind) with density \rho and volume A v t through cross-sectional area A, then K = \frac{1}{2} At\rho v^3.

Work is the result of a force F applied over a displacement \mathbf{s} and is given by the line integral

\displaystyle W = \int_C \mathbf{F} . \mathrm{d}\mathbf{s} = \int_{t_1}^{t_2} \mathbf{F} . \frac{\mathrm{d}\mathbf{s}}{\mathrm{d}t} \ \mathrm{d}t= \int_{t_1}^{t_2} \mathbf{F}.\mathbf{v}\ \mathrm{d}t .

This has the simple form W = Fs \cos \theta when force is constant and displacement is linear where \theta is the angle between the force and displacement vectors.

Using Newton’s 2nd law and the relation \frac{\mathrm{d}}{\mathrm{d}t} (\mathbf{v}^2) = 2\mathbf{v}.\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} this can be written as

\displaystyle W = m\int_{t_1}^{t_2} \frac{d\mathbf{v}}{dt} . \mathbf{v} \mathrm{d}t = \frac{1}{2}m\int_{t_1}^{t_2} \frac{\mathrm{d}}{\mathrm{d}t} (\mathbf{v}^2) \mathrm{d}t = \frac{1}{2}m\int_{v_1^2}^{v_2^2} \mathrm{d}(\mathbf{v}^2) = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2.

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 W = \int_{v_1}^{v_2} m \mathbf{v}.\mathrm{d}\mathbf{v} = \int_{p_1}^{p_2} \mathbf{v}.d\mathbf{p} where \mathbf{p} = m\mathbf{v} is momentum.

The above has the rotational analogue K = \frac{1}{2} I \omega^2 where I is moment of inertia and \omega is angular velocity and the equation for work becomes

\displaystyle W = \int_{t_1}^{t_2} \mathbf{T} . \mathbf{\omega}\ \mathrm{d}t,

where \mathbf{T} is a torque vector.

This has the simple form W = Fr \omega = \tau \omega in the special case of a constant magnitude tangential force where \tau = Fr is the torque resulting from force F applied at distance r 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:

W = \int P(t)\  \mathrm{d}t = \int_{t_1}^{t_2} \mathbf{F}.\mathbf{v}\ \mathrm{d}t.

If the work done by a force field \mathbf{F} 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 U satisfying \mathbf{F} = -\nabla U. 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: \mathbf{F} = -(GMm/r^2) \hat{r}, U = -GMm/r, where M, m are the masses of two bodies, r the distance between their centre of masses and G is Newton’s gravitation constant.
  • due to earth’s gravity at the surface: \mathbf{F} = -mg, U = mgh where g \approx 9.8 ms^{-2} and h is the object’s height above ground (small compared with the size of the earth).
  • due to a spring obeying Hooke’s law: \mathbf{F} = -kx, U = kx^2/2 where k is the spring constant and x the displacement from an equilibrium position.
  • due to an electrostatic field: \mathbf{F} = q\mathbf{E} = (k_e qQ/r^2) \hat{r}, U = k_e qQ/r where k_e is Coulomb’s constant 1/(4\pi \epsilon_0) and q, Q are charges. This can be written as U = qV where V is a potential function measured in volts.
  • for a system of point charges: \displaystyle U =k_e \sum_{1 \leq i < j \leq n} \frac{q_i q_j}{r_{ij}}.
  • for a system of conductors: U = \frac{1}{2} \sum_{i=1}^n Q_i V_i where the charge on conductor i is Q_i and its potential is V_i.
  • for a charged dielectric: the above may be generalised to the volume integral U = \frac{1}{2} \int_V \rho \Phi \ \mathrm{d}v where \rho is charge density and \Phi is the potential corresponding to the electric field.
  • for an electric dipole in an electric field: U = -\mathbf{p}.\mathbf{E} where \mathbf{p} 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: U = -\mathbf{\mu}.\mathbf{B} where \mathbf{\mu} 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 L is v = L di/dt and the current though a capacitance C is i = C dv/dt. These inserted into the relationship E = \int i(t)v(t) \ \mathrm{d}t lead to the formulas E = \frac{1}{2}L(\Delta I)^2 and E = \frac{1}{2}C(\Delta V)^2 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 \mathbf{S} = \mathbf{E} \times \mathbf{H}, the cross product of the electric and magnetising field vectors. The electromagnetic energy in a volume V is given by ([1])

\displaystyle \frac{1}{2}\int_V \mathbf{B}.\mathbf{H} + \mathbf{E}.\mathbf{D}  \ \mathrm{d}v,

where \mathbf{D} is the electric displacement field and \mathbf{B} is the magnetic field. This is more commonly written as \displaystyle \frac{1}{2} \int_V \epsilon_0 |E|^2 + |B^2|/\mu_0 \ \mathrm{d}v when the relationships \mathbf{D} = \epsilon_0\mathbf{E}, \mathbf{B} = \mu_0 \mathbf{H} 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

E^2 = (pc)^2 + (m_0c^2)^2.

This leads to E = pc for massless particles (such as photons) and more generally E = \gamma m_0 c^2 , the mass-energy equivalence relation (here \gamma = (1 - (v/c)^2)^{-1/2} and m_0 is rest mass).

In quantum mechanics the energy of a photon is also written as E = hf = hc/\lambda (Planck-Einstein relation) where h is Planck’s constant and f, \lambda are frequency and wavelength respectively. Energies of quantum systems are based on the eigenstates of the Hamiltonian operator, an example of which is \displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla^2+V(x).

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

\displaystyle U = \int  \left(T \text{d}S - p \mathrm{d}V + \sum_i \mu_i \mathrm{d}N_i \right)

where T is temperature, S is entropy, N_i is the number of particles and \mu_i the chemical potential of species i. 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 \bar{E} = \frac{3}{2}k_B T where k_B is Boltzmann’s constant.

In thermal transfer the change in internal energy is given by \Delta U = m C \Delta T where m is mass and C 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 P = A \epsilon \sigma T^4 where \epsilon is the emissivity (=1 for black body radiation) and \sigma 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 m of a substance with specific latent heat constant L is given by Q = mL.

Finally, the energy of a single wavelength of a mechanical wave is \displaystyle \frac{1}{2} m\omega^2 A^2 where m the mass of a wavelength, A the amplitude and \omega 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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