Energy (Physics)

physics

Definition

Energy (Physics)

Energy is the conserved scalar quantity associated with the capacity of a physical system to do work. It is neither created nor destroyed; it is only transferred or converted between forms.

Formally, for a force $\mathbf{F}$ applied along a path $\gamma$, the work done on the system is

$$W={\int }_{\gamma }\mathbf{F}\cdot d\mathbf{x},$$

and the change in the system’s energy equals this work:

$$\Delta E=W.$$

This yields the conservation law. For an isolated system with no exchange of work or heat with its surroundings, $\Delta E=0$, so $E$ is a constant of motion.

Joule

Definition

Joule

The joule ($\mathrm{J}$) is the SI unit of energy. It is defined as the work done when a force of one newton acts over a displacement of one metre,

$$1\mathrm{J}=1\mathrm{N}\cdot \mathrm{m}.$$

In SI base units,

$$1\mathrm{J}=1\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-2}.$$

It is also the unit of work and of heat, consistent with the realisation that these are all forms of energy.

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Forms

Kinetic Eenrgy

Definition

Kinetic Energy

Kinetic energy is the form of energy a body possesses by virtue of its motion. For a point particle of mass $m$ moving at speed $v$, it is

$$K=\frac{1}{2}m{v}^2.$$

More generally, for a body with momentum $\mathbf{p}$, $K={\mathbf{p}}^2/2m$. Kinetic energy is always non-negative and is zero only when the body is at rest.

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Potential Energy

Definition

Potential Energy

Potential energy is the form of energy stored in the configuration of a physical system. It is a function $U(\mathbf{x})$ of the system’s position in configuration space, such that the force derived from it is

$$\mathbf{F}=-\nabla U.$$

The minus sign encodes that the force points towards decreasing potential: the system accelerates in the direction that lowers its stored energy. Because only differences $\Delta U$ produce observable forces, $U$ is defined up to an additive constant.

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Thermal Energy

Definition

Thermal Energy

Thermal energy is the energy contained in the disordered microscopic motion of the particles of a physical system. It is the internal energy associated with random translations, rotations, and vibrations of atoms and molecules, as opposed to the ordered motion of the whole body.

Formally, for a system of $N$ particles with total kinetic energy $K_{\text{tot}}$, the thermal energy is the kinetic energy measured in the centre-of-mass frame,

$$E_{\text{thermal}}=\sum _{i=1}^N\frac{1}{2}{m}_i⟨{\mathbf{v}}_i^{\prime }\cdot {\mathbf{v}}_i^{\prime }⟩,$$

where ${\mathbf{v}}_i^{\prime }={\mathbf{v}}_i-{\mathbf{v}}_{\text{cm}}$ is the velocity relative to the centre of mass. Flow of thermal energy between systems is heat.

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Chemical Energy

Definition

Chemical Energy

Chemical energy is the form of energy stored in the bonds of a molecule, released or absorbed when the bonds are rearranged in a chemical reaction. It is the potential energy of the electronic configuration: a function $U(\mathbf{x})$ of the positions of the constituent atoms and their electrons.

Formally, for a reaction converting reactants $R$ into products $P$,

$$R⟶P,\Delta U=U(P)-U(R).$$

The free-energy change $\Delta G=\Delta U-T\Delta S$ determines the direction: exergonic reactions ($\Delta G<0$) release chemical energy, endergonic reactions ($\Delta G>0$) require an input of energy.

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Electromagnetic Energy

Definition

Electromagnetic Energy

Electromagnetic energy is the form of energy carried by electric and magnetic fields. For a field configuration with electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$, the energy density is

$$u=\frac{1}{2}{\epsilon }_0{\mathbf{E}}^2+\frac{1}{2{\mu }_0}{\mathbf{B}}^2,$$

so the total energy stored in a region $V$ is $E_{\text{EM}}={\int }_{V}udV$. When the fields oscillate together they form electromagnetic radiation, which transports this energy at the speed of light.

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Nuclear Energy

Definition

Nuclear Energy

Nuclear energy is the form of energy stored in the nucleus of an atom, released when nucleons are rearranged into a more tightly bound configuration. It derives from the binding energy $B(Z,A)$ of a nucleus with $Z$ protons and $A$ nucleons, related to the mass deficit by Einstein’s

$$E=m{c}^2,B(Z,A)=[Z{m}_p+(A-Z)m_n-m_{\text{nuc}}]c^2.$$

A reaction that increases the binding energy per nucleon — either fission of heavy nuclei or fusion of light ones — releases the difference as kinetic energy of the products and radiation.

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