Permittivity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Permittivity 

 

A dielectric medium showing orientation of charged particles creating polarization effects. Such a medium can have a lower ratio of electric flux to charge (more permittivity) than empty space

In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ε (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor.

In the simplest case, the electric displacement field D resulting from an applied electric field E is

${\displaystyle \mathbf{D}=\epsilon \mathbf{E}.}$

More generally, the permittivity is a thermodynamic function of state. It can depend on the frequency, magnitude, and direction of the applied field. The SI unit for permittivity is farad per meter (F/m).

The permittivity is often represented by the relative permittivity ε_r which is the ratio of the absolute permittivity ε and the vacuum permittivity ε₀

${\displaystyle \kappa ={\epsilon }_{\mathrm{r}}=\frac{\epsilon }{{\epsilon }_0}}$.

This dimensionless quantity is also often and ambiguously referred to as the permittivity. Another common term encountered for both absolute and relative permittivity is the dielectric constant which has been deprecated in physics and engineering as well as in chemistry.

By definition, a perfect vacuum has a relative permittivity of exactly 1 whereas at standard temperature and pressure, air has a relative permittivity of κ_air ≈ 1.0006.

Relative permittivity is directly related to electric susceptibility (χ) by

${\displaystyle \chi =\kappa -1}$

otherwise written as

${\displaystyle \epsilon ={\epsilon }_{\mathrm{r}}{\epsilon }_0=(1+\chi ){\epsilon }_0}$

The term "permittivity" was introduced in the 1880s by Oliver Heaviside to complement Thomson's (1872) "permeability". Formerly written as p, the designation with ε has been in common use since the 1950s.

Units

The standard SI unit for permittivity is farad per meter (F/m or F·m⁻¹).

${\displaystyle \frac{\text{F}}{\text{m}}=\frac{\text{C}}{\text{V}\cdot \text{m}}=\frac{{\text{C}}^2}{\text{N}\cdot {\text{m}}^2}=\frac{{\text{C}}^2\cdot {\text{s}}^2}{\text{kg}\cdot {\text{m}}^3}}$

Explanation

In electromagnetism, the electric displacement field D represents the distribution of electric charges in a given medium resulting from the presence of an electric field E. This distribution includes charge migration and electric dipole reorientation. Its relation to permittivity in the very simple case of linear, homogeneous, isotropic materials with "instantaneous" response to changes in electric field is:

${\displaystyle \mathbf{D}=\epsilon \mathbf{E}}$

where the permittivity ε is a scalar. If the medium is anisotropic, the permittivity is a second rank tensor.

In general, permittivity is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity, temperature, and other parameters. In a nonlinear medium, the permittivity can depend on the strength of the electric field. Permittivity as a function of frequency can take on real or complex values.

In SI units, permittivity is measured in farads per meter (F/m or A²·s⁴·kg⁻¹·m⁻³). The displacement field D is measured in units of coulombs per square meter (C/m²), while the electric field E is measured in volts per meter (V/m). D and E describe the interaction between charged objects. D is related to the charge densities associated with this interaction, while E is related to the forces and potential differences.

Vacuum permittivity

Main article: Vacuum permittivity

The vacuum permittivity ε₀ (also called permittivity of free space or the electric constant) is the ratio D/E in free space. It also appears in the Coulomb force constant,

${\displaystyle k_{\text{e}}=\frac{1}{4\pi {\epsilon }_0}}$

Its value is

${\displaystyle {\epsilon }_0\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{c_0^2{\mu }_0}\approx 8.8541878128(13)\times {10}^{-12}\text{ F/m }}$

where

• c₀ is the speed of light in free space,
• µ₀ is the vacuum permeability.

The constants c₀ and μ₀ were both defined in SI units to have exact numerical values until the 2019 redefinition of the SI base units. Therefore, until that date, ε₀ could be also stated exactly as a fraction, ${\displaystyle \frac{1}{c_0^2{\mu }_0}=\frac{1}{35950207149.4727056\pi }\text{ F/m}}$ even if the result was irrational (because the fraction contained π). In contrast, the ampere was a measured quantity before 2019, but since then the ampere is now exactly defined and it is μ₀ that is an experimentally measured quantity (with consequent uncertainty) and therefore so is the new 2019 definition of ε₀ (c₀ remains exactly defined before and since 2019).

Relative permittivity

Main article: Relative permittivity

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity ε_r (also called dielectric constant, although this term is deprecated and sometimes only refers to the static, zero-frequency relative permittivity). In an anisotropic material, the relative permittivity may be a tensor, causing birefringence. The actual permittivity is then calculated by multiplying the relative permittivity by ε₀:

${\displaystyle \epsilon ={\epsilon }_{\mathrm{r}}{\epsilon }_0=(1+\chi ){\epsilon }_0,}$

where χ (frequently written χ_e) is the electric susceptibility of the material.

The susceptibility is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that

${\displaystyle \mathbf{P}={\epsilon }_0\chi \mathbf{E},}$

where ε₀ is the electric permittivity of free space.

The susceptibility of a medium is related to its relative permittivity ε_r by

${\displaystyle \chi ={\epsilon }_{\mathrm{r}}-1.}$

So in the case of a vacuum,

${\displaystyle \chi =0.}$

The susceptibility is also related to the polarizability of individual particles in the medium by the Clausius-Mossotti relation.

The electric displacement D is related to the polarization density P by

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}={\epsilon }_0(1+\chi )\mathbf{E}={\epsilon }_{\mathrm{r}}{\epsilon }_0\mathbf{E}.}$

The permittivity ε and permeability µ of a medium together determine the phase velocity v = c/n of electromagnetic radiation through that medium:

${\displaystyle \epsilon \mu =\frac{1}{v^2}.}$

Practical applications

Determining capacitance

The capacitance of a capacitor is based on its design and architecture, meaning it will not change with charging and discharging. The formula for capacitance in a parallel plate capacitor is written as

${\displaystyle C=\epsilon \frac{A}{d}}$

where ${\displaystyle A}$ is the area of one plate, ${\displaystyle d}$ is the distance between the plates, and ${\displaystyle \epsilon }$ is the permittivity of the medium between the two plates. For a capacitor with relative permittivity ${\displaystyle \kappa }$, it can be said that

${\displaystyle C=\kappa {\epsilon }_0\frac{A}{d}}$

Gauss's law

Permittivity is connected to electric flux (and by extension electric field) through Gauss's law. Gauss's law states that for a closed Gaussian surface, S

${\displaystyle {\mathrm{\Phi }}_E=\frac{Q_{\text{enc}}}{{\epsilon }_0}={\oint }_S\mathbf{E}\cdot \mathrm{d}\mathbf{A}}$

where ${\displaystyle {\mathrm{\Phi }}_E}$ is the net electric flux passing through the surface, ${\displaystyle Q_{\text{enc}}}$ is the charge enclosed in the Gaussian surface, ${\displaystyle \mathbf{E}}$ is the electric field vector at a given point on the surface, and ${\displaystyle \mathrm{d}\mathbf{A}}$ is a differential area vector on the Gaussian surface.

If the Gaussian surface uniformly encloses an insulated, symmetrical charge arrangement, the formula can be simplified to

${\displaystyle EA\cos (\theta )=\frac{Q_{\text{enc}}}{{\epsilon }_0}}$

where ${\displaystyle \theta }$ represents the angle between the electric field lines and the normal (perpendicular) to S.

If all of the electric field lines cross the surface at 90°, the formula can be further simplified to

${\displaystyle E=\frac{Q_{\text{enc}}}{{\epsilon }_{0}A}}$

Because the surface area of a sphere is ${\displaystyle 4\pi r^2}$, the electric field a distance ${\displaystyle r}$ away from a uniform, spherical charge arrangement is

${\displaystyle E=\frac{Q}{{\epsilon }_{0}A}=\frac{Q}{{\epsilon }_0\left(4\pi r^2\right)}=\frac{Q}{4\pi {\epsilon }_0{r}^2}=\frac{kQ}{r^2}}$

where ${\displaystyle k}$ is the Coulomb constant (${\displaystyle \sim 9.0\times {10}^9\text{m}/\text{F}}$). This formula applies to the electric field due to a point charge, outside of a conducting sphere or shell, outside of a uniformly charged insulating sphere, or between the plates of a spherical capacitor.

Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is

${\displaystyle \mathbf{P}(t)={\epsilon }_0{\int }_{-\mathrm{\infty }}^t\chi \left(t-t^{\prime }\right)\mathbf{E}\left(t^{\prime }\right)d{t}^{\prime }.}$

That is, the polarization is a convolution of the electric field at previous times with time-dependent susceptibility given by χ(Δt). The upper limit of this integral can be extended to infinity as well if one defines χ(Δt) = 0 for Δt < 0. An instantaneous response would correspond to a Dirac delta function susceptibility χ(Δt) = χδ(Δt).

It is convenient to take the Fourier transform with respect to time and write this relationship as a function of frequency. Because of the convolution theorem, the integral becomes a simple product,

${\displaystyle \mathbf{P}(\omega )={\epsilon }_0\chi (\omega )\mathbf{E}(\omega ).}$

This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. effectively χ(Δt) = 0 for Δt < 0), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility χ(0).

Complex permittivity

A dielectric permittivity spectrum over a wide range of frequencies. ε′ and ε″ denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.

As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the frequency of the field. This frequency dependence reflects the fact that a material's polarization does not change instantaneously when an electric field is applied. The response must always be causal (arising after the applied field), which can be represented by a phase difference. For this reason, permittivity is often treated as a complex function of the (angular) frequency ω of the applied field:

${\displaystyle \epsilon \to \hat{\epsilon }(\omega )}$

(since complex numbers allow specification of magnitude and phase). The definition of permittivity therefore becomes

${\displaystyle D_0{e}^{-i\omega t}=\hat{\epsilon }(\omega )E_0{e}^{-i\omega t},}$

where

• D₀ and E₀ are the amplitudes of the displacement and electric fields, respectively,
• i is the imaginary unit, i² = −1.

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity ε_s (also ε_DC):

${\displaystyle {\epsilon }_{\mathrm{s}}=\underset{\omega \to 0}{lim}\hat{\epsilon }(\omega ).}$

At the high-frequency limit (meaning optical frequencies), the complex permittivity is commonly referred to as ε_∞ (or sometimes ε_opt). At the plasma frequency and below, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for alternating fields of low frequencies, and as the frequency increases a measurable phase difference δ emerges between D and E. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate field strength (E₀), D and E remain proportional, and

${\displaystyle \hat{\epsilon }=\frac{D_0}{E_0}=|\epsilon |e^{-i\delta }.}$

Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:

${\displaystyle \hat{\epsilon }(\omega )={\epsilon }^{\prime }(\omega )-i{\epsilon }^{″}(\omega )=\left|\frac{D_0}{E_0}\right|\left(\cos \delta -i\sin \delta \right).}$

where

• ε′ is the real part of the permittivity;
• ε″ is the imaginary part of the permittivity;
• δ is the loss angle.

The choice of sign for time-dependence, e^−iωt, dictates the sign convention for the imaginary part of permittivity. The signs used here correspond to those commonly used in physics, whereas for the engineering convention one should reverse all imaginary quantities.

The complex permittivity is usually a complicated function of frequency ω, since it is a superimposed description of dispersion phenomena occurring at multiple frequencies. The dielectric function ε(ω) must have poles only for frequencies with positive imaginary parts, and therefore satisfies the Kramers–Kronig relations. However, in the narrow frequency ranges that are often studied in practice, the permittivity can be approximated as frequency-independent or by model functions.

At a given frequency, the imaginary part, ε″, leads to absorption loss if it is positive (in the above sign convention) and gain if it is negative. More generally, the imaginary parts of the eigenvalues of the anisotropic dielectric tensor should be considered.

In the case of solids, the complex dielectric function is intimately connected to band structure. The primary quantity that characterizes the electronic structure of any crystalline material is the probability of photon absorption, which is directly related to the imaginary part of the optical dielectric function ε(ω). The optical dielectric function is given by the fundamental expression:

${\displaystyle \epsilon (\omega )=1+\frac{8{\pi }^2{e}^2}{m^2}\sum _{c,v}\int W_{c,v}(E)(\phi (\hslash \omega -E)-\phi (\hslash \omega +E))dx.}$

In this expression, W_c,v(E) represents the product of the Brillouin zone-averaged transition probability at the energy E with the joint density of states, J_c,v(E); φ is a broadening function, representing the role of scattering in smearing out the energy levels. In general, the broadening is intermediate between Lorentzian and Gaussian; for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

Tensorial permittivity

According to the Drude model of magnetized plasma, a more general expression which takes into account the interaction of the carriers with an alternating electric field at millimeter and microwave frequencies in an axially magnetized semiconductor requires the expression of the permittivity as a non-diagonal tensor. (see also Electro-gyration).

${\displaystyle \mathbf{D}(\omega )=\left|\begin{array}{ccc}{\epsilon }_1& -i{\epsilon }_2& 0\\ i{\epsilon }_2& {\epsilon }_1& 0\\ 0& 0& {\epsilon }_z\end{array}\right|\mathbf{E}(\omega )}$

If ε₂ vanishes, then the tensor is diagonal but not proportional to the identity and the medium is said to be a uniaxial medium, which has similar properties to a uniaxial crystal.

Classification of materials

Classification of materials based on permittivity

εr″/εr′	Current conduction	Field propagation
0		perfect dielectric lossless medium
≪ 1	low-conductivity material poor conductor	low-loss medium good dielectric
≈ 1	lossy conducting material	lossy propagation medium
≫ 1	high-conductivity material good conductor	high-loss medium poor dielectric
∞	perfect conductor

Materials can be classified according to their complex-valued permittivity ε, upon comparison of its real ε′ and imaginary ε″ components (or, equivalently, conductivity, σ, when accounted for in the latter). A perfect conductor has infinite conductivity, σ = ∞, while a perfect dielectric is a material that has no conductivity at all, σ = 0; this latter case, of real-valued permittivity (or complex-valued permittivity with zero imaginary component) is also associated with the name lossless media. Generally, when σ/ωε′ ≪ 1 we consider the material to be a low-loss dielectric (although not exactly lossless), whereas σ/ωε′ ≫ 1 is associated with a good conductor; such materials with non-negligible conductivity yield a large amount of loss that inhibits the propagation of electromagnetic waves, thus are also said to be lossy media. Those materials that do not fall under either limit are considered to be general media.

Lossy medium

In the case of a lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:

${\displaystyle J_{\text{tot}}=J_{\mathrm{c}}+J_{\mathrm{d}}=\sigma E+i\omega {\epsilon }^{\prime }E=i\omega \hat{\epsilon }E}$

where

• σ is the conductivity of the medium;
• ${\displaystyle {\epsilon }^{\prime }={\epsilon }_0{\epsilon }_r}$ is the real part of the permittivity.
• ${\displaystyle \hat{\epsilon }={\epsilon }^{\prime }-i{\epsilon }^{″}}$is the complex permittivity

Note that this is using the electrical engineering convention of the Complex conjugate ambiguity; the physics/chemistry convention involves the complex conjugate of these equations.

The size of the displacement current is dependent on the frequency ω of the applied field E; there is no displacement current in a constant field.

In this formalism, the complex permittivity is defined as:

${\displaystyle \hat{\epsilon }={\epsilon }^{\prime }\left(1-i\frac{\sigma }{\omega {\epsilon }^{\prime }}\right)={\epsilon }^{\prime }-i\frac{\sigma }{\omega }}$

In general, the absorption of electromagnetic energy by dielectrics is covered by a few different mechanisms that influence the shape of the permittivity as a function of frequency:

• First are the relaxation effects associated with permanent and induced molecular dipoles. At low frequencies the field changes slowly enough to allow dipoles to reach equilibrium before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field because of the viscosity of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called dielectric relaxation and for ideal dipoles is described by classic Debye relaxation.
• Second are the resonance effects, which arise from the rotations or vibrations of atoms, ions, or electrons. These processes are observed in the neighborhood of their characteristic absorption frequencies.

The above effects often combine to cause non-linear effects within capacitors. For example, dielectric absorption refers to the inability of a capacitor that has been charged for a long time to completely discharge when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, real capacitors will develop a small voltage, a phenomenon that is also called soakage or battery action. For some dielectrics, such as many polymer films, the resulting voltage may be less than 1–2% of the original voltage. However, it can be as much as 15–25% in the case of electrolytic capacitors or supercapacitors.

Quantum-mechanical interpretation

In terms of quantum mechanics, permittivity is explained by atomic and molecular interactions.

At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at the microwave frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break hydrogen bonds. The field does work against the bonds and the energy is absorbed by the material as heat. This is why microwave ovens work very well for materials containing water. There are two maxima of the imaginary component (the absorptive index) of water, one at the microwave frequency, and the other at far ultraviolet (UV) frequency. Both of these resonances are at higher frequencies than the operating frequency of microwave ovens.

At moderate frequencies, the energy is too high to cause rotation, yet too low to affect electrons directly, and is absorbed in the form of resonant molecular vibrations. In water, this is where the absorptive index starts to drop sharply, and the minimum of the imaginary permittivity is at the frequency of blue light (optical regime).

At high frequencies (such as UV and above), molecules cannot relax, and the energy is purely absorbed by atoms, exciting electron energy levels. Thus, these frequencies are classified as ionizing radiation.

While carrying out a complete ab initio (that is, first-principles) modelling is now computationally possible, it has not been widely applied yet. Thus, a phenomenological model is accepted as being an adequate method of capturing experimental behaviors. The Debye model and the Lorentz model use a first-order and second-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).

Measurement

Main article: Dielectric spectroscopy

The relative permittivity of a material can be found by a variety of static electrical measurements. The complex permittivity is evaluated over a wide range of frequencies by using different variants of dielectric spectroscopy, covering nearly 21 orders of magnitude from 10⁻⁶ to 10¹⁵ hertz. Also, by using cryostats and ovens, the dielectric properties of a medium can be characterized over an array of temperatures. In order to study systems for such diverse excitation fields, a number of measurement setups are used, each adequate for a special frequency range.

Various microwave measurement techniques are outlined in Chen et al.. Typical errors for the Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%.

• Low-frequency time domain measurements (10⁻⁶ to 10³ Hz)
• Low-frequency frequency domain measurements (10⁻⁵ to 10⁶ Hz)
• Reflective coaxial methods (10⁶ to 10¹⁰ Hz)
• Transmission coaxial method (10⁸ to 10¹¹ Hz)
• Quasi-optical methods (10⁹ to 10¹⁰ Hz)
• Terahertz time-domain spectroscopy (10¹¹ to 10¹³ Hz)
• Fourier-transform methods (10¹¹ to 10¹⁵ Hz)

At infrared and optical frequencies, a common technique is ellipsometry. Dual polarisation interferometry is also used to measure the complex refractive index for very thin films at optical frequencies.

For the 3D measurement of dielectric tensors at optical frequency, Dielectric tensor tomography can be used.

Electric potential energy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electric_potential_energy 

Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged objects.

The term "electric potential energy" is used to describe the potential energy in systems with time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields.

Definition

The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration.

The electrostatic potential energy, U_E, of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position r_ref to that position r.

${\displaystyle U_{\mathrm{E}}(\mathbf{r})=-W_{r_{\mathrm{r}\mathrm{e}\mathrm{f}}\to r}=-{\int }_{{\mathbf{r}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}^{\mathbf{r}}q\mathbf{E}({\mathbf{r}}^{\prime })\cdot \mathrm{d}{\mathbf{r}}^{\prime }}$,

where E is the electrostatic field and dr' is the displacement vector in a curve from the reference position r_ref to the final position r.

The electrostatic potential energy can also be defined from the electric potential as follows:

The electrostatic potential energy, U_E, of one point charge q at position r in the presence of an electric potential ${\displaystyle \mathrm{\Phi }}$ is defined as the product of the charge and the electric potential.

${\displaystyle U_{\mathrm{E}}(\mathbf{r})=q\mathrm{\Phi }(\mathbf{r})}$,

where ${\displaystyle \mathrm{\Phi }}$ is the electric potential generated by the charges, which is a function of position r.

Units

The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule). In the CGS system the erg is the unit of energy, being equal to 10⁻⁷ Joules. Also electronvolts may be used, 1 eV = 1.602×10⁻¹⁹ Joules.

Electrostatic potential energy of one point charge

One point charge q in the presence of another point charge Q

A point charge q in the electric field of another charge Q.

The electrostatic potential energy, U_E, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is:

${\displaystyle U_E(r)=k_{\text{e}}\frac{qQ}{r},}$

where ${\displaystyle k_{\text{e}}=\frac{1}{4\pi {\epsilon }_0}}$ is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula.

Outline of proof

The electrostatic force F acting on a charge q can be written in terms of the electric field E as

$${\displaystyle \mathbf{F}=q\mathbf{E},}$$

By definition, the change in electrostatic potential energy, U_E, of a point charge q that has moved from the reference position r_ref to position r in the presence of an electric field E is the negative of the work done by the electrostatic force to bring it from the reference position r_ref to that position r.

$${\displaystyle U_E(r)-U_E(r_{\mathrm{r}\mathrm{e}\mathrm{f}})=-W_{r_{\mathrm{r}\mathrm{e}\mathrm{f}}\to r}=-{\int }_{r_{\mathrm{r}\mathrm{e}\mathrm{f}}}^{r}q\mathbf{E}\cdot \mathrm{d}\mathbf{s}.}$$

where:

• r = position in 3d space of the charge q, using cartesian coordinates r = (x, y, z), taking the position of the Q charge at r = (0,0,0), the scalar r = |r| is the norm of the position vector,
• ds = differential displacement vector along a path C going from r_ref to r,
• ${\displaystyle W_{r_{\mathrm{r}\mathrm{e}\mathrm{f}}\to r}}$ is the work done by the electrostatic force to bring the charge from the reference position r_ref to r,

Usually U_E is set to zero when r_ref is infinity:

$${\displaystyle U_E(r_{\mathrm{r}\mathrm{e}\mathrm{f}}=\mathrm{\infty })=0}$$

so

$${\displaystyle U_E(r)=-{\int }_{\mathrm{\infty }}^{r}q\mathbf{E}\cdot \mathrm{d}\mathbf{s}}$$

When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. This happens in time-invariant electric fields. When talking about electrostatic potential energy, time-invariant electric fields are always assumed so, in this case, the electric field is conservative and Coulomb's law can be used.

Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. By the definition of the position vector r and the displacement vector s, it follows that r and s are also radially directed from Q. So, E and ds must be parallel:

$${\displaystyle \mathbf{E}\cdot \mathrm{d}\mathbf{s}=|\mathbf{E}|\cdot |\mathrm{d}\mathbf{s}|\cos (0)=E\mathrm{d}s}$$

Using Coulomb's law, the electric field is given by

$${\displaystyle |\mathbf{E}|=E=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{s^2}}$$

and the integral can be easily evaluated:

$${\displaystyle U_E(r)=-{\int }_{\mathrm{\infty }}^{r}q\mathbf{E}\cdot \mathrm{d}\mathbf{s}=-{\int }_{\mathrm{\infty }}^r\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{s^2}\mathrm{d}s=\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{r}=k_e\frac{qQ}{r}}$$

One point charge q in the presence of n point charges Q_i

Electrostatic potential energy of q due to Q₁ and Q₂ charge system:${\displaystyle U_E=q\frac{1}{4\pi {\epsilon }_0}\left(\frac{Q_1}{r_1}+\frac{Q_2}{r_2}\right)}$

The electrostatic potential energy, U_E, of one point charge q in the presence of n point charges Q_i, taking an infinite separation between the charges as the reference position, is:

${\displaystyle U_E(r)=k_{\text{e}}q\sum _{i=1}^n\frac{Q_i}{r_i},}$

where ${\displaystyle k_{\text{e}}=\frac{1}{4\pi {\epsilon }_0}}$ is the Coulomb constant, r_i is the distance between the point charges q and Q_i, and q and Q_i are the assigned values of the charges.

Electrostatic potential energy stored in a system of point charges

The electrostatic potential energy U_E stored in a system of N charges q₁, q₂, …, q_N at positions r₁, r₂, …, r_N respectively, is:

${\displaystyle U_{\mathrm{E}}=\frac{1}{2}\sum _{i=1}^N{q}_i\mathrm{\Phi }({\mathbf{r}}_i)=\frac{1}{2}{k}_e\sum _{i=1}^N{q}_i\sum _{\stackrel{j=1}{j\ne i}}^N\frac{q_j}{r_{ij}},}$

(1)

where, for each i value, Φ(r_i) is the electrostatic potential due to all point charges except the one at r_i,^{[note 2]} and is equal to:

$${\displaystyle \mathrm{\Phi }({\mathbf{r}}_i)=k_e\sum _{\stackrel{j=1}{j\ne i}}^N\frac{q_j}{{\mathbf{r}}_{ij}},}$$

where r_ij is the distance between q_i and q_j.

Outline of proof

The electrostatic potential energy U_E stored in a system of two charges is equal to the electrostatic potential energy of a charge in the electrostatic potential generated by the other. That is to say, if charge q₁ generates an electrostatic potential Φ₁, which is a function of position r, then

$${\displaystyle U_{\mathrm{E}}=q_2{\mathrm{\Phi }}_1({\mathbf{r}}_2).}$$

Doing the same calculation with respect to the other charge, we obtain

$${\displaystyle U_{\mathrm{E}}=q_1{\mathrm{\Phi }}_2({\mathbf{r}}_1).}$$

The electrostatic potential energy is mutually shared by ${\displaystyle q_1}$ and ${\displaystyle q_2}$, so the total stored energy is

$${\displaystyle U_E=\frac{1}{2}\left[q_2{\mathrm{\Phi }}_1({\mathbf{r}}_2)+q_1{\mathrm{\Phi }}_2({\mathbf{r}}_1)\right]}$$

This can be generalized to say that the electrostatic potential energy U_E stored in a system of N charges q₁, q₂, …, q_N at positions r₁, r₂, …, r_N respectively, is:

$${\displaystyle U_{\mathrm{E}}=\frac{1}{2}\sum _{i=1}^N{q}_i\mathrm{\Phi }({\mathbf{r}}_i).}$$

Energy stored in a system of one point charge

The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic force against which an external agent must do work in moving the point charge from infinity to its final location.

A common question arises concerning the interaction of a point charge with its own electrostatic potential. Since this interaction doesn't act to move the point charge itself, it doesn't contribute to the stored energy of the system.

Energy stored in a system of two point charges

Consider bringing a point charge, q, into its final position near a point charge, Q₁. The electric potential Φ(r) due to Q₁ is

$${\displaystyle \mathrm{\Phi }(r)=k_e\frac{Q_1}{r}}$$

Hence we obtain, the electrostatic potential energy of q in the potential of Q₁ as

$${\displaystyle U_E=\frac{1}{4\pi {\epsilon }_0}\frac{q{Q}_1}{r_1}}$$

where r₁ is the separation between the two point charges.

Energy stored in a system of three point charges

The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q₁ due to two charges Q₂ and Q₃, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q₂ and Q₃.

The electrostatic potential energy stored in the system of three charges is:

$${\displaystyle U_{\mathrm{E}}=\frac{1}{4\pi {\epsilon }_0}\left[\frac{Q_1{Q}_2}{r_{12}}+\frac{Q_1{Q}_3}{r_{13}}+\frac{Q_2{Q}_3}{r_{23}}\right]}$$

Outline of proof

Using the formula given in (1), the electrostatic potential energy of the system of the three charges will then be:

$${\displaystyle U_{\mathrm{E}}=\frac{1}{2}\left[Q_1\mathrm{\Phi }({\mathbf{r}}_1)+Q_2\mathrm{\Phi }({\mathbf{r}}_2)+Q_3\mathrm{\Phi }({\mathbf{r}}_3)\right]}$$

Where ${\displaystyle \mathrm{\Phi }({\mathbf{r}}_1)}$ is the electric potential in r₁ created by charges Q₂ and Q₃, ${\displaystyle \mathrm{\Phi }({\mathbf{r}}_2)}$ is the electric potential in r₂ created by charges Q₁ and Q₃, and ${\displaystyle \mathrm{\Phi }({\mathbf{r}}_3)}$ is the electric potential in r₃ created by charges Q₁ and Q₂. The potentials are:

$${\displaystyle \mathrm{\Phi }({\mathbf{r}}_1)={\mathrm{\Phi }}_2({\mathbf{r}}_1)+{\mathrm{\Phi }}_3({\mathbf{r}}_1)=\frac{1}{4\pi {\epsilon }_0}\frac{Q_2}{r_{12}}+\frac{1}{4\pi {\epsilon }_0}\frac{Q_3}{r_{13}}}$$

$${\displaystyle \mathrm{\Phi }({\mathbf{r}}_2)={\mathrm{\Phi }}_1({\mathbf{r}}_2)+{\mathrm{\Phi }}_3({\mathbf{r}}_2)=\frac{1}{4\pi {\epsilon }_0}\frac{Q_1}{r_{21}}+\frac{1}{4\pi {\epsilon }_0}\frac{Q_3}{r_{23}}}$$

$${\displaystyle \mathrm{\Phi }({\mathbf{r}}_3)={\mathrm{\Phi }}_1({\mathbf{r}}_3)+{\mathrm{\Phi }}_2({\mathbf{r}}_3)=\frac{1}{4\pi {\epsilon }_0}\frac{Q_1}{r_{31}}+\frac{1}{4\pi {\epsilon }_0}\frac{Q_2}{r_{32}}}$$

Where r_ij is the distance between charge Q_i and Q_j.

If we add everything:

$${\displaystyle U_{\mathrm{E}}=\frac{1}{2}\frac{1}{4\pi {\epsilon }_0}\left[\frac{Q_1{Q}_2}{r_{12}}+\frac{Q_1{Q}_3}{r_{13}}+\frac{Q_2{Q}_1}{r_{21}}+\frac{Q_2{Q}_3}{r_{23}}+\frac{Q_3{Q}_1}{r_{31}}+\frac{Q_3{Q}_2}{r_{32}}\right]}$$

Finally, we get that the electrostatic potential energy stored in the system of three charges:

$${\displaystyle U_{\mathrm{E}}=\frac{1}{4\pi {\epsilon }_0}\left[\frac{Q_1{Q}_2}{r_{12}}+\frac{Q_1{Q}_3}{r_{13}}+\frac{Q_2{Q}_3}{r_{23}}\right]}$$

Energy stored in an electrostatic field distribution in vacuum

The energy density, or energy per unit volume, $\frac{dU}{dV}$, of the electrostatic field of a continuous charge distribution is:

$${\displaystyle u_e=\frac{dU}{dV}=\frac{1}{2}{\epsilon }_0{\left|\mathbf{E}\right|}^2.}$$

Outline of proof

One may take the equation for the electrostatic potential energy of a continuous charge distribution and put it in terms of the electrostatic field.

Since Gauss's law for electrostatic field in differential form states

$${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}}$$

where

• ${\displaystyle \mathbf{E}}$ is the electric field vector
• ${\displaystyle \rho }$ is the total charge density including dipole charges bound in a material
• ${\displaystyle {\epsilon }_0}$ is the permittivity of free space,

then,

$${\displaystyle \begin{array}{rl}U& =\frac{1}{2}\underset{\text{all space}}{\int }\rho (r)\mathrm{\Phi }(r)dV\\ & =\frac{1}{2}\underset{\text{all space}}{\int }{\epsilon }_0(\mathrm{\nabla }\cdot \mathbf{E})\mathrm{\Phi }dV\end{array}}$$

so, now using the following divergence vector identity

$${\displaystyle \mathrm{\nabla }\cdot (\mathbf{A}B)=(\mathrm{\nabla }\cdot \mathbf{A})B+\mathbf{A}\cdot (\mathrm{\nabla }B)\Rightarrow (\mathrm{\nabla }\cdot \mathbf{A})B=\mathrm{\nabla }\cdot (\mathbf{A}B)-\mathbf{A}\cdot (\mathrm{\nabla }B)}$$

we have

$${\displaystyle U=\frac{{\epsilon }_0}{2}\underset{\text{all space}}{\int }\mathrm{\nabla }\cdot (\mathbf{E}\mathrm{\Phi })dV-\frac{{\epsilon }_0}{2}\underset{\text{all space}}{\int }(\mathrm{\nabla }\mathrm{\Phi })\cdot \mathbf{E}dV}$$

using the divergence theorem and taking the area to be at infinity where ${\displaystyle \mathrm{\Phi }(\mathrm{\infty })=0}$

$${\displaystyle \begin{array}{rl}U& =\stackrel{0}{\overbrace{\frac{{\epsilon }_0}{2}\underset{{}_{\text{ of space}}^{\text{boundary}}}{\int }\mathrm{\Phi }\mathbf{E}\cdot d\mathbf{A}}}-\frac{{\epsilon }_0}{2}\underset{\text{all space}}{\int }(-\mathbf{E})\cdot \mathbf{E}dV\\ & =\underset{\text{all space}}{\int }\frac{1}{2}{\epsilon }_0{\left|\mathbf{E}\right|}^{2}dV.\end{array}}$$

So, the energy density, or energy per unit volume $\frac{dU}{dV}$ of the electrostatic field is:

$${\displaystyle u_e=\frac{1}{2}{\epsilon }_0{\left|\mathbf{E}\right|}^2.}$$

Energy stored in electronic elements

The electric potential energy stored in a capacitor is U_E=1/2 CV²

Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat. This is known as the Joule effect. A capacitor stores it in its electric field. The total electrostatic potential energy stored in a capacitor is given by

$${\displaystyle U_E=\frac{1}{2}QV=\frac{1}{2}C{V}^2=\frac{Q^2}{2C}}$$

where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.

Outline of proof

One may assemble charges to a capacitor in infinitesimal increments, ${\displaystyle dq\to 0}$, such that the amount of work done to assemble each increment to its final location may be expressed as

$${\displaystyle W_q=Vdq=\frac{q}{C}dq.}$$

The total work done to fully charge the capacitor in this way is then

$${\displaystyle W=\int dW={\int }_0^{Q}Vdq=\frac{1}{C}{\int }_0^{Q}qdq=\frac{Q^2}{2C}.}$$

where ${\displaystyle Q}$ is the total charge on the capacitor. This work is stored as electrostatic potential energy, hence,

$${\displaystyle W=U_E=\frac{Q^2}{2C}.}$$

Notably, this expression is only valid if ${\displaystyle dq\to 0}$, which holds for many-charge systems such as large capacitors having metallic electrodes. For few-charge systems the discrete nature of charge is important. The total energy stored in a few-charge capacitor is

$${\displaystyle U_E=\frac{Q^2}{C}}$$

which is obtained by a method of charge assembly utilizing the smallest physical charge increment ${\displaystyle \mathrm{\Delta }q=e}$ where ${\displaystyle e}$ is the elementary unit of charge and ${\displaystyle Q=Ne}$ where ${\displaystyle N}$ is the total number of charges in the capacitor.

The total electrostatic potential energy may also be expressed in terms of the electric field in the form

$${\displaystyle U_E=\frac{1}{2}{\int }_V\mathrm{E}\cdot \mathrm{D}dV}$$

where ${\displaystyle \mathrm{D}}$ is the electric displacement field within a dielectric material and integration is over the entire volume of the dielectric.

(A virtual experiment based on the energy transfert between capacitor plates reveals that an additional term must be taken into account when the electrostatic energy is expressed in terms of the electric field and displacement vectors ^[3].

While this extra energy cancels when dealing with insulators, in general it cannot be ignored, as for instance with semiconductors.)

The total electrostatic potential energy stored within a charged dielectric may also be expressed in terms of a continuous volume charge, ${\displaystyle \rho }$,

$${\displaystyle U_E=\frac{1}{2}{\int }_V\rho \mathrm{\Phi }dV}$$

where integration is over the entire volume of the dielectric.

These latter two expressions are valid only for cases when the smallest increment of charge is zero (${\displaystyle dq\to 0}$) such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.