Dipole

Potential of static dipoles

In electromagnetism, the calculation of the electric and magnetic fields are often made simpler by first calculating the scaler and vector potentials, Φ and A respectively.

The electrostatic potential at position r due to an electric dipole at the origin is given by:

$${\displaystyle {\mathrm{\Phi }}_{dip}=\frac{1}{4\pi {\epsilon }_0}\frac{\mathbf{p}\cdot \hat{\mathbf{r}}}{r^2},}$$ where p is the electric dipole moment, and ε₀ is the permittivity of free space. This term appears as the second term in the multipole expansion of an arbitrary electrostatic potential Φ. This term will dominate at large distances if there is no net charge (and if p≠0).

The vector potential A_dip at position r of a magnetic dipole moment m at the origin is

$${\displaystyle {\mathbf{A}}_{dip}=\frac{{\mu }_0}{4\pi }\frac{\mathbf{m}\times \hat{\mathbf{r}}}{r^2},}$$

where μ₀ is the permeability of free space. This term appears as the second term (first non-zero term) in the multipole expansion of an arbitrary vector potential A in terms of the current density J that created it. This term dominates at large distances if m≠0.

Field of a static dipole

The electric field, E_dip, and magnetic field, B_dip, at a location, r, due to a dipole at the origin with dipole moment, p for the electric dipole or m for the magnetic dipole, can be determined from the scalar, Φ_dip, and vector, A_dip, potential respectively as:

$${\displaystyle \begin{array}{cccc}{\mathbf{E}}_{dip}& =-\mathrm{\nabla }{\mathrm{\Phi }}_{dip}& =\frac{1}{4\pi {\epsilon }_0}\frac{3(\mathbf{p}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{r^3}& -{\delta }^3(\mathbf{r})\frac{\mathbf{p}}{3{\epsilon }_0},\\ {\mathbf{B}}_{dip}& =\mathrm{\nabla }\times {\mathbf{A}}_{dip}& =\frac{{\mu }_0}{4\pi }\frac{3(\mathbf{m}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}}{r^3}& +{\delta }^3(\mathbf{r})\frac{2{\mu }_0\mathbf{m}}{3},\end{array}}$$

where ε₀ is the permitivity of free space and μ₀ is the permeability of free space (both constants). The last term in the equations (containing the dirac delta function) only contributes at the origin and in most cases can be ignored. Note the similarity in the two equations making them near identical except for the last term.

These equations are exactly the field of a point (ideal) dipole, exactly the dipole term in the multipole expansion of an arbitrary field, and approximately the field of any dipole-like configuration at large distances. The field of a real (physical) dipole is continuous everywhere and will be different close to the origin. The delta function represents the strong field pointing in the opposite direction between the point charges for the electric dipole case (and in same direction for magnetic dipole), which is often omitted since one is rarely interested in the field at the dipole's position.

For further discussions about the internal field of dipoles, see or Magnetic moment § Internal magnetic field of a dipole.

Alternative formulation in spherical coordinates

An alternative formulation that simplifies the equations by orienting the z-axis in the direction of the dipole moment is:

$${\displaystyle \begin{array}{c}{\mathbf{E}}_{dip}=\frac{p}{4\pi {\epsilon }_0{r}^3}\left(2\cos \theta \hat{\mathbf{r}}+\sin \theta \hat{\theta }\right),\\ {\mathbf{B}}_{dip}=\frac{{\mu }_{0}m}{4\pi r^3}\left(2\cos \theta \hat{\mathbf{r}}+\sin \theta \hat{\theta }\right),\end{array}}$$ where θ is the angle from the z-axis (the direction of the dipole moment), r is the distance from the dipole to where the field is, ${\displaystyle \hat{r}}$ is the direction of r, and ${\displaystyle \hat{\theta }}$ is the direction perpendicular to ${\displaystyle \hat{r}}$ and pointing away from the z-axis.

Magnitude

The magnitude of the dipole field strengths, E_dip and E_dip, at a distance, r, from the dipoles at the origin are: $${\displaystyle \begin{array}{ccc}{E}_{dip}& =& \frac{1}{4\pi {\epsilon }_0}\frac{p}{r^3}\sqrt{1+3{\cos }^2(\theta )},\\ B_{dip}& =& \frac{{\mu }_0}{4\pi }\frac{m}{r^3}\sqrt{1+3{\cos }^2(\theta )},\end{array}}$$

where θ is the angle between the direction that the dipole is pointing (typically chosen as the z-axis) and the direction of r, ε₀ is the permittivity of free space, and μ₀ is permeability of free space.

Conversion to cylindrical coordinates is achieved using r² = z² + ρ² and ${\displaystyle \theta =\arccos \left(\frac{z}{\sqrt{z^2+{\rho }^2}}\right)}$ where ρ is the perpendicular distance from the z-axis gives: $${\displaystyle \begin{array}{ccc}{E}_{dip}& =& \frac{p}{4\pi {\epsilon }_0{\left(z^2+{\rho }^2\right)}^{\frac{3}{2}}}\sqrt{1+\frac{3z^2}{z^2+{\rho }^2}},\\ B_{dip}& =& \frac{{\mu }_{0}m}{4\pi {\left(z^2+{\rho }^2\right)}^{\frac{3}{2}}}\sqrt{1+\frac{3z^2}{z^2+{\rho }^2}}.\end{array}}$$

Torques and forces on static dipoles

Since the direction of an electric field is defined as the direction of the force on a positive charge, electric field lines point away from a positive charge and toward a negative charge.

When placed in a homogeneous electric or magnetic field, equal but opposite forces arise on each side of the dipole creating a torque τ: $${\displaystyle \mathit{\tau }=\mathbf{p}\times \mathbf{E}}$$ for an electric dipole moment p (in coulomb-meters), or $${\displaystyle \mathit{\tau }=\mathbf{m}\times \mathbf{B}}$$ for a magnetic dipole moment m (in ampere-square meters).

The resulting torque will tend to align the dipole with the applied field, which in the case of an electric dipole, yields a potential energy of $${\displaystyle U=-\mathbf{p}\cdot \mathbf{E}.}$$

The energy of a magnetic dipole is similarly $${\displaystyle U=-\mathbf{m}\cdot \mathbf{B}.}$$

Molecular electric dipoles

Electric dipole moments are responsible for the behavior of a substance in the presence of external electric fields. The dipoles tend to be aligned to the external field which can be constant or time-dependent. This effect forms the basis of a modern experimental technique called dielectric spectroscopy.

Dipole moments can be found in common molecules such as water and also in biomolecules such as proteins due to non-uniform distributions of positive and negative charges on the various atoms. Therefore, a molecule's dipole is an electric dipole with an inherent electric field that should not be confused with a magnetic dipole, which generates a magnetic field. The physical chemist Peter J. W. Debye was the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in the non-SI unit named debye in his honor.

For molecules there are three types of dipoles:

Permanent dipoles

These occur when two atoms in a molecule have substantially different electronegativity : One atom attracts electrons more than another, becoming more negative, while the other atom becomes more positive. A molecule with a permanent dipole moment is called a polar molecule. See Intermolecular force § Dipole–dipole interactions.

Instantaneous dipoles

These occur due to chance when electrons happen to be more concentrated in one place than another in a molecule, creating a temporary dipole. These dipoles are smaller in magnitude than permanent dipoles, but still play a large role in chemistry and biochemistry due to their prevalence. See instantaneous dipole.

Induced dipoles

These can occur when one molecule with a permanent dipole repels another molecule's electrons, inducing a dipole moment in that molecule. A molecule is polarized when it carries an induced dipole. See induced-dipole attraction.

It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix. Such values however are not directly comparable to experiment due to the potential presence of nuclear quantum effects, which can be substantial for even simple systems like the ammonia molecule. Coupled cluster theory (especially CCSD(T)) can give very accurate dipole moments, although it is possible to get reasonable estimates (within about 5%) from density functional theory, especially if hybrid or double hybrid functionals are employed. The dipole moment of a molecule can also be calculated based on the molecular structure using the concept of group contribution methods.

By means of the total dipole moment of some material one can compute the dielectric constant. If ${\displaystyle {\mathcal{M}}_{\mathrm{T}\mathrm{o}\mathrm{t}}}$ is the total dipole moment of the sample, then the dielectric constant is given by $${\displaystyle \epsilon =1+k⟨{\mathcal{M}}_{\text{Tot}}^2⟩}$$ where k is a constant and ${\displaystyle ⟨{\mathcal{M}}_{\text{Tot}}^2⟩=⟨{\mathcal{M}}_{\text{Tot}}(t=0){\mathcal{M}}_{\text{Tot}}(t=0)⟩}$ is the time correlation function of the total dipole moment. In general the total dipole moment have contributions coming from translations and rotations of the molecules in the sample, $${\displaystyle {\mathcal{M}}_{\text{Tot}}={\mathcal{M}}_{\text{Trans}}+{\mathcal{M}}_{\text{Rot}}.}$$

Therefore, the dielectric constant has contributions from both terms. This approach can be generalized to compute the frequency dependent dielectric function.

Alternatively, electric dipole moment values for a given material can be calculated by the measured value of the dielectric constant of that material.

Quantum-mechanical dipole operator

Consider a collection of N particles with charges q_i and position vectors r_i. For instance, this collection may be a molecule consisting of electrons, all with charge −e, and nuclei with charge eZ_i, where Z_i is the atomic number of the i th nucleus. The dipole observable (physical quantity) has the quantum mechanical dipole operator: $${\displaystyle \mathfrak{p}=\sum _{i=1}^N{q}_i{\mathbf{r}}_i.}$$

Notice that this definition is valid only for neutral atoms or molecules, i.e., total charge equal to zero. In the ionized case, we have $${\displaystyle \mathfrak{p}=\sum _{i=1}^N{q}_i({\mathbf{r}}_i-{\mathbf{r}}_c),}$$ where ${\displaystyle {\mathbf{r}}_c}$ is the center of mass of the molecule/group of particles.

Atomic dipoles

A non-degenerate (S-state) atom can have only a zero permanent dipole. This fact follows quantum mechanically from the inversion symmetry of atoms. All 3 components of the dipole operator are antisymmetric under inversion with respect to the nucleus, $${\displaystyle \mathfrak{I}\mathfrak{p}{\mathfrak{I}}^{-1}=-\mathfrak{p},}$$ where ${\displaystyle \mathfrak{p}}$ is the dipole operator and ${\displaystyle \mathfrak{I}}$ is the inversion operator.

The permanent dipole moment of an atom in a non-degenerate state (see Degenerate energy level) is given as the expectation (average) value of the dipole operator, $${\displaystyle ⟨\mathfrak{p}⟩=⟨S|\mathfrak{p}|S⟩,}$$ where ${\displaystyle |S⟩}$ is an S-state, non-degenerate, wavefunction, which is symmetric or anti-symmetric under inversion: ${\displaystyle \mathfrak{I}|S⟩=\pm |S⟩}$. Since the product of the wavefunction (in the ket) and its complex conjugate (in the bra) is always symmetric under inversion and its inverse, $${\displaystyle ⟨\mathfrak{p}⟩=⟨{\mathfrak{I}}^{-1}S|\mathfrak{p}|{\mathfrak{I}}^{-1}S⟩=⟨S|\mathfrak{I}\mathfrak{p}{\mathfrak{I}}^{-1}|S⟩=-⟨\mathfrak{p}⟩}$$ it follows that the expectation value changes sign under inversion. We used here the fact that ${\displaystyle \mathfrak{I}}$, being a symmetry operator, is unitary: ${\displaystyle {\mathfrak{I}}^{-1}={\mathfrak{I}}^{\ast }}$ and by definition the Hermitian adjoint ${\displaystyle {\mathfrak{I}}^{\ast }}$ may be moved from bra to ket and then becomes ${\displaystyle {\mathfrak{I}}^{\ast \ast }=\mathfrak{I}}$. Since the only quantity that is equal to minus itself is the zero, the expectation value vanishes, $${\displaystyle ⟨\mathfrak{p}⟩=0.}$$

In the case of open-shell atoms with degenerate energy levels, one could define a dipole moment by the aid of the first-order Stark effect. This gives a non-vanishing dipole (by definition proportional to a non-vanishing first-order Stark shift) only if some of the wavefunctions belonging to the degenerate energies have opposite parity; i.e., have different behavior under inversion. This is a rare occurrence, but happens for the excited H-atom, where 2s and 2p states are "accidentally" degenerate (see article Laplace–Runge–Lenz vector for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).

Atomic Dipole Moment in QTAIM

In the Quantum Theory of Atoms in Molecules (QTAIM), an atomic dipole moment is defined from the electron density by partitioning a molecule or crystal into atomic basins bounded by zero-flux surfaces in the gradient vector field of the electron density. The atomic dipole consists of two contributions: (1) the intrinsic polarization of the atomic basin itself, corresponding to the displacement of the electronic charge centroid relative to the nucleus, and (2) a charge-transfer contribution arising from the transfer of electronic charge across interatomic surfaces shared with neighboring atoms. In this manner, the total molecular or crystalline dipole moment is exactly additive, being equal to the sum of all atomic contributions, thereby providing a real-space decomposition of polarization in molecules and solids.

Dipole radiation

In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to spherical wave radiation.

In particular, consider a harmonically oscillating electric dipole, with angular frequency ω and a dipole moment p₀ along the ẑ direction of the form $${\displaystyle \mathbf{p}(\mathbf{r},t)=\mathbf{p}(\mathbf{r})e^{-i\omega t}=p_0\hat{\mathbf{z}}{e}^{-i\omega t}.}$$

In vacuum, the exact field produced by this oscillating dipole can be derived using the retarded potential formulation as: $${\displaystyle \begin{array}{cc}\mathbf{E}& =\frac{1}{4\pi {\epsilon }_0}\left[\frac{{\omega }^2}{c^{2}r}\left(\hat{\mathbf{r}}\times \mathbf{p}\right)\times \hat{\mathbf{r}}+\left(\frac{1}{r^3}-\frac{i\omega }{c{r}^2}\right)\left(3\hat{\mathbf{r}}\left[\hat{\mathbf{r}}\cdot \mathbf{p}\right]-\mathbf{p}\right)\right]e^{i\omega (r/c-t)}\\ \mathbf{B}& =\frac{{\omega }^2}{4\pi {\epsilon }_0{c}^3}(\hat{\mathbf{r}}\times \mathbf{p})\left(1-\frac{c}{i\omega r}\right)\frac{e^{i\omega (r/c-t)}}{r}.\end{array}}$$

For ⁠rω/c⁠ ≫ 1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product: $${\displaystyle \begin{array}{cc}\mathbf{B}& =\frac{{\omega }^2}{4\pi {\epsilon }_0{c}^3}(\hat{\mathbf{r}}\times \mathbf{p})\frac{e^{i\omega (r/c-t)}}{r}\\ & =\frac{{\omega }^2{\mu }_0{p}_0}{4\pi c}(\hat{\mathbf{r}}\times \hat{\mathbf{z}})\frac{e^{i\omega (r/c-t)}}{r}=-\frac{{\omega }^2{\mu }_0{p}_0}{4\pi c}\sin (\theta )\frac{e^{i\omega (r/c-t)}}{r}\hat{\mathit{\varphi }}\\ \mathbf{E}& =c\mathbf{B}\times \hat{\mathbf{r}}\\ & =-\frac{{\omega }^2{\mu }_0{p}_0}{4\pi }\sin (\theta )\left(\hat{\mathit{\varphi }}\times \hat{\mathbf{r}}\right)\frac{e^{i\omega (r/c-t)}}{r}=-\frac{{\omega }^2{\mu }_0{p}_0}{4\pi }\sin (\theta )\frac{e^{i\omega (r/c-t)}}{r}\hat{\mathit{\theta }}.\end{array}}$$

The time-averaged Poynting vector $${\displaystyle ⟨\mathbf{S}⟩=\frac{{\mu }_0{p}_0^2{\omega }^4}{32{\pi }^{2}c}\frac{{\sin }^2(\theta )}{r^2}\hat{\mathbf{r}}}$$ is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the spherical harmonic function (sin θ) responsible for such toroidal angular distribution is precisely the l = 1 "p" wave.

The total time-average power radiated by the field can then be derived from the Poynting vector as $${\displaystyle P=\frac{{\mu }_0{\omega }^4{p}_0^2}{12\pi c}.}$$

Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the Rayleigh scattering, and the underlying effects why the sky consists of mainly blue colour.

A circular polarized dipole is described as a superposition of two linear dipoles.