Environmental impact of electricity generation

From Wikipedia, the free encyclopedia

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

Greenhouse gas emissions per energy source.

Coal power is being phased out because of its pollution - such as Navajo Generating Station

Electric power systems consist of generation plants of different energy sources, transmission networks, and distribution lines. Each of these components can have environmental impacts at multiple stages of their development and use including in their construction, during the generation of electricity, and in their decommissioning and disposal. These impacts can be split into operational impacts (fuel sourcing, global atmospheric and localized pollution) and construction impacts (manufacturing, installation, decommissioning, and disposal). All forms of electricity generation have some form of environmental impact, but coal-fired power is the dirtiest. This page is organized by energy source and includes impacts such as water usage, emissions, local pollution, and wildlife displacement.

Greenhouse gas emissions

This section is an excerpt from Life-cycle greenhouse gas emissions of energy sources.[edit]

Greenhouse gas emissions are one of the environmental impacts of electricity generation. Measurement of life-cycle greenhouse gas emissions involves calculating the global warming potential of energy sources through life-cycle assessment. These are usually sources of only electrical energy but sometimes sources of heat are evaluated. The findings are presented in units of global warming potential per unit of electrical energy generated by that source. The scale uses the global warming potential unit, the carbon dioxide equivalent (CO₂e), and the unit of electrical energy, the kilowatt hour (kWh). The goal of such assessments is to cover the full life of the source, from material and fuel mining through construction to operation and waste management.

In 2014, the Intergovernmental Panel on Climate Change harmonized the carbon dioxide equivalent (CO₂e) findings of the major electricity generating sources in use worldwide. This was done by analyzing the findings of hundreds of individual scientific papers assessing each energy source. Coal is by far the worst emitter, followed by natural gas, with solar, wind and nuclear all low-carbon. Hydropower, biomass, geothermal and ocean power may generally be low-carbon, but poor design or other factors could result in higher emissions from individual power stations.

For all technologies, advances in efficiency, and therefore reductions in CO₂e since the time of publication, have not been included. For example, the total life cycle emissions from wind power may have lessened since publication. Similarly, due to the time frame over which the studies were conducted, nuclear Generation II reactor's CO₂e results are presented and not the global warming potential of Generation III reactors. Other limitations of the data include: a) missing life cycle phases, and, b) uncertainty as to where to define the cut-off point in the global warming potential of an energy source. The latter is important in assessing a combined electrical grid in the real world, rather than the established practice of simply assessing the energy source in isolation.

Water usage

Water usage is one of the main environmental impacts of electricity generation. All thermal power plants (coal, natural gas, nuclear, geothermal, and biomass) use water as a cooling fluid to drive the thermodynamic cycles that allow electricity to be extracted from heat energy. Solar uses water for cleaning equipment, while hydroelectricity has water usage from evaporation from the reservoirs. The amount of water usage is often of great concern for electricity generating systems as populations increase and droughts become a concern. In addition, changes in water resources may impact the reliability of electricity generation.

Discussions of water usage of electricity generation distinguish between water withdrawal and water consumption. According to the United States Geological Survey, "withdrawal" is defined as the amount of water removed from the ground or diverted from a water source for use, while "consumption" refers to the amount of water that is evaporated, transpired, incorporated into products or crops, or otherwise removed from the immediate water environment. Both water withdrawal and consumption are important environmental impacts to evaluate.

General numbers for fresh water usage of different power sources are shown below.

	Water Consumption (gal/MW-h)
Power source	Low case	Medium/average case	High case
Nuclear power	100 (once-through cooling)	270 once-through, 650 (tower and pond)	845 (cooling tower)
Coal	58	500	1,100 (cooling tower, generic combustion)
Natural gas	100 (once-through cycle)	800 (steam-cycle, cooling towers)	1,170 (steam-cycle with cooling towers)
Hydroelectricity	1,430	4,491	18,000
Solar thermal	53 (dry cooling)	800	1,060 (Trough)
Geothermal	1,800		4,000
Biomass	300		480
Solar photovoltaic	0	26	33
Wind power	0	0	1

Steam-cycle plants (nuclear, coal, NG, solar thermal) require a great deal of water for cooling, to remove the heat at the steam condensers. The amount of water needed relative to plant output will be reduced with increasing boiler temperatures. Coal- and gas-fired boilers can produce high steam temperatures and so are more efficient, and require less cooling water relative to output. Nuclear boilers are limited in steam temperature by material constraints, and solar thermal is limited by concentration of the energy source.

Thermal cycle plants near the ocean have the option of using seawater. Such a site will not have cooling towers and will be much less limited by environmental concerns of the discharge temperature since dumping heat will have very little effect on water temperatures. This will also not deplete the water available for other uses. Nuclear power in Japan for instance, uses no cooling towers at all because all plants are located on the coast. If dry cooling systems are used, significant water from the water table will not be used. Other, more novel, cooling solutions exist, such as sewage cooling at the Palo Verde Nuclear Generating Station.

Hydroelectricity's main cause of water usage is both evaporation and seepage into the water table.

While water usage is still a major necessity for the production of electricity, since 2015 the use of water has decreased. In 2015 the total water withdrawals from thermoelectric power plants was just over 60 trillion gallons, but in 2020 it decreased to just under 50 trillion gallons. The water use has gone down because of the increase in the use of renewable energy sources.

80% of the decrease in water use is due to the use of natural gas and the use of renewables instead of just producing energy through coal-fired plants. And the other 20% of the decrease in water use comes from the implementation of closed loop recirculating and hybrid cooling systems rather than once through cooling systems. Once through cooling systems has an excessive amount of water withdrawals, so the water is only used once then released. While the closed loop water is reused several times so the water withdrawals is much lower.

Fossil fuels

See also: Environmental impact of the energy industry § Fossil fuel use, Environmental impact of the oil shale industry, and Health and environmental impact of the coal industry

Most electricity today is generated by burning fossil fuels and producing steam which is then used to drive a steam turbine that, in turn, drives an electrical generator.

More serious are concerns about the emissions that result from fossil fuel burning. Fossil fuels constitute a significant repository of carbon buried deep underground. Burning them results in the conversion of this carbon to carbon dioxide, which is then released into the atmosphere. The estimated CO₂ emission from the world's electrical power industry is 10 billion tonnes yearly. This results in an increase in the Earth's levels of atmospheric carbon dioxide, which enhances the greenhouse effect and contributes to global warming.

Coal power

Depending on the particular fossil fuel and the method of burning, other emissions may be produced as well. Ozone, sulfur dioxide, NO₂ and other gases are often released, as well as particulate matter. Sulfur and nitrogen oxides contribute to smog and acid rain. In the past, plant owners addressed this problem by building very tall flue-gas stacks, so that the pollutants would be diluted in the atmosphere. While this helps reduce local contamination, it does not help at all with global issues.

Fossil fuels, particularly coal, also contain dilute radioactive material, and burning them in very large quantities releases this material into the environment, leading to low levels of local and global radioactive contamination, the levels of which are, ironically, higher than a nuclear power station as their radioactive contaminants are controlled and stored.

Coal also contains traces of toxic heavy elements such as mercury, arsenic and others. Mercury vaporized in a power plant's boiler may stay suspended in the atmosphere and circulate around the world. While a substantial inventory of mercury exists in the environment, as other man-made emissions of mercury become better controlled, power plant emissions become a significant fraction of the remaining emissions. Power plant emissions of mercury in the United States are thought to be about 50 tons per year in 2003, and several hundred tons per year in China. Power plant designers can fit equipment to power stations to reduce emissions.

Coal mining practices in the United States have also included strip mining and removing mountain tops. Mill tailings are left out bare and have been leached into local rivers and resulted in most or all of the rivers in coal producing areas to run red year round with sulfuric acid that kills all life in the rivers.

Fossil gas power

In 2022 the IEA said that greenhouse gas emissions from gas-fired power plants had increased by nearly 3% the previous year and that more efforts were needed to reduce them.

As well as greenhouse gases, these power plants emit nitrogen oxides (NOx) but this is less dangerous than NOx from gas appliances in houses.

The efficiency of gas-fired power plants can be improved by co-generation and geothermal (combined heat and power) methods. Process steam can be extracted from steam turbines. Waste heat produced by thermal generating stations can be used for space heating of nearby buildings. By combining electric power production and heating, less fuel is consumed, thereby reducing the environmental effects compared with separate heat and power systems.

Fuel oil and diesel

Dirty oil is burnt in power plants in a few oil producing countries such as Iran. Diesel is often used in backup generators, which can cause air pollution.

Switching from fuels to electricity

Clean energy is mostly generated in the form of electricity, such as renewable energy or nuclear power. Switching to these energy sources requires that end uses, such as transport and heating, be electrified for the world's energy systems to be sustainable. Recent work has shown that in the U.S. and Canada the use of heat pumps (HP) is economic if powered with solar photovoltaic (PV) devices to offset propane heating in rural areas and natural gas heating in cities. A 2023 study investigated: (1) a residential natural gas-based heating system and grid electricity, (2) a residential natural gas-based heating system with PV to serve the electric load, (3) a residential HP system with grid electricity, and (4) a residential HP+PV system. It found that under typical inflation conditions, the lifecycle cost of natural gas and reversible, air-source heat pumps are nearly identical, which in part explains why heat pump sales have surpassed gas furnace sales in the U.S. for the first time during a period of high inflation. With higher rates of inflation or lower PV capital costs, PV becomes a hedge against rising prices and encourages the adoption of heat pumps by also locking in both electricity and heating cost growth. The study concludes: "The real internal rate of return for such prosumer technologies is 20x greater than a long-term certificate of deposit, which demonstrates the additional value PV and HP technologies offer prosumers over comparably secure investment vehicles while making substantive reductions in carbon emissions." This approach can be improved by integrating a thermal battery into the heat pump+solar energy heating system.

It is easier to sustainably produce electricity than it is to sustainably produce liquid fuels. Therefore, adoption of electric vehicles is a way to make transport more sustainable. Hydrogen vehicles may be an option for larger vehicles which have not yet been widely electrified, such as long distance lorries. Many of the techniques needed to lower emissions from shipping and aviation are still early in their development.

A large fraction of the world population cannot afford sufficient cooling for their homes. In addition to air conditioning, which requires electrification and additional power demand, passive building design and urban planning will be needed to ensure cooling needs are met in a sustainable way. Similarly, many households in the developing and developed world suffer from fuel poverty and cannot heat their houses enough. Existing heating practices are often polluting.

A key sustainable solution to heating is electrification (heat pumps, or the less efficient electric heater). The IEA estimates that heat pumps currently provide only 5% of space and water heating requirements globally, but could provide over 90%. Use of ground source heat pumps not only reduces total annual energy loads associated with heating and cooling, it also flattens the electric demand curve by eliminating the extreme summer peak electric supply requirements. However, heat pumps and resistive heating alone will not be sufficient for the electrification of industrial heat. This because in several processes higher temperatures are required which cannot be achieved with these types of equipment. For example, for the production of ethylene via steam cracking temperatures as high as 900 °C are required. Hence, drastically new processes are required. Nevertheless, power-to-heat is expected to be the first step in the electrification of the chemical industry with an expected large-scale implementation by 2025.

Some cities in the United States have started prohibiting gas hookups for new houses, with state laws passed and under consideration to either require electrification or prohibit local requirements. The UK government is experimenting with electrification for home heating to meet its climate goals. Ceramic and Induction heating for cooktops as well as industrial applications (for instance steam crackers) are examples of technologies that can be used to transition away from natural gas.

Nuclear power

See also: Nuclear power debate

Nuclear power activities involving the environment; mining, enrichment, generation and geological disposal.

Nuclear power has various environmental impacts, both positive and negative, including the construction and operation of the plant, the nuclear fuel cycle, and the effects of nuclear accidents. Nuclear power plants do not burn fossil fuels and so do not directly emit carbon dioxide. The carbon dioxide emitted during mining, enrichment, fabrication and transport of fuel is small when compared with the carbon dioxide emitted by fossil fuels of similar energy yield, however, these plants still produce other environmentally damaging wastes. Nuclear energy and renewable energy have reduced environmental costs by decreasing CO₂ emissions resulting from energy consumption.

There is a catastrophic risk potential if containment fails, which in nuclear reactors can be brought about by overheated fuels melting and releasing large quantities of fission products into the environment. In normal operation, nuclear power plants release less radioactive material than coal power plants whose fly ash contains significant amounts of thorium, uranium and their daughter nuclides.

A large nuclear power plant may reject waste heat to a natural body of water; this can result in undesirable increase of the water temperature with adverse effect on aquatic life. Alternatives include cooling towers.

Mining of uranium ore can disrupt the environment around the mine. However, with modern in-situ leaching technology this impact can be reduced compared to "classical" underground or open-pit mining. Disposal of spent nuclear fuel is controversial, with many proposed long-term storage schemes under intense review and criticism. Diversion of fresh- or low-burnup spent fuel to weapons production presents a risk of nuclear proliferation, however all nuclear weapons states derived the material for their first nuclear weapon from (non-power) research reactors or dedicated "production reactors" and/or uranium enrichment. Finally, some parts the structure of the reactor itself becomes radioactive through neutron activation and will require decades of storage before it can be economically dismantled and in turn disposed of as waste. Measures like reducing the cobalt content in steel to decrease the amount of cobalt-60 produced by neutron capture can reduce the amount of radioactive material produced and the radiotoxicity that originates from this material. However, part of the issue is not radiological but regulatory as most countries assume any given object that originates from the "hot" (radioactive) area of a nuclear power plant or a facility in the nuclear fuel cycle is ipso facto radioactive, even if no contamination or neutron irradiation induced radioactivity is detectable.

Renewable energy

Renewable power technologies can have significant environmental benefits. Unlike coal and natural gas, they can generate electricity and fuels without releasing significant quantities of CO₂ and other greenhouse gases that contribute to climate change, however the greenhouse gas savings from a number of biofuels have been found to be much less than originally anticipated, as discussed in the article Indirect land use change impacts of biofuels.

Both solar and wind have been criticized from an aesthetic point of view. However, methods and opportunities exist to deploy these renewable technologies efficiently and unobtrusively: fixed solar collectors can double as noise barriers along highways, and extensive roadway, parking lot, and roof-top area is currently available; amorphous photovoltaic cells can also be used to tint windows and produce energy.

Hydroelectricity

The major advantage of conventional hydroelectric dams with reservoirs is their ability to store potential power for later electrical production. The combination of a natural supply of energy and production on demand has made hydro power the largest source of renewable energy by far. Other advantages include longer life than fuel-fired generation, low operating costs, and the provision of facilities for water sports. Some dams also operate as pumped-storage plants balancing supply and demand in the generation system. Overall, hydroelectric power can be less expensive than electricity generated from fossil fuels or nuclear energy, and areas with abundant hydroelectric power attract industry.

However, in addition to the advantages above, there are several disadvantages to dams that create large reservoirs. These may include: dislocation of people living where the reservoirs are planned, release of significant amounts of carbon dioxide at construction and flooding of the reservoir, disruption of aquatic ecosystems and bird life, adverse impacts on the river environment, and in rare cases catastrophic failure of the dam wall.

Some other disadvantages of the construction of hydroelectric dams is having to build access roads to get to the dam which disrupt the land ecosystem and not just the water ecosystems. Also with the increase in carbon dioxide, there is an increase in methane. This is from the flooding during the creation of the dams, when plants are submerged underwater and decay, they release methane gas. Another disadvantage is the upfront cost to build the dam and the amount of time it takes to build it.

Some dams only generate power and serve no other purpose, but in many places large reservoirs are needed for flood control and/or irrigation, adding a hydroelectric portion is a common way to pay for a new reservoir. Flood control protects life/property and irrigation supports increased agriculture.

Small hydro and run-of-the-river are two low impact alternatives to hydroelectric reservoirs, although they may produce intermittent power due to a lack of stored water.

Tidal

Tidal power can affect marine life. The turbines' rotating blades can accidentally kill swimming sea life. Projects such as the one in Strangford include a safety mechanism that turns off the turbine when marine animals approach. However, this feature causes a major loss in energy because of the amount of marine life that passes through the turbines. Some fish may avoid the area if threatened by a constantly rotating or noisy object. Marine life is a huge factor when siting tidal power energy generators, and precautions are taken to ensure that as few marine animals as possible are affected by it. The Tethys database provides access to scientific literature and general information on the potential environmental effects of tidal energy. In terms of global warming potential (i.e. carbon footprint), the impact of tidal power generation technologies ranges between 15 and 37 gCO₂-eq/kWhe, with a median value of 23.8 gCO₂-eq/kWhe. This is in line with the impact of other renewables like wind and solar power, and significantly better than fossil-based technologies.

Biomass

Further information: Biomass § Environmental impact

Electrical power can be generated by burning anything which will combust. Some electrical power is generated by burning crops which are grown specifically for the purpose. Usually this is done by fermenting plant matter to produce ethanol, which is then burned. This may also be done by allowing organic matter to decay, producing biogas, which is then burned. Also, when burned, wood is a form of biomass fuel.

Burning biomass produces many of the same emissions as burning fossil fuels. However, growing biomass captures carbon dioxide out of the air, so that the net contribution to global atmospheric carbon dioxide levels is small.

The process of growing biomass is subject to the same environmental concerns as any kind of agriculture. It uses a large amount of land, and fertilizers and pesticides may be necessary for cost-effective growth. Biomass that is produced as a by-product of agriculture shows some promise, but most such biomass is currently being used, for plowing back into the soil as fertilizer if nothing else.

Wind power

Livestock grazing near a wind turbine.

The environmental impact of electricity generation from wind power is minor when compared to that of fossil fuel power. Wind turbines have some of the lowest global warming potential per unit of electricity generated: far less greenhouse gas is emitted than for the average unit of electricity, so wind power helps limit climate change. Wind power consumes no fuel, and emits no air pollution, unlike fossil fuel power sources. The energy consumed to manufacture and transport the materials used to build a wind power plant is equal to the new energy produced by the plant within a few months.

Onshore (on-land) wind farms can have a significant visual impact and impact on the landscape. Due to a very low surface power density and spacing requirements, wind farms typically need to be spread over more land than other power stations. Their network of turbines, access roads, transmission lines, and substations can result in "energy sprawl"; although land between the turbines and roads can still be used for agriculture.

Conflicts arise especially in scenic and culturally-important landscapes. Siting restrictions (such as setbacks) may be implemented to limit the impact. The land between the turbines and access roads can still be used for farming and grazing. y can lead to "industrialization of the countryside". Some wind farms are opposed for potentially spoiling protected scenic areas, archaeological landscapes and heritage sites. A report by the Mountaineering Council of Scotland concluded that wind farms harmed tourism in areas known for natural landscapes and panoramic views.

Habitat loss and fragmentation are the greatest potential impacts on wildlife of onshore wind farms, but they are small and can be mitigated if proper monitoring and mitigation strategies are implemented. The worldwide ecological impact is minimal. Thousands of birds and bats, including rare species, have been killed by wind turbine blades, as around other manmade structures, though wind turbines are responsible for far fewer bird deaths than fossil-fuel infrastructure. This can be mitigated with proper wildlife monitoring.

Many wind turbine blades are made of fiberglass and some only had a lifetime of 10 to 20 years. Previously, there was no market for recycling these old blades, and they were commonly disposed of in landfills. Because blades are hollow, they take up a large volume compared to their mass. Since 2019, some landfill operators have begun requiring blades to be crushed before being landfilled. Blades manufactured in the 2020s are more likely to be designed to be completely recyclable.

Wind turbines also generate noise. At a distance of 300 metres (980 ft) this may be around 45 dB, which is slightly louder than a refrigerator. At 1.5 km (1 mi) distance they become inaudible. There are anecdotal reports of negative health effects on people who live very close to wind turbines. Peer-reviewed research has generally not supported these claims. Pile-driving to construct non-floating wind farms is noisy underwater, but in operation offshore wind is much quieter than ships.

Geothermal power

Main article: Geothermal power

Geothermal energy is the heat of the Earth, which can be tapped into to produce electricity in power plants. Warm water produced from geothermal sources can be used for industry, agriculture, bathing and cleansing. Where underground steam sources can be tapped, the steam is used to run a steam turbine. Geothermal steam sources have a finite life as underground water is depleted. Arrangements that circulate surface water through rock formations to produce hot water or steam are, on a human-relevant time scale, renewable.

While a geothermal power plant does not burn any fuel, it will still have emissions due to substances other than steam which come up from the geothermal wells. These may include hydrogen sulfide, and carbon dioxide. Some geothermal steam sources entrain non-soluble minerals that must be removed from the steam before it is used for generation; this material must be properly disposed. Any (closed cycle) steam power plant requires cooling water for condensers; diversion of cooling water from natural sources, and its increased temperature when returned to streams or lakes, may have a significant impact on local ecosystems.

Removal of ground water and accelerated cooling of rock formations can cause earth tremors. Enhanced geothermal systems (EGS) fracture underground rock to produce more steam; such projects can cause earthquakes. Certain geothermal projects (such as one near Basel, Switzerland in 2006) have been suspended or canceled owing to objectionable seismicity induced by geothermal recovery. However, risks associated with "hydrofracturing induced seismicity are low compared to that of natural earthquakes, and can be reduced by careful management and monitoring" and "should not be regarded as an impediment to further development of the Hot Rock geothermal energy resource".

Solar power

Part of the Senftenberg Solarpark, a solar photovoltaic power plant located on former open-pit mining areas close to the city of Senftenberg, in Eastern Germany. The 78 MW Phase 1 of the plant was completed within three months.

Solar power is cleaner than electricity from fossil fuels, so can be good for the environment when it replaces that. Solar power does not lead to any harmful emissions during operation, but the production of the panels leads to some amount of pollution. A 2021 study estimated the carbon footprint of manufacturing monocrystalline panels at 515 g CO₂/kWp in the US and 740 g CO₂/kWp in China, but this is expected to fall as manufacturers use more clean electricity and recycled materials. Solar power carries an upfront cost to the environment via production with a carbon payback time of several years as of 2022, but offers clean energy for the remainder of their 30-year lifetime.

The life-cycle greenhouse-gas emissions of solar farms are less than 50 gram (g) per kilowatt-hour (kWh), but with battery storage could be up to 150 g/kWh. In contrast, a combined cycle gas-fired power plant without carbon capture and storage emits around 500 g/kWh, and a coal-fired power plant about 1000 g/kWh. Similar to all energy sources where their total life cycle emissions are mostly from construction, the switch to low carbon power in the manufacturing and transportation of solar devices would further reduce carbon emissions.

Lifecycle surface power density of solar power varies a lot but averages about 7 W/m2, compared to about 240 for nuclear power and 480 for gas. However when the land required for gas extraction and processing is accounted for gas power is estimated to have not much higher power density than solar. PV requires much larger amounts of land surface to produce the same nominal amount of energy as sources with higher surface power density and capacity factor. According to a 2021 study, obtaining 25 to 80% of electricity from solar farms in their own territory by 2050 would require the panels to cover land ranging from 0.5 to 2.8% of the European Union, 0.3 to 1.4% in India, and 1.2 to 5.2% in Japan and South Korea. Occupation of such large areas for PV farms could drive residential opposition as well as lead to deforestation, removal of vegetation and conversion of farm land. However some countries, such as South Korea and Japan, use land for agriculture under PV, or floating solar, together with other low-carbon power sources. Worldwide land use has minimal ecological impact. Land use can be reduced to the level of gas power by installing on buildings and other built up areas.

Harmful materials are used in the production of solar panels, but in generally in small amounts. As of 2022 the environmental impact of perovskite is hard to estimate, but there is some concern that lead may become a problem. A 2021 International Energy Agency study projects the demand for copper will double by 2040. The study cautions that supply needs to increase rapidly to match demand from large-scale deployment of solar and required grid upgrades. More tellurium and indium may also be needed and recycling may help.

As solar panels are sometimes replaced with more efficient panels, the second-hand panels are sometimes reused in developing countries, for example in Africa. Several countries have specific regulations for the recycling of solar panels. Although maintenance cost is already low compared to other energy sources, some academics have called for solar power systems to be designed to be more repairable.

A very small proportion of solar power is concentrated solar power. Concentrated solar power may use much more water than gas-fired power. This can be a problem, as this type of solar power needs strong sunlight so is often built in deserts.

Ampère's circuital law

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law

 

In classical electromagnetism, Ampère's circuital law (not to be confused with Ampère's force law) relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop.

James Clerk Maxwell derived it using hydrodynamics in his 1861 published paper "On Physical Lines of Force". In 1865 he generalized the equation to apply to time-varying currents by adding the displacement current term, resulting in the modern form of the law, sometimes called the Ampère–Maxwell law, which is one of Maxwell's equations which form the basis of classical electromagnetism.

Ampère's original circuital law

In 1820 Danish physicist Hans Christian Ørsted discovered that an electric current creates a magnetic field around it, when he noticed that the needle of a compass next to a wire carrying current turned so that the needle was perpendicular to the wire. He investigated and discovered the rules which govern the field around a straight current-carrying wire:

• The magnetic field lines encircle the current-carrying wire.
• The magnetic field lines lie in a plane perpendicular to the wire.
• If the direction of the current is reversed, the direction of the magnetic field reverses.
• The strength of the field is directly proportional to the magnitude of the current.
• The strength of the field at any point is inversely proportional to the distance of the point from the wire.

This sparked a great deal of research into the relation between electricity and magnetism. André-Marie Ampère investigated the magnetic force between two current-carrying wires, discovering Ampère's force law. In the 1850s Scottish mathematical physicist James Clerk Maxwell generalized these results and others into a single mathematical law. The original form of Maxwell's circuital law, which he derived as early as 1855 in his paper "On Faraday's Lines of Force" based on an analogy to hydrodynamics, relates magnetic fields to electric currents that produce them. It determines the magnetic field associated with a given current, or the current associated with a given magnetic field.

The original circuital law only applies to a magnetostatic situation, to continuous steady currents flowing in a closed circuit. For systems with electric fields that change over time, the original law (as given in this section) must be modified to include a term known as Maxwell's correction (see below).

Equivalent forms

The original circuital law can be written in several different forms, which are all ultimately equivalent:

• An "integral form" and a "differential form". The forms are exactly equivalent, and related by the Kelvin–Stokes theorem (see the "proof" section below).
• Forms using SI units, and those using cgs units. Other units are possible, but rare. This section will use SI units, with cgs units discussed later.
• Forms using either B or H magnetic fields. These two forms use the total current density and free current density, respectively. The B and H fields are related by the constitutive equation: B = μ₀H in non-magnetic materials where μ₀ is the magnetic constant.

Explanation

The integral form of the original circuital law is a line integral of the magnetic field around some closed curve C (arbitrary but must be closed). The curve C in turn bounds both a surface S which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by S), and encloses the current. The mathematical statement of the law is a relation between the circulation of the magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral).

In terms of total current, (which is the sum of both free current and bound current) the line integral of the magnetic B-field (in teslas, T) around closed curve C is proportional to the total current I_enc passing through a surface S (enclosed by C). In terms of free current, the line integral of the magnetic H-field (in amperes per metre, A·m⁻¹) around closed curve C equals the free current I_f,enc through a surface S.

Forms of the original circuital law written in SI units

	Integral form	Differential form
Using B-field and total current	${\displaystyle {\oint }_C\mathbf{B}\cdot \mathrm{d}\mathit{l}={\mu }_0{\iint }_S\mathbf{J}\cdot \mathrm{d}\mathbf{S}={\mu }_0{I}_{\mathrm{e}\mathrm{n}\mathrm{c}}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}}$
Using H-field and free current	${\displaystyle {\oint }_C\mathbf{H}\cdot \mathrm{d}\mathit{l}={\iint }_S{\mathbf{J}}_{\mathrm{f}}\cdot \mathrm{d}\mathbf{S}=I_{\mathrm{f},\mathrm{e}\mathrm{n}\mathrm{c}}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\mathrm{f}}}$

• J is the total current density (in amperes per square metre, A·m⁻²),
• J_f is the free current density only,
• ∮_C is the closed line integral around the closed curve C,
• ∬_S denotes a 2-D surface integral over S enclosed by C,
• · is the vector dot product,
• dl is an infinitesimal element (a differential) of the curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C)
• dS is the vector area of an infinitesimal element of surface S (that is, a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface S. The direction of the normal must correspond with the orientation of C by the right hand rule), see below for further explanation of the curve C and surface S.
• ∇ × is the curl operator.

Ambiguities and sign conventions

There are a number of ambiguities in the above definitions that require clarification and a choice of convention.

1. First, three of these terms are associated with sign ambiguities: the line integral ∮_C could go around the loop in either direction (clockwise or counterclockwise); the vector area dS could point in either of the two directions normal to the surface; and I_enc is the net current passing through the surface S, meaning the current passing through in one direction, minus the current in the other direction—but either direction could be chosen as positive. These ambiguities are resolved by the right-hand rule: With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area dS. Also the current passing in the same direction as dS must be counted as positive. The right hand grip rule can also be used to determine the signs.
2. Second, there are infinitely many possible surfaces S that have the curve C as their border. (Imagine a soap film on a wire loop, which can be deformed by blowing on the film). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter: in the magnetostatic case, the current density is solenoidal (see next section), so the divergence theorem and continuity equation imply that the flux through any surface with boundary C, with the same sign convention, is the same. In practice, one usually chooses the most convenient surface (with the given boundary) to integrate over.

Free current versus bound current

The electric current that arises in the simplest textbook situations would be classified as "free current"—for example, the current that passes through a wire or battery. In contrast, "bound current" arises in the context of bulk materials that can be magnetized and/or polarized. (All materials can to some extent.)

When a material is magnetized (for example, by placing it in an external magnetic field), the electrons remain bound to their respective atoms, but behave as if they were orbiting the nucleus in a particular direction, creating a microscopic current. When the currents from all these atoms are put together, they create the same effect as a macroscopic current, circulating perpetually around the magnetized object. This magnetization current J_M is one contribution to "bound current".

The other source of bound current is bound charge. When an electric field is applied, the positive and negative bound charges can separate over atomic distances in polarizable materials, and when the bound charges move, the polarization changes, creating another contribution to the "bound current", the polarization current J_P.

The total current density J due to free and bound charges is then:

${\displaystyle \mathbf{J}={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{M}}+{\mathbf{J}}_{\mathrm{P}},}$

with J_f  the "free" or "conduction" current density.

All current is fundamentally the same, microscopically. Nevertheless, there are often practical reasons for wanting to treat bound current differently from free current. For example, the bound current usually originates over atomic dimensions, and one may wish to take advantage of a simpler theory intended for larger dimensions. The result is that the more microscopic Ampère's circuital law, expressed in terms of B and the microscopic current (which includes free, magnetization and polarization currents), is sometimes put into the equivalent form below in terms of H and the free current only. For a detailed definition of free current and bound current, and the proof that the two formulations are equivalent, see the "proof" section below.

Shortcomings of the original formulation of the circuital law

There are two important issues regarding the circuital law that require closer scrutiny. First, there is an issue regarding the continuity equation for electrical charge. In vector calculus, the identity for the divergence of a curl states that the divergence of the curl of a vector field must always be zero. Hence

${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{B})=0,}$

and so the original Ampère's circuital law implies that

${\displaystyle \mathrm{\nabla }\cdot \mathbf{J}=0,}$

i.e. that the current density is solenoidal.

But in general, reality follows the continuity equation for electric charge:

${\displaystyle \mathrm{\nabla }\cdot \mathbf{J}=-\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t},}$

which is nonzero for a time-varying charge density. An example occurs in a capacitor circuit where time-varying charge densities exist on the plates.

Second, there is an issue regarding the propagation of electromagnetic waves. For example, in free space, where

${\displaystyle \mathbf{J}=\mathbf{0},}$

the circuital law implies that

${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\mathbf{0},}$

i.e. that the magnetic field is irrotational, but to maintain consistency with the continuity equation for electric charge, we must have

${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\frac{1}{c^2}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}.}$

To treat these situations, the contribution of displacement current must be added to the current term in the circuital law.

James Clerk Maxwell conceived of displacement current as a polarization current in the dielectric vortex sea, which he used to model the magnetic field hydrodynamically and mechanically. He added this displacement current to Ampère's circuital law at equation 112 in his 1861 paper "On Physical Lines of Force".

Displacement current

Main article: Displacement current

In free space, the displacement current is related to the time rate of change of electric field.

In a dielectric the above contribution to displacement current is present too, but a major contribution to the displacement current is related to the polarization of the individual molecules of the dielectric material. Even though charges cannot flow freely in a dielectric, the charges in molecules can move a little under the influence of an electric field. The positive and negative charges in molecules separate under the applied field, causing an increase in the state of polarization, expressed as the polarization density P. A changing state of polarization is equivalent to a current.

Both contributions to the displacement current are combined by defining the displacement current as:

${\displaystyle {\mathbf{J}}_{\mathrm{D}}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{D}(\mathbf{r},t),}$

where the electric displacement field is defined as:

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}={\epsilon }_0{\epsilon }_{\mathrm{r}}\mathbf{E},}$

where ε₀ is the electric constant, ε_r the relative static permittivity, and P is the polarization density. Substituting this form for D in the expression for displacement current, it has two components:

${\displaystyle {\mathbf{J}}_{\mathrm{D}}={\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t}.}$

The first term on the right hand side is present everywhere, even in a vacuum. It doesn't involve any actual movement of charge, but it nevertheless has an associated magnetic field, as if it were an actual current. Some authors apply the name displacement current to only this contribution.

The second term on the right hand side is the displacement current as originally conceived by Maxwell, associated with the polarization of the individual molecules of the dielectric material.

Maxwell's original explanation for displacement current focused upon the situation that occurs in dielectric media. In the modern post-aether era, the concept has been extended to apply to situations with no material media present, for example, to the vacuum between the plates of a charging vacuum capacitor. The displacement current is justified today because it serves several requirements of an electromagnetic theory: correct prediction of magnetic fields in regions where no free current flows; prediction of wave propagation of electromagnetic fields; and conservation of electric charge in cases where charge density is time-varying. For greater discussion see Displacement current.

Extending the original law: the Ampère–Maxwell equation

Next, the circuital equation is extended by including the polarization current, thereby remedying the limited applicability of the original circuital law.

Treating free charges separately from bound charges, the equation including Maxwell's correction in terms of the H-field is (the H-field is used because it includes the magnetization currents, so J_M does not appear explicitly, see H-field and also Note):

${\displaystyle {\oint }_C\mathbf{H}\cdot \mathrm{d}\mathit{l}={\iint }_S\left({\mathbf{J}}_{\mathrm{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}\right)\cdot \mathrm{d}\mathbf{S}}$

(integral form), where H is the magnetic H field (also called "auxiliary magnetic field", "magnetic field intensity", or just "magnetic field"), D is the electric displacement field, and J_f is the enclosed conduction current or free current density. In differential form,

${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\mathrm{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}.}$

On the other hand, treating all charges on the same footing (disregarding whether they are bound or free charges), the generalized Ampère's equation, also called the Maxwell–Ampère equation, is in integral form (see the "proof" section below):

${\displaystyle {\oint }_C\mathbf{B}\cdot \mathrm{d}\mathit{l}={\iint }_S\left({\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\cdot \mathrm{d}\mathbf{S}}$

In differential form,

${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}}$

In both forms J includes magnetization current density as well as conduction and polarization current densities. That is, the current density on the right side of the Ampère–Maxwell equation is:

${\displaystyle {\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{D}}+{\mathbf{J}}_{\mathrm{M}}={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{P}}+{\mathbf{J}}_{\mathrm{M}}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}=\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t},}$

where current density J_D is the displacement current, and J is the current density contribution actually due to movement of charges, both free and bound. Because ∇ ⋅ D = ρ, the charge continuity issue with Ampère's original formulation is no longer a problem. Because of the term in ε₀∂E/∂t, wave propagation in free space now is possible.

With the addition of the displacement current, Maxwell was able to hypothesize (correctly) that light was a form of electromagnetic wave. See electromagnetic wave equation for a discussion of this important discovery.

Proof of equivalence

Proof that the formulations of the circuital law in terms of free current are equivalent to the formulations involving total current

In this proof, we will show that the equation

${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\mathrm{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}}$

is equivalent to the equation

${\displaystyle \frac{1}{{\mu }_0}(\mathrm{\nabla }\times \mathbf{B})=\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}.}$

Note that we are only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the Kelvin–Stokes theorem.

We introduce the polarization density P, which has the following relation to E and D:

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}.}$

Next, we introduce the magnetization density M, which has the following relation to B and H:

${\displaystyle \frac{1}{{\mu }_0}\mathbf{B}=\mathbf{H}+\mathbf{M}}$

and the following relation to the bound current:

${\displaystyle \begin{array}{rl}{\mathbf{J}}_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}& =\mathrm{\nabla }\times \mathbf{M}+\frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t}\\ & ={\mathbf{J}}_{\mathrm{M}}+{\mathbf{J}}_{\mathrm{P}},\end{array}}$

where

${\displaystyle {\mathbf{J}}_{\mathrm{M}}=\mathrm{\nabla }\times \mathbf{M},}$

is called the magnetization current density, and

${\displaystyle {\mathbf{J}}_{\mathrm{P}}=\frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t},}$

is the polarization current density. Taking the equation for B:

${\displaystyle \begin{array}{rl}\frac{1}{{\mu }_0}(\mathrm{\nabla }\times \mathbf{B})& =\mathrm{\nabla }\times \left(\mathbf{H}+\mathbf{M}\right)\\ & =\mathrm{\nabla }\times \mathbf{H}+{\mathbf{J}}_{\mathrm{M}}\\ & ={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{P}}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}+{\mathbf{J}}_{\mathrm{M}}.\end{array}}$

Consequently, referring to the definition of the bound current:

${\displaystyle \begin{array}{rl}\frac{1}{{\mu }_0}(\mathrm{\nabla }\times \mathbf{B})& ={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\\ & =\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t},\end{array}}$

as was to be shown.

Ampère's circuital law in cgs units

In cgs units, the integral form of the equation, including Maxwell's correction, reads

${\displaystyle {\oint }_C\mathbf{B}\cdot \mathrm{d}\mathit{l}=\frac{1}{c}{\iint }_S\left(4\pi \mathbf{J}+\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\cdot \mathrm{d}\mathbf{S},}$

where c is the speed of light.

The differential form of the equation (again, including Maxwell's correction) is

${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\frac{1}{c}\left(4\pi \mathbf{J}+\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right).}$