Supramolecular chemistry

From Wikipedia, the free encyclopedia

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

Supramolecular chemistry is the branch of chemistry concerning chemical systems composed of discrete numbers of molecules. The strength of the forces responsible for spatial organization of the system ranges from weak intermolecular forces, electrostatic charge, or hydrogen bonding to strong covalent bonding, provided that the electronic coupling strength remains small relative to the energy parameters of the component. While traditional chemistry concentrates on the covalent bond, supramolecular chemistry examines the weaker and reversible non-covalent interactions between molecules. These forces include hydrogen bonding, metal coordination, hydrophobic forces, van der Waals forces, pi–pi interactions and electrostatic effects.

Important concepts advanced by supramolecular chemistry include molecular self-assembly, molecular folding, molecular recognition, host–guest chemistry, mechanically-interlocked molecular architectures, and dynamic covalent chemistry. The study of non-covalent interactions is crucial to understanding many biological processes that rely on these forces for structure and function. Biological systems are often the inspiration for supramolecular research.

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  Self-assembly of a circular double helicate
  
  
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  Host–guest complex within another host (cucurbituril)
   
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  Mechanically-interlocked molecules (rotaxane)
  
  
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  An example of a host–guest chemistry
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History

18-crown-6 can be synthesized from using potassium ion as the template cation

The existence of intermolecular forces was first postulated by Johannes Diderik van der Waals in 1873. However, Nobel laureate Hermann Emil Fischer developed supramolecular chemistry's philosophical roots. In 1894, Fischer suggested that enzyme–substrate interactions take the form of a "lock and key", the fundamental principles of molecular recognition and host–guest chemistry. In the early twentieth century non-covalent bonds were understood in gradually more detail, with the hydrogen bond being described by Latimer and Rodebush in 1920.

With the deeper understanding of the non-covalent interactions, for example, the clear elucidation of DNA structure, chemists started to emphasize the importance of non-covalent interactions. In 1967, Charles J. Pedersen discovered crown ethers, which are ring-like structures capable of chelating certain metal ions. Then, in 1969, Jean-Marie Lehn discovered a class of molecules similar to crown ethers, called cryptands. After that, Donald J. Cram synthesized many variations to crown ethers, on top of separate molecules capable of selective interaction with certain chemicals. The three scientists were awarded the Nobel Prize in Chemistry in 1987 for "development and use of molecules with structure-specific interactions of high selectivity". In 2016, Bernard L. Feringa, Sir J. Fraser Stoddart, and Jean-Pierre Sauvage were awarded the Nobel Prize in Chemistry, "for the design and synthesis of molecular machines".

Carboxylic acid dimers

The term supermolecule (or supramolecule) was introduced by Karl Lothar Wolf et al. (Übermoleküle) in 1937 to describe hydrogen-bonded acetic acid dimers. The term supermolecule is also used in biochemistry to describe complexes of biomolecules, such as peptides and oligonucleotides composed of multiple strands.

Eventually, chemists applied these concepts to synthetic systems. One breakthrough came in the 1960s with the synthesis of the crown ethers by Charles J. Pedersen. Following this work, other researchers such as Donald J. Cram, Jean-Marie Lehn and Fritz Vögtle reported a variety of three-dimensional receptors, and throughout the 1980s research in the area gathered a rapid pace with concepts such as mechanically interlocked molecular architectures emerging.

The influence of supramolecular chemistry was established by the 1987 Nobel Prize for Chemistry which was awarded to Donald J. Cram, Jean-Marie Lehn, and Charles J. Pedersen in recognition of their work in this area. The development of selective "host–guest" complexes in particular, in which a host molecule recognizes and selectively binds a certain guest, was cited as an important contribution.

Concepts

A ribosome is a biological machine that uses protein dynamics on nanoscales.

Molecular self-assembly

Molecular self-assembly is the construction of systems without guidance or management from an outside source (other than to provide a suitable environment). The molecules are directed to assemble through non-covalent interactions. Self-assembly may be subdivided into intermolecular self-assembly (to form a supramolecular assembly), and intramolecular self-assembly (or folding as demonstrated by foldamers and polypeptides). Molecular self-assembly also allows the construction of larger structures such as micelles, membranes, vesicles, liquid crystals, and is important to crystal engineering.

Molecular recognition and complexation

Molecular recognition is the specific binding of a guest molecule to a complementary host molecule to form a host–guest complex. Often, the definition of which species is the "host" and which is the "guest" is arbitrary. The molecules are able to identify each other using non-covalent interactions. Key applications of this field are the construction of molecular sensors and catalysis.

Template-directed synthesis

Molecular recognition and self-assembly may be used with reactive species in order to pre-organize a system for a chemical reaction (to form one or more covalent bonds). It may be considered a special case of supramolecular catalysis. Non-covalent bonds between the reactants and a "template" hold the reactive sites of the reactants close together, facilitating the desired chemistry. This technique is particularly useful for situations where the desired reaction conformation is thermodynamically or kinetically unlikely, such as in the preparation of large macrocycles. This pre-organization also serves purposes such as minimizing side reactions, lowering the activation energy of the reaction, and producing desired stereochemistry. After the reaction has taken place, the template may remain in place, be forcibly removed, or may be "automatically" decomplexed on account of the different recognition properties of the reaction product. The template may be as simple as a single metal ion or may be extremely complex.

Mechanically interlocked molecular architectures

Mechanically interlocked molecular architectures consist of molecules that are linked only as a consequence of their topology. Some non-covalent interactions may exist between the different components (often those that were used in the construction of the system), but covalent bonds do not. Supramolecular chemistry, and template-directed synthesis in particular, is key to the efficient synthesis of the compounds. Examples of mechanically interlocked molecular architectures include catenanes, rotaxanes, molecular knots, molecular Borromean rings, 2D [c2]daisy chain polymer and ravels.

Dynamic covalent chemistry

In dynamic covalent chemistry covalent bonds are broken and formed in a reversible reaction under thermodynamic control. While covalent bonds are key to the process, the system is directed by non-covalent forces to form the lowest energy structures.

Biomimetics

Many synthetic supramolecular systems are designed to copy functions of biological systems. These biomimetic architectures can be used to learn about both the biological model and the synthetic implementation. Examples include photoelectrochemical systems, catalytic systems, protein design and self-replication.

Imprinting

Molecular imprinting describes a process by which a host is constructed from small molecules using a suitable molecular species as a template. After construction, the template is removed leaving only the host. The template for host construction may be subtly different from the guest that the finished host binds to. In its simplest form, imprinting uses only steric interactions, but more complex systems also incorporate hydrogen bonding and other interactions to improve binding strength and specificity.

Molecular machinery

Molecular machines are molecules or molecular assemblies that can perform functions such as linear or rotational movement, switching, and entrapment. These devices exist at the boundary between supramolecular chemistry and nanotechnology, and prototypes have been demonstrated using supramolecular concepts. Jean-Pierre Sauvage, Sir J. Fraser Stoddart and Bernard L. Feringa shared the 2016 Nobel Prize in Chemistry for the 'design and synthesis of molecular machines'.

Building blocks

Supramolecular systems are rarely designed from first principles. Rather, chemists have a range of well-studied structural and functional building blocks that they are able to use to build up larger functional architectures. Many of these exist as whole families of similar units, from which the analog with the exact desired properties can be chosen.

Synthetic recognition motifs

• The pi-pi charge-transfer interactions of bipyridinium with dioxyarenes or diaminoarenes have been used extensively for the construction of mechanically interlocked systems and in crystal engineering.
• The use of crown ether binding with metal or ammonium cations is ubiquitous in supramolecular chemistry.
• The formation of carboxylic acid dimers and other simple hydrogen bonding interactions.
• The complexation of bipyridines or terpyridines with ruthenium, silver or other metal ions is of great utility in the construction of complex architectures of many individual molecules.
• Anion complexation provides a means of linking modules.

Macrocycles

Macrocycles are a traditional component in supramolecular chemistry. The macrocyclic effect enhances otherwise weak interactions. Cyclodextrins, calixarenes, cucurbiturils, and crown ethers allow the incorporation of alkali metal cations. More complex, 3-dimenesion receptors include cyclophanes, and cryptands. Supramolecular metallocycles and metallacrowns are related components. Common metallocycle shapes in these types of applications include triangles, squares, and pentagons, each bearing functional groups that connect the pieces via "self-assembly."

Structural units

Many supramolecular systems require their components to have suitable spacing and conformations relative to each other, and therefore easily employed structural units are required.

• Commonly used spacers and connecting groups include polyether chains, biphenyls and terphenyls, and simple alkyl chains. The chemistry for creating and connecting these units is very well understood.
• nanoparticles, nanorods, fullerenes and dendrimers offer nanometer-sized structure and encapsulation units.
• Surfaces can be used as scaffolds for the construction of complex systems and also for interfacing electrochemical systems with electrodes. Regular surfaces can be used for the construction of self-assembled monolayers and multilayers.
• The understanding of intermolecular interactions in solids has undergone a major renaissance via inputs from different experimental and computational methods in the last decade. This includes high-pressure studies in solids and "in situ" crystallization of compounds which are liquids at room temperature along with the use of electron density analysis, crystal structure prediction and DFT calculations in solid state to enable a quantitative understanding of the nature, energetics and topological properties associated with such interactions in crystals.

Photo-chemically and electro-chemically active units

• Porphyrins, and phthalocyanines have highly tunable photochemical and electrochemical activity as well as the potential to form complexes.
• Photochromic and photoisomerizable groups can change their shapes and properties, including binding properties, upon exposure to light.
• Tetrathiafulvalene (TTF) and quinones have multiple stable oxidation states, and therefore can be used in redox reactions and electrochemistry.
• Other units, such as benzidine derivatives, viologens, and fullerenes, are useful in supramolecular electrochemical devices.

Biologically-derived units

• The extremely strong complexation between avidin and biotin is instrumental in blood clotting, and has been used as the recognition motif to construct synthetic systems.
• The binding of enzymes with their cofactors has been used as a route to produce modified enzymes, electrically contacted enzymes, and even photoswitchable enzymes.
• DNA has been used both as a structural and as a functional unit in synthetic supramolecular systems.

Applications

Supramolecular chemistry per se has found few applications, but underpins some useful phenomena.

Catalysis

Main article: Supramolecular catalysis

The concepts of supramolecular chemistry can inspire the design of catalyst. Non-covalent interactions influence the binding reactants.

Medicine

Design based on supramolecular chemistry has inspired the design of functional biomaterials and therapeutics. Supramolecular biomaterials afford a number of modular and generalizable platforms with tunable mechanical, chemical and biological properties. These include systems based on supramolecular assembly of peptides, host–guest macrocycles, high-affinity hydrogen bonding, and metal–ligand interactions.

A supramolecular approach has been used extensively to create artificial ion channels for the transport of sodium and potassium ions into and out of cells.

Supramolecular interactions influence drug-target binding. In the area of drug delivery, supramolecular chemistry could provide encapsulation and targeted release mechanisms. In addition, supramolecular systems have been designed to disrupt protein–protein interactions that are important to cellular function.

Sensors

Supramolecular interactions have been proposed to detect analytes.

Volcanic winter

From Wikipedia, the free encyclopedia

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

The conversion of sulfur dioxide to sulfuric acid, which condenses rapidly in the stratosphere to form fine sulfate aerosols.

A volcanic winter is a reduction in global temperatures caused by droplets of sulfuric acid obscuring the Sun and raising Earth's albedo (increasing the reflection of solar radiation) after a large, sulfur-rich, particularly explosive volcanic eruption. Climate effects are primarily dependent upon the amount of injection of SO₂ and H₂S into the stratosphere where they react with OH and H₂O to form H₂SO₄ on a timescale of a week, and the resulting H₂SO₄ aerosols produce the dominant radiative effect. Volcanic stratospheric aerosols cool the surface by reflecting solar radiation and warm the stratosphere by absorbing terrestrial radiation for several years. Moreover, the cooling trend can be further extended by atmosphere–ice–ocean feedback mechanisms. These feedbacks can continue to maintain the cool climate long after the volcanic aerosols have dissipated.

Physical process

An explosive volcanic eruption releases magma materials in the form of volcanic ash and gases into the atmosphere. While most volcanic ash settles to the ground within a few weeks after the eruption, impacting only the local area for a short duration, the emitted SO₂ can lead to the formation of H₂SO₄ aerosols in the stratosphere. These aerosols can circle the hemisphere of the eruption source in a matter of weeks and persist with an e-folding decay time of about a year. As a result, they have a radiative impact that can last for several years.

The subsequent dispersal of a volcanic cloud in the stratosphere and its impact on climate are strongly influenced by several factors, including the season of the eruption, the latitude of the source volcano, and the injection height. If the SO₂ injection height remains confined to the troposphere, the resulting H₂SO₄ aerosols have a residence time of only a few days due to efficient removal through precipitation. The lifetime of H₂SO₄ aerosols resulting from extratropical eruptions is shorter compared to those from tropical eruptions, due to a longer transport path from the tropics to removal across the mid- or high-latitude tropopause, but extratropical eruptions strengthens the hemispheric climate impact by confining the aerosol to a single hemisphere. Injections in the winter are also much less radiatively efficient than injections during the summer for high-latitude volcanic eruptions, when the removal of stratospheric aerosols in polar regions is enhanced.

The sulfate aerosol interacts strongly with solar radiation through scattering, giving rise to remarkable atmospheric optical phenomena in the stratosphere. These phenomena include solar dimming, coronae or Bishop's rings, peculiar twilight coloration, and dark total lunar eclipses. Historical records that documented these atmospheric events are indications of volcanic winters and date back to periods preceding the Common Era.

Surface temperature observations following historic eruptions show that there is no correlation between eruption size, as represented by the VEI or eruption volume, and the severity of the climate cooling. This is because eruption size does not correlate with the amount of SO₂ emitted.

Long-term positive feedback

It has been proposed that the cooling effects of volcanic eruptions can extend beyond the initial several years, lasting for decades to possibly even millennia. This prolonged impact is hypothesized to be a result of positive feedback mechanisms involving ice and ocean dynamics, even after the H₂SO₄ aerosols have dissipated.

During the first few years following a volcanic eruption, the presence of H₂SO₄ aerosols can induce a significant cooling effect. This cooling can lead to a widespread lowering of snowline, enabling the rapid expansion of sea ice, ice caps and continental glacier. As a result, ocean temperatures decrease, and surface albedo increases, further reinforcing the expansion of sea ice, ice caps, and glacier. These processes create a strong positive feedback loop, allowing the cooling trend to persist over centennial-scale or even longer periods of time.

It has been proposed that a cluster of closely spaced, large volcanic eruptions triggered or amplified the Little Ice Age, Late Antique Little Ice Age, stadials, Younger Dryas, Heinrich events, and Dansgaard-Oeschger events through the atmosphere-ice-ocean positive feedbacks.

Weathering effects

Timescales of various volcanic cooling mechanisms on climate

The weathering of a sufficiently large volume of rapidly erupted volcanic materials has been proposed as an important factor in Earth's silicate weathering cycle, which operates on a timescale of tens of millions of years. During this process, weathered silicate minerals react with carbon dioxide and water, resulting in the formation of magnesium carbonate and calcium carbonate. These carbonates are then removed from the atmosphere and sequestrated on the ocean floor. The eruption of a large volume of volcanic materials can enhance weathering processes, thereby lowering atmospheric CO₂ levels and contributing to global temperature reduction.

The rapid emplacement of mafic large igneous provinces has the potential to cause a swift decline in atmospheric CO₂ content, leading to a multi-million-year-long icehouse climate. A notable example is the Sturtian glaciation, which is considered the most severe and widespread known glacial event in Earth's history. This glaciation is believed to have been caused by the weathering of erupted Franklin Large Igneous Province.

Past volcanic coolings

Tree-ring-based temperature reconstructions, historical records of dust veils, and ice cores studies have confirmed that some of the coldest years during the last five millennia were directly caused by massive volcanic injections of SO₂.

Northern Hemisphere coolings are observed following major volcanic eruptions, and temperatures are reconstructed from tree-ring data.

Hemispheric temperature anomalies resulting from volcanic eruptions have primarily been reconstructed based on tree-ring data for the past two millennia. For earlier periods in the Holocene, the identification of frost rings that coincide with large ice core sulfate spikes serves as an indicator of severe volcanic winters. The quantification of volcanic coolings further back in time during the Last Glacial Period is made possible by annually resolved δ¹⁸O records. This is a non-exhaustive compilation of notable and consequential coolings that have been definitively attributed to volcanic aerosols, although the source volcanos of the aerosols are rarely identified.

Northern Hemisphere cooling episodes definitively attributed to volcanic eruptions

Cooling episode (CE/BCE)	Volcanic eruptions	N.H. peak temperature anomaly	Notes
1991–1993	1991 eruption of Mount Pinatubo	−0.5 K
1883–1886	1883 eruption of Krakatoa	−0.3 K
1809–1820	1808 mystery eruptions, 1815 eruption of Mount Tambora	−1.7 K	Year Without a Summer
1453–1460	1452 N.H. mystery eruption, 1458 S.H. mystery eruption	−1.2 K	The attribution of the 1458 eruption to Kuwae Caldera remains controversial.
1258–1260	1257 Samalas eruption	−1.3 K	The single largest sulfur injection of the Common Era.
536–546	535 N.H. mystery eruptions, 540 tropical mystery eruption	−1.4 K	The first phase of Late Antique Little Ice Age.
−43–41	Okmok II	−2–3 K

During the Last Glacial Period, volcanic coolings comparable to the largest volcanic coolings during the Common Era (e.g. Tambora, Samalas) are inferred based on the magnitudes of δ¹⁸O anomalies. In particular, in the period 12,000–32,000 years ago, the peak δ¹⁸O cooling anomaly of the eruptions exceeds the anomaly after the largest eruptions in the Common Era. One Last Glacial Period eruption that have gained significant attention is the eruption of the Youngest Toba Tuff (YTT), which has sparked vigorous debates regarding its climate effects.

Youngest Toba Tuff

See also: Toba catastrophe theory

The eruption of YTT from Toba Caldera, 74,000 years ago, is regarded as the largest known Quaternary eruption and two orders of magnitude greater than the magma volume of the largest historical eruption, Tambora. The exceptional magnitude of this eruption has prompted sustained debate as to its global and regional impact on climate.

Sulfate concentration and isotope measurements from polar ice cores taken around the time of 74,000 years BP have identified four atmospheric aerosol events that could potentially be attributed to YTT. The calculated stratospheric sulfate loadings for these four events range from 219 to 535 million tonnes, which is 1 to 3 times greater than that of the Samalas eruption in 1257 CE. Global climate models simulate peak global mean cooling of 2.3 to 4.1 K for this amount of erupted sulfate aerosols, and complete temperature recovery does not occur within 10 years.

Empirical evidence for cooling induced by YTT, however, is mixed. YTT coincides with the onset of Greenland Stadial 20 (GS-20), which is characterized by a 1,500-year cooling period. GS-20 is considered the most isotopically extreme and coldest stadial, as well as having the weakest Asian monsoon, in the last 100,000 years. This timing has led some to speculate on the relation between YTT and GS-20. The stratigraphic position of YTT in relation to the GS-20 transition suggests that the stadial would have occurred without YTT, as the cooling was already underway. There is the possibility that YTT contributed to the extremity of GS-20. The South China Sea shows a 1 K cooling over 1,000 years following the deposition of YTT, while the Arabian Sea shows no discernible impact. In India and the Bay of Bengal, initial cooling and prolonged desiccation are observed above the YTT ash layer, but it is argued that these environmental changes were already occurring prior to YTT. Lake Malawi sediments do not provide evidence supporting a volcanic winter within a few years after the eruption of YTT, but the resolution of the sediments is questioned due to sediment mixing. Directly above the YTT layer in Lake Malawi, there is evidence of a 2,000-year-long megadrought and cooling period. Greenland ice cores identify a 110-year period of accelerated cooling immediately following what is likely the YTT aerosol event.

Sturtian glaciation

The enhanced weathering of continental flood basalts, which erupted just prior to the onset of the Sturtian glaciation at 717 million years ago, is recognized as the trigger for the most severe glaciation in Earth's history. During this period, Earth's surface temperatures dropped below the freezing point of water everywhere, and ice rapidly advanced from low latitudes to the equator, covering a worldwide extent. This glaciation lasted almost 60 million years, from 717 to 659 million years ago.

Geochronology dates the rapid emplacement of 5,000,000 km² (1,900,000 mi²) Franklin large igneous province just 1 million year before the onset of Sturtian glaciation. Multiple large igneous provinces on the scale of 1,000,000 km² (390,000 mi²) were also emplaced on Rodinia between 850 and 720 million years ago. Weathering of massive amount of fresh mafic materials initiated runaway cooling and ice-albedo feedback after 1 million year. Chemical isotopic compositions show a massive flux of weathered freshly erupted materials entering the ocean, coinciding with the eruptions of large igneous provinces.  Simulations demonstrate that the increased weatherability led to drop in atmospheric CO₂ of the order of 1,320 ppm and an 8 K cooling of global temperatures, triggering the most extraordinary episode of climate change in the geologic record.

Effects on life

Further information: 1991 eruption of Mount Pinatubo § Effects on agriculture

The supervolcano caldera Lake Toba

Population bottlenecks—a sharp decrease in the population of a species population—have been attributed to volcanic winters by some researchers. Such events may diminish populations to "levels low enough for evolutionary changes, which occur much faster in small populations, to produce rapid population differentiation". With the Lake Toba bottleneck, many species showed the massive genetic effects of this decrease in the gene pool; Toba may have reduced the human population to between 15,000 and 40,000, or even fewer.

Electromagnetic wave equation

From Wikipedia, the free encyclopedia

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

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

$${\displaystyle \begin{array}{rl}\left(v_{\mathrm{p}\mathrm{h}}^2{\mathrm{\nabla }}^2-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)\mathbf{E}& =\mathbf{0}\\ \left(v_{\mathrm{p}\mathrm{h}}^2{\mathrm{\nabla }}^2-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)\mathbf{B}& =\mathbf{0}\end{array}}$$

where

$${\displaystyle v_{\mathrm{p}\mathrm{h}}=\frac{1}{\sqrt{\mu \epsilon }}}$$

is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇² is the Laplace operator. In a vacuum, v_ph = c₀ = 299792458 m/s, a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In most older literature, B is called the magnetic flux density or magnetic induction. The following equations$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\end{array}}$$predicate that any electromagnetic wave must be a transverse wave, where the electric field E and the magnetic field B are both perpendicular to the direction of wave propagation.

The origin of the electromagnetic wave equation

A postcard from Maxwell to Peter Tait.

In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, James Clerk Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In Part VI of his 1864 paper titled Electromagnetic Theory of Light, Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are:

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\times \mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\\ \mathrm{\nabla }\times \mathbf{B}& ={\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\end{array}}$$

These are the general Maxwell's equations specialized to the case with charge and current both set to zero. Taking the curl of the curl equations gives:

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{E}\right)& =\mathrm{\nabla }\times \left(-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\right)=-\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\times \mathbf{B}\right)=-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}\\ \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{B}\right)& =\mathrm{\nabla }\times \left({\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)={\mu }_0{\epsilon }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\times \mathbf{E}\right)=-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}\end{array}}$$

We can use the vector identity

$${\displaystyle \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{V}\right)=\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{V}\right)-{\mathrm{\nabla }}^2\mathbf{V}}$$

where V is any vector function of space. And

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{V}=\mathrm{\nabla }\cdot \left(\mathrm{\nabla }\mathbf{V}\right)}$$

where ∇V is a dyadic which when operated on by the divergence operator ∇ ⋅ yields a vector. Since

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\end{array}}$$

then the first term on the right in the identity vanishes and we obtain the wave equations:

$${\displaystyle \begin{array}{rl}\frac{1}{c_0^2}\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{E}& =\mathbf{0}\\ \frac{1}{c_0^2}\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{B}& =\mathbf{0}\end{array}}$$

where

$${\displaystyle c_0=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=2.99792458\times {10}^8\text{m/s}}$$

is the speed of light in free space.

Covariant form of the homogeneous wave equation

Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of Special Relativity.

These relativistic equations can be written in contravariant form as

$${\displaystyle ◻A^{\mu }=0}$$

where the electromagnetic four-potential is

$${\displaystyle A^{\mu }=\left(\frac{\varphi }{c},\mathbf{A}\right)}$$

with the Lorenz gauge condition:

$${\displaystyle {\mathrm{\partial }}_{\mu }{A}^{\mu }=0,}$$

and where

$${\displaystyle ◻={\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}}$$

is the d'Alembert operator.

Homogeneous wave equation in curved spacetime

Main article: Maxwell's equations in curved spacetime

The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

$${\displaystyle -{A^{\alpha ;\beta }}_{;\beta }+{R^{\alpha }}_{\beta }{A}^{\beta }=0}$$

where ${\displaystyle {R^{\alpha }}_{\beta }}$ is the Ricci curvature tensor and the semicolon indicates covariant differentiation.

The generalization of the Lorenz gauge condition in curved spacetime is assumed:

$${\displaystyle {A^{\mu }}_{;\mu }=0.}$$

Inhomogeneous electromagnetic wave equation

Main article: Inhomogeneous electromagnetic wave equation

Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.

Solutions to the homogeneous electromagnetic wave equation

Main article: Wave equation

The general solution to the electromagnetic wave equation is a linear superposition of waves of the form

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r},t)& =g(\varphi (\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot \mathbf{r})\\ \mathbf{B}(\mathbf{r},t)& =g(\varphi (\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot \mathbf{r})\end{array}}$$

for virtually any well-behaved function g of dimensionless argument φ, where ω is the angular frequency (in radians per second), and k = (k_x, k_y, k_z) is the wave vector (in radians per meter).

Although the function g can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, g cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.

In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation:

$${\displaystyle k=|\mathbf{k}|=\frac{\omega }{c}=\frac{2\pi }{\lambda }}$$

where k is the wavenumber and λ is the wavelength. The variable c can only be used in this equation when the electromagnetic wave is in a vacuum.

Monochromatic, sinusoidal steady-state

The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:

$${\displaystyle \mathbf{E}(\mathbf{r},t)=\mathrm{\Re }\left\{\mathbf{E}(\mathbf{r})e^{i\omega t}\right\}}$$

where

• i is the imaginary unit,
• ω = 2π f  is the angular frequency in radians per second,
•  f  is the frequency in hertz, and
• ${\displaystyle e^{i\omega t}=\cos (\omega t)+i\sin (\omega t)}$ is Euler's formula.

Plane wave solutions

Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation

Consider a plane defined by a unit normal vector

$${\displaystyle \mathbf{n}=\frac{\mathbf{k}}{k}.}$$

Then planar traveling wave solutions of the wave equations are

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r})& ={\mathbf{E}}_0{e}^{-i\mathbf{k}\cdot \mathbf{r}}\\ \mathbf{B}(\mathbf{r})& ={\mathbf{B}}_0{e}^{-i\mathbf{k}\cdot \mathbf{r}}\end{array}}$$

where r = (x, y, z) is the position vector (in meters).

These solutions represent planar waves traveling in the direction of the normal vector n. If we define the z direction as the direction of n, and the x direction as the direction of E, then the magnetic field lies in the y direction and is related to the electric field by the relation

$${\displaystyle c^2\frac{\mathrm{\partial }B}{\mathrm{\partial }z}=\frac{\mathrm{\partial }E}{\mathrm{\partial }t}.}$$

Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.

This solution is the linearly polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.

Spectral decomposition

Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r},t)& ={\mathbf{E}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\\ \mathbf{B}(\mathbf{r},t)& ={\mathbf{B}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\end{array}}$$

where

• t is time (in seconds),
• ω is the angular frequency (in radians per second),
• k = (k_x, k_y, k_z) is the wave vector (in radians per meter), and
• ${\displaystyle {\varphi }_0}$ is the phase angle (in radians).

The wave vector is related to the angular frequency by

$${\displaystyle k=|\mathbf{k}|=\frac{\omega }{c}=\frac{2\pi }{\lambda }}$$

where k is the wavenumber and λ is the wavelength.

The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.

Multipole expansion

Assuming monochromatic fields varying in time as ${\displaystyle e^{-i\omega t}}$, if one uses Maxwell's Equations to eliminate B, the electromagnetic wave equation reduces to the Helmholtz equation for E:

$${\displaystyle ({\mathrm{\nabla }}^2+k^2)\mathbf{E}=0,\mathbf{B}=-\frac{i}{k}\mathrm{\nabla }\times \mathbf{E},}$$

with k = ω/c as given above. Alternatively, one can eliminate E in favor of B to obtain:

$${\displaystyle ({\mathrm{\nabla }}^2+k^2)\mathbf{B}=0,\mathbf{E}=-\frac{i}{k}\mathrm{\nabla }\times \mathbf{B}.}$$

A generic electromagnetic field with frequency ω can be written as a sum of solutions to these two equations. The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. However, applying this expansion to each vector component of E or B will give solutions that are not generically divergence-free (∇ ⋅ E = ∇ ⋅ B = 0), and therefore require additional restrictions on the coefficients.

The multipole expansion circumvents this difficulty by expanding not E or B, but r ⋅ E or r ⋅ B into spherical harmonics. These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇² (r ⋅ F) = r ⋅ (∇² F). The resulting expressions for a generic electromagnetic field are:

$${\displaystyle \begin{array}{rl}\mathbf{E}& =e^{-i\omega t}\sum _{l,m}\sqrt{l(l+1)}\left[a_E(l,m){\mathbf{E}}_{l,m}^{(E)}+a_M(l,m){\mathbf{E}}_{l,m}^{(M)}\right]\\ \mathbf{B}& =e^{-i\omega t}\sum _{l,m}\sqrt{l(l+1)}\left[a_E(l,m){\mathbf{B}}_{l,m}^{(E)}+a_M(l,m){\mathbf{B}}_{l,m}^{(M)}\right],\end{array}}$$

where ${\displaystyle {\mathbf{E}}_{l,m}^{(E)}}$ and ${\displaystyle {\mathbf{B}}_{l,m}^{(E)}}$ are the electric multipole fields of order (l, m), and ${\displaystyle {\mathbf{E}}_{l,m}^{(M)}}$ and ${\displaystyle {\mathbf{B}}_{l,m}^{(M)}}$ are the corresponding magnetic multipole fields, and a_E(l, m) and a_M(l, m) are the coefficients of the expansion. The multipole fields are given by

$${\displaystyle \begin{array}{rl}{\mathbf{B}}_{l,m}^{(E)}& =\sqrt{l(l+1)}\left[B_l^{(1)}{h}_l^{(1)}(kr)+B_l^{(2)}{h}_l^{(2)}(kr)\right]{\mathbf{\Phi }}_{l,m}\\ {\mathbf{E}}_{l,m}^{(E)}& =\frac{i}{k}\mathrm{\nabla }\times {\mathbf{B}}_{l,m}^{(E)}\\ {\mathbf{E}}_{l,m}^{(M)}& =\sqrt{l(l+1)}\left[E_l^{(1)}{h}_l^{(1)}(kr)+E_l^{(2)}{h}_l^{(2)}(kr)\right]{\mathbf{\Phi }}_{l,m}\\ {\mathbf{B}}_{l,m}^{(M)}& =-\frac{i}{k}\mathrm{\nabla }\times {\mathbf{E}}_{l,m}^{(M)},\end{array}}$$

where h_l^(1,2)(x) are the spherical Hankel functions, E_l^(1,2) and B_l^(1,2) are determined by boundary conditions, and

$${\displaystyle {\mathbf{\Phi }}_{l,m}=\frac{1}{\sqrt{l(l+1)}}(\mathbf{r}\times \mathrm{\nabla })Y_{l,m}}$$

are vector spherical harmonics normalized so that

$${\displaystyle \int {\mathbf{\Phi }}_{l,m}^{\ast }\cdot {\mathbf{\Phi }}_{l^{\prime },m^{\prime }}d\mathrm{\Omega }={\delta }_{l,l^{\prime }}{\delta }_{m,m^{\prime }}.}$$

The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae radiation patterns, or nuclear gamma decay. In these applications, one is often interested in the power radiated in the far-field. In this regions, the E and B fields asymptotically approach

$${\displaystyle \begin{array}{rl}\mathbf{B}& \approx \frac{e^{i(kr-\omega t)}}{kr}\sum _{l,m}(-i{)}^{l+1}\left[a_E(l,m){\mathbf{\Phi }}_{l,m}+a_M(l,m)\hat{\mathbf{r}}\times {\mathbf{\Phi }}_{l,m}\right]\\ \mathbf{E}& \approx \mathbf{B}\times \hat{\mathbf{r}}.\end{array}}$$

The angular distribution of the time-averaged radiated power is then given by

$${\displaystyle \frac{dP}{d\mathrm{\Omega }}\approx \frac{1}{2k^2}{\left|\sum _{l,m}(-i{)}^{l+1}\left[a_E(l,m){\mathbf{\Phi }}_{l,m}\times \hat{\mathbf{r}}+a_M(l,m){\mathbf{\Phi }}_{l,m}\right]\right|}^2.}$$