If You Want to Save the Planet, Drop the Campaign Against Capitalism

written by Andrew Glover

Original link:  https://quillette.com/2018/10/29/if-you-want-to-save-the-planet-drop-the-campaign-against-capitalism/?fbclid=IwAR2hcTpn4LgsgBddDPSmJ14HHEb9EAbDQBqEzIZUhbm7lbNwgtA4NBGeyaQ 

 

But if you go carrying pictures of chairman Mao,
You ain’t going to make it with anyone anyhow

—The Beatles, 1968

This month, the Intergovernmental Panel on Climate Change (IPCC) issued a report concluding that it is all but inevitable that overall global warming will exceed the 1.5 degree Celsius limit dictated in the 2015 Paris Agreement. The report also discusses the potentially catastrophic consequences of this warming, which include extreme weather events, an accelerated rise in sea levels, and shrinking Arctic sea ice.

In keeping with the well-established trend, political conservatives generally have exhibited skepticism of these newly published IPCC conclusions. That includes U.S. President Donald Trump, who told 60 Minutes, “We have scientists that disagree with [anthropogenic global warming]. You’d have to show me the [mainstream] scientists because they have a very big political agenda.” On Fox News, a commentator argued that “the planet has largely stopped warming over the past 15 years, data shows—and [the IPCC report] could not explain why the Mercury had stopped rising.” Conservative YouTuber Ian Miles Cheong declared flatly that:

"Climate change is a hoax invented by neo-Marxists within the scientific community to destabilize the world economy and dismantle what they call “systems of oppression” and what the rest of us call capitalism."

This pattern of conservative skepticism on climate change is so well-established that many of us now take it for granted. But given conservatism’s natural impulse toward protecting our heritage, one might think that conservatives would be just as concerned with preserving order in the natural environment as they are with preserving order in our social and political environments. Ensuring that subsequent generations can live well is ordinarily a core concern for conservatives.

To this, conservatives might (and do) counter that they are merely pushing back against environmental extremists who seek to leverage the cause of global warming as a means to expand government, eliminate hierarchies of wealth, and reorganize society along social lines. And while most environmentally conscious citizens harbor no such ambitions, there is a substantial basis for this claim. Indeed, some environmentalists are forthright in seeking to implement the principles of “ecosocialism.” Meteorologist and self-described ecosocialist Eric Holthaus, for instance, responded to the IPCC report by declaring that:

"The world's top scientists just gave rigorous backing to systematically dismantle capitalism as a key requirement to maintaining civilization and a habitable planet."

One of the most prominent voices in this space has been Canadian writer Naomi Klein, whose 2015 book, This Changes Everything: Capitalism vs the Climate, argued that capitalism must be dismantled for the world to avert catastrophe. While I am sympathetic with some of the critiques that Klein directs at corporations and “free market fundamentalism,” her argument doesn’t hold water—because mitigating climate risks is a project whose enormous scope, cost and complexity can only be managed by regulated capitalist welfare states. Moreover, it’s difficult to see how she isn’t simply using the crisis of climate change as a veneer to agitate for her preferred utopian socio-economic system. As has been pointed out by Jonathan Chait of New York magazine, Klein appears to be adapting a mirror image of the same strategy she critiqued in her previous book, The Shock Doctrine, wherein she claimed that cynical politicians, pundits and corporations seize on crises to lock in economic restructuring along radical free market principles.

Simply put, describing the call for climate action in economically or politically revolutionary terms is always going to be counterproductive, because the vast majority of ordinary people in most countries don’t want a revolution. Environmentalists such as Klein are correct, however, in their more limited claim that market mechanisms alone can’t prevent global warming, since such mechanisms don’t impute the environmental costs associated with the way we produce goods and live our lives. Without some means of capturing the social price of environmentally destructive practices—resource extraction, in particular—we will invariably embrace wasteful and damaging practices.

Consider, for instance, the vast quantities of natural gas that are flared at oil wells simply because it’s seen as too costly to build gas pipelines to these facilities. This is a context in which we’d urge government to exercise its regulatory power; or to impose some kind of pricing mechanism that, either by carrot or stick, incentivizes the capture of the flared gas. Public policy has a necessary role in guiding capitalist decision makers toward the long-term sustainability of the environment. Unfortunately, this outcome is hard to achieve in a political environment characterized by tribalism, polarization and blame-shifting.

It is true that when it comes to climate change, the political left is more closely grounded in science than the right (even if both sides often tend to deny inconvenient truths more generally). But the left also has proven to be blinkered when it comes to appropriate responses, a tendency that has seeped into the latest IPCC report. While it’s not surprising that the report advocates support for renewable energy, its authors fail to acknowledge the warming effect that scaled up renewable-energy generation would have on land use due to their low energy density (think of the enormous footprint of solar farms). Likewise, the pro-environmental left’s distaste for nuclear power persists, despite its status as a geographically dense, safe, virtually carbon-free energy source.

The whole issue has become a sort of microcosm of the blind spots and dogmas embraced by both sides. As Jonathan Haidt argues, conservatives tend to be skeptical of top-down governance, preferring to focus on smaller nested structures that are less ambitious in scope, and hence easier to manage. This general principle takes form in conservative philosopher Roger Scruton’s approach to environmentalism, which argues that activism on issues such as climate change should be undertaken by communities at the local level, rather than by national (or international) bureaucrats and politicians—because the local level is where “people protect things which they know and love, things which are necessary for their life, and which will elicit in them the kind of disposition to make sacrifices, which, after all, is what it’s all about.”

While Scruton’s environmentalism gives us a reason to protect our local environments, the reality is that the effects of many environmentally damaging practices are not just experienced locally. A community may be motivated to protect a nearby forest from logging because it forms part of their love of home, but greenhouse gas emissions are displaced and dispersed into the shared atmosphere, contributing to global atmospheric degradation. Because of this, any approach that dismisses broader policy initiatives is unlikely to succeed in bringing down global carbon emissions. But at the very least, Scruton’s analysis awakens us to the reality that such policies will gain popular support only if they are justified and implemented in a manner that takes into consideration the views and sentiments of conservatives and liberals alike. Wind and solar farms will face less opposition if local communities get a greater say in where they are located. And while carbon taxes are effective in reducing emissions in some jurisdictions, conservatives will usually oppose them unless they are structured in a revenue-neutral manner, by legislating them alongside equivalent reductions in income tax, for instance.

Environmentalists also should acknowledge that some conservative objections to large-scale, top-down global instruments such as the Paris Agreement are perfectly legitimate. The provisions in such treaties typically are non-binding and require the good faith of all signatories. With many authoritarian countries seemingly misleading the rest of the world about their levels of economic activity, it’s not unreasonable to assume they would do the same when it comes to reporting carbon emissions. Moreover, those countries without the means to enforce reductions in carbon emissions domestically can’t be regarded as reliable participants in a global agreement to voluntarily decarbonize their economies.

This isn’t to say we shouldn’t be discussing climate change at a global level, or that international agreements don’t have any value. But environmentalists’ tendency to treat these documents as holy writ comes off as naïve, and thereby tends to undermine their cause.

Overall, our best hope for dealing with the emissions of developing countries is likely to assist them in managing their energy infrastructure so as to bypass high-emissions technologies. China, despite often being lauded for the amount of renewable energy it produces, now emits more carbon dioxide than the U.S. and Europe combined. With technologies such as large-scale solar generation becoming cost competitive with coal, progress is possible, but far from guaranteed without Western support.

These measures aren’t revolutionary. But that’s the point: In the environmental sector, just as in every other arena, there’s an opportunity cost to adopting revolutionary postures—since these revolutionaries tend to make more enemies than allies. If this project is really about saving the planet, rather than destroying capitalism, cooling the earth will mean cooling our rhetoric as well.

Andrew Glover is a sociologist who tweets at @theandrewglover.

Electrical resistance and conductance

From Wikipedia, the free encyclopedia

The electrical resistance of an object is a measure of its opposition to the flow of electric current. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S). 

 

The resistance of an object depends in large part on the material it is made of—objects made of electrical insulators like rubber tend to have very high resistance and low conductivity, while objects made of electrical conductors like metals tend to have very low resistance and high conductivity. This material dependence is quantified by resistivity or conductivity. However, resistance and conductance are extensive rather than bulk properties, meaning that they also depend on the size and shape of an object. For example, a wire's resistance is higher if it is long and thin, and lower if it is short and thick. All objects show some resistance, except for superconductors, which have a resistance of zero.

The resistance (R) of an object is defined as the ratio of voltage across it (V) to current through it (I), while the conductance (G) is the inverse:

${\displaystyle R={V \over I},\qquad G={I \over V}={\frac {1}{R}}}$

For a wide variety of materials and conditions, V and I are directly proportional to each other, and therefore R and G are constants (although they will depend on the size and shape of the object, the material it is made of, and other factors like temperature or strain). This proportionality is called Ohm's law, and materials that satisfy it are called ohmic materials.

In other cases, such as a transformer, diode or battery, V and I are not directly proportional. The ratio V/I is sometimes still useful, and is referred to as a "chordal resistance" or "static resistance", since it corresponds to the inverse slope of a chord between the origin and an I–V curve. In other situations, the derivative ${\displaystyle {\frac {dV}{dI}}\,\!}$ may be most useful; this is called the "differential resistance".

Introduction

The hydraulic analogy compares electric current flowing through circuits to water flowing through pipes. When a pipe (left) is filled with hair (right), it takes a larger pressure to achieve the same flow of water. Pushing electric current through a large resistance is like pushing water through a pipe clogged with hair: It requires a larger push (electromotive force) to drive the same flow (electric current).

In the hydraulic analogy, current flowing through a wire (or resistor) is like water flowing through a pipe, and the voltage drop across the wire is like the pressure drop that pushes water through the pipe. Conductance is proportional to how much flow occurs for a given pressure, and resistance is proportional to how much pressure is required to achieve a given flow. (Conductance and resistance are reciprocals.)

The voltage drop (i.e., difference between voltages on one side of the resistor and the other), not the voltage itself, provides the driving force pushing current through a resistor. In hydraulics, it is similar: The pressure difference between two sides of a pipe, not the pressure itself, determines the flow through it. For example, there may be a large water pressure above the pipe, which tries to push water down through the pipe. But there may be an equally large water pressure below the pipe, which tries to push water back up through the pipe. If these pressures are equal, no water flows. (In the image at right, the water pressure below the pipe is zero.)

The resistance and conductance of a wire, resistor, or other element is mostly determined by two properties:

• geometry (shape), and
• material

Geometry is important because it is more difficult to push water through a long, narrow pipe than a wide, short pipe. In the same way, a long, thin copper wire has higher resistance (lower conductance) than a short, thick copper wire.

Materials are important as well. A pipe filled with hair restricts the flow of water more than a clean pipe of the same shape and size. Similarly, electrons can flow freely and easily through a copper wire, but cannot flow as easily through a steel wire of the same shape and size, and they essentially cannot flow at all through an insulator like rubber, regardless of its shape. The difference between copper, steel, and rubber is related to their microscopic structure and electron configuration, and is quantified by a property called resistivity.

In addition to geometry and material, there are various other factors that influence resistance and conductance, such as temperature; see below.

Conductors and resistors

A 6.5 MΩ resistor, as identified by its electronic color code (blue–green–black-yellow-red). An ohmmeter could be used to verify this value.

Substances in which electricity can flow are called conductors. A piece of conducting material of a particular resistance meant for use in a circuit is called a resistor. Conductors are made of high-conductivity materials such as metals, in particular copper and aluminium. Resistors, on the other hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount of energy that it needs to dissipate, precision, and costs.

Ohm's law

The current-voltage characteristics of four devices: Two

resistors, a diode, and a battery. The horizontal axis is

voltage drop, the vertical axis is current. Ohm's law is satisfied

when the graph is a straight line through the origin. Therefore,

the two resistors are ohmic, but the diode and battery are not.

For many materials, the current I through the material is proportional to the voltage V applied across it:

${\displaystyle I\propto V}$

over a wide range of voltages and currents. Therefore, the resistance and conductance of objects or electronic components made of these materials is constant. This relationship is called Ohm's law, and materials which obey it are called ohmic materials. Examples of ohmic components are wires and resistors. The current-voltage (IV) graph of an ohmic device consists of a straight line through the origin with positive slope.

Other components and materials used in electronics do not obey Ohm's law; the current is not proportional to the voltage, so the resistance varies with the voltage and current through them. These are called nonlinear or nonohmic. Examples include diodes and fluorescent lamps. The IV curve of a nonohmic device is a curved line.

Relation to resistivity and conductivity

A piece of resistive material with electrical contacts on both ends.

The resistance of a given object depends primarily on two factors: What material it is made of, and its shape. For a given material, the resistance is inversely proportional to the cross-sectional area; for example, a thick copper wire has lower resistance than an otherwise-identical thin copper wire. Also, for a given material, the resistance is proportional to the length; for example, a long copper wire has higher resistance than an otherwise-identical short copper wire. The resistance R and conductance G of a conductor of uniform cross section, therefore, can be computed as

${\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}},\\G&=\sigma {\frac {A}{\ell }}.\end{aligned}}}$

where ${\displaystyle \ell }$ is the length of the conductor, measured in metres [m], A is the cross-sectional area of the conductor measured in square metres [m²], σ (sigma) is the electrical conductivity measured in siemens per meter (S·m⁻¹), and ρ (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres (Ω·m). The resistivity and conductivity are proportionality constants, and therefore depend only on the material the wire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals: ${\displaystyle \rho =1/\sigma }$. Resistivity is a measure of the material's ability to oppose electric current.

This formula is not exact, as it assumes the current density is totally uniform in the conductor, which is not always true in practical situations. However, this formula still provides a good approximation for long thin conductors such as wires.

Another situation for which this formula is not exact is with alternating current (AC), because the skin effect inhibits current flow near the center of the conductor. For this reason, the geometrical cross-section is different from the effective cross-section in which current actually flows, so resistance is higher than expected. Similarly, if two conductors near each other carry AC current, their resistances increase due to the proximity effect. At commercial power frequency, these effects are significant for large conductors carrying large currents, such as busbars in an electrical substation, or large power cables carrying more than a few hundred amperes.

What determines resistivity?

The resistivity of different materials varies by an enormous amount: For example, the conductivity of teflon is about 10³⁰ times lower than the conductivity of copper. Why is there such a difference? Loosely speaking, a metal has large numbers of "delocalized" electrons that are not stuck in any one place, but free to move across large distances, whereas in an insulator (like teflon), each electron is tightly bound to a single molecule, and a great force is required to pull it away. Semiconductors lie between these two extremes. More details can be found in the article: Electrical resistivity and conductivity. For the case of electrolyte solutions, see the article: Conductivity (electrolytic).

Resistivity varies with temperature. In semiconductors, resistivity also changes when exposed to light. See below.

Measuring resistance

An instrument for measuring resistance is called an ohmmeter. Simple ohmmeters cannot measure low resistances accurately because the resistance of their measuring leads causes a voltage drop that interferes with the measurement, so more accurate devices use four-terminal sensing.

Typical resistances

Component	Resistance (Ω)
1 meter of copper wire with 1 mm diameter	0.02
1 km overhead power line (typical)	0.03
AA battery (typical internal resistance)	0.1
Incandescent light bulb filament (typical)	200–1000
Human body	1000 to 100,000

Static and differential resistance

The IV curve of a non-ohmic device (purple). The static resistance at point A is the inverse slope of line B through the origin. The differential resistance at A is the inverse slope of tangent line C.

The IV curve of a component with negative differential resistance, an unusual phenomenon where the IV curve is non-monotonic.

Many electrical elements, such as diodes and batteries do not satisfy Ohm's law. These are called non-ohmic or non-linear, and their I–V curves are not straight lines through the origin.

Resistance and conductance can still be defined for non-ohmic elements. However, unlike ohmic resistance, non-linear resistance is not constant but varies with the voltage or current through the device; i.e., its operating point. There are two types of resistance

• Static resistance (also called chordal or DC resistance) – This corresponds to the usual definition of resistance; the voltage divided by the current

${\displaystyle R_{\mathrm {static} }={\frac {V}{I}}\,}$.

It is the slope of the line (chord) from the origin through the point on the curve. Static resistance determines the power dissipation in an electrical component. Points on the IV curve located in the 2nd or 4th quadrants, for which the slope of the chordal line is negative, have negative static resistance. Passive devices, which have no source of energy, cannot have negative static resistance. However active devices such as transistors or op-amps can synthesize negative static resistance with feedback, and it is used in some circuits such as gyrators.

• Differential resistance (also called dynamic, incremental or small signal resistance) – Differential resistance is the derivative of the voltage with respect to the current; the slope of the IV curve at a point

${\displaystyle R_{\mathrm {diff} }={\frac {dV}{dI}}\,}$.

If the IV curve is nonmonotonic (with peaks and troughs), the curve has a negative slope in some regions—so in these regions the device has negative differential resistance. Devices with negative differential resistance can amplify a signal applied to them, and are used to make amplifiers and oscillators. These include tunnel diodes, Gunn diodes, IMPATT diodes, magnetron tubes, and unijunction transistors.

AC circuits

Impedance and admittance

The voltage (red) and current (blue) versus time (horizontal axis) for a capacitor (top) and inductor (bottom). Since the amplitude of the current and voltage sinusoids are the same, the absolute value of impedance is 1 for both the capacitor and the inductor (in whatever units the graph is using). On the other hand, the phase difference between current and voltage is −90° for the capacitor; therefore, the complex phase of the impedance of the capacitor is −90°. Similarly, the phase difference between current and voltage is +90° for the inductor; therefore, the complex phase of the impedance of the inductor is +90°.

When an alternating current flows through a circuit, the relation between current and voltage across a circuit element is characterized not only by the ratio of their magnitudes, but also the difference in their phases. For example, in an ideal resistor, the moment when the voltage reaches its maximum, the current also reaches its maximum (current and voltage are oscillating in phase). But for a capacitor or inductor, the maximum current flow occurs as the voltage passes through zero and vice versa (current and voltage are oscillating 90° out of phase, see image at right). Complex numbers are used to keep track of both the phase and magnitude of current and voltage:

${\displaystyle V(t)={\text{Re}}(V_{0}e^{j\omega t}),\quad I(t)={\text{Re}}(I_{0}e^{j\omega t}),\quad Z={\frac {V_{0}}{I_{0}}},\quad Y={\frac {I_{0}}{V_{0}}}}$

where:

• t is time,
• V(t) and I(t) are, respectively, voltage and current as a function of time,
• V₀, I₀, Z, and Y are complex numbers,
• Z is called impedance,
• Y is called admittance,
• Re indicates real part,
• ${\displaystyle \omega }$ is the angular frequency of the AC current,
• ${\displaystyle j={\sqrt {-1}}}$ is the imaginary unit.

The impedance and admittance may be expressed as complex numbers that can be broken into real and imaginary parts:

${\displaystyle Z=R+jX,\quad Y=G+jB}$

where R and G are resistance and conductance respectively, X is reactance, and B is susceptance. For ideal resistors, Z and Y reduce to R and G respectively, but for AC networks containing capacitors and inductors, X and B are nonzero.

${\displaystyle Z=1/Y}$ for AC circuits, just as ${\displaystyle R=1/G}$ for DC circuits.

Frequency dependence of resistance

Another complication of AC circuits is that the resistance and conductance can be frequency-dependent. One reason, mentioned above is the skin effect (and the related proximity effect). Another reason is that the resistivity itself may depend on frequency (see Drude model, deep-level traps, resonant frequency, Kramers–Kronig relations, etc.)

Energy dissipation and Joule heating

Running current through a material with high resistance creates heat, in a phenomenon called Joule heating. In this picture, a cartridge heater, warmed by Joule heating, is glowing red hot.

Resistors (and other elements with resistance) oppose the flow of electric current; therefore, electrical energy is required to push current through the resistance. This electrical energy is dissipated, heating the resistor in the process. This is called Joule heating (after James Prescott Joule), also called ohmic heating or resistive heating.

The dissipation of electrical energy is often undesired, particularly in the case of transmission losses in power lines. High voltage transmission helps reduce the losses by reducing the current for a given power.

On the other hand, Joule heating is sometimes useful, for example in electric stoves and other electric heaters (also called resistive heaters). As another example, incandescent lamps rely on Joule heating: the filament is heated to such a high temperature that it glows "white hot" with thermal radiation (also called incandescence).

The formula for Joule heating is:

${\displaystyle P=I^{2}R}$

where P is the power (energy per unit time) converted from electrical energy to thermal energy, R is the resistance, and I is the current through the resistor.

Dependence of resistance on other conditions

Temperature dependence

Near room temperature, the resistivity of metals typically increases as temperature is increased, while the resistivity of semiconductors typically decreases as temperature is increased. The resistivity of insulators and electrolytes may increase or decrease depending on the system. For the detailed As a consequence, the resistance of wires, resistors, and other components often change with temperature. This effect may be undesired, causing an electronic circuit to malfunction at extreme temperatures. In some cases, however, the effect is put to good use. When temperature-dependent resistance of a component is used purposefully, the component is called a resistance thermometer or thermistor. (A resistance thermometer is made of metal, usually platinum, while a thermistor is made of ceramic or polymer.)

Resistance thermometers and thermistors are generally used in two ways. First, they can be used as thermometers: By measuring the resistance, the temperature of the environment can be inferred. Second, they can be used in conjunction with Joule heating (also called self-heating): If a large current is running through the resistor, the resistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in a circuit-protection role similar to fuses, or for feedback in circuits, or for many other purposes. In general, self-heating can turn a resistor into a nonlinear and hysteretic circuit element.

If the temperature T does not vary too much, a linear approximation is typically used:

${\displaystyle R(T)=R_{0}[1+\alpha (T-T_{0})]}$

where ${\displaystyle \alpha }$ is called the temperature coefficient of resistance, ${\displaystyle T_{0}}$ is a fixed reference temperature (usually room temperature), and ${\displaystyle R_{0}}$ is the resistance at temperature ${\displaystyle T_{0}}$. The parameter ${\displaystyle \alpha }$ is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, ${\displaystyle \alpha }$ is different for different reference temperatures. For this reason it is usual to specify the temperature that ${\displaystyle \alpha }$ was measured at with a suffix, such as ${\displaystyle \alpha _{15}}$, and the relationship only holds in a range of temperatures around the reference.

The temperature coefficient ${\displaystyle \alpha }$ is typically +3×10⁻³ K⁻¹ to +6×10⁻³ K⁻¹ for metals near room temperature. It is usually negative for semiconductors and insulators, with highly variable magnitude.

Strain dependence

Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon strain. By placing a conductor under tension (a form of stress that leads to strain in the form of stretching of the conductor), the length of the section of conductor under tension increases and its cross-sectional area decreases. Both these effects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on strain gauges for details about devices constructed to take advantage of this effect.

Light illumination dependence

Some resistors, particularly those made from semiconductors, exhibit photoconductivity, meaning that their resistance changes when light is shining on them. Therefore, they are called photoresistors (or light dependent resistors). These are a common type of light detector.

Superconductivity

Superconductors are materials that have exactly zero resistance and infinite conductance, because they can have V=0 and I≠0. This also means there is no joule heating, or in other words no dissipation of electrical energy. Therefore, if superconductive wire is made into a closed loop, current flows around the loop forever. Superconductors require cooling to temperatures near 4 K with liquid helium for most metallic superconductors like niobium–tin alloys, or cooling to temperatures near 77K with liquid nitrogen for the expensive, brittle and delicate ceramic high temperature superconductors. Nevertheless, there are many technological applications of superconductivity, including superconducting magnets.

Capacitance

From Wikipedia, the free encyclopedia

Common symbols	C
SI unit	farad
Other units	μF, nF, pF
In SI base units	F = A2 s4 kg−1 m−2
Derivations from other quantities	C = charge / voltage
Dimension	M−1 L−2 T4 I2

Capacitance is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. There are two closely related notions of capacitance: self capacitance and mutual capacitance. Any object that can be electrically charged exhibits self capacitance. A material with a large self capacitance holds more electric charge at a given voltage than one with low capacitance. The notion of mutual capacitance is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components (along with resistors and inductors).

The capacitance is a function only of the geometry of the design (e.g. area of the plates and the distance between them) and the permittivity of the dielectric material between the plates of the capacitor. For many dielectric materials, the permittivity and thus the capacitance, is independent of the potential difference between the conductors and the total charge on them.

The SI unit of capacitance is the farad (symbol: F), named after the English physicist Michael Faraday. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates. The reciprocal of capacitance is called elastance.

Self-capacitance

In electrical circuits, the term capacitance is usually a shorthand for the mutual capacitance between two adjacent conductors, such as the two plates of a capacitor. However, for an isolated conductor, there also exists a property called self-capacitance, which is the amount of electric charge that must be added to an isolated conductor to raise its electric potential by one unit (i.e. one volt, in most measurement systems). The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, with the conductor centered inside this sphere.
Mathematically, the self-capacitance of a conductor is defined by

${\displaystyle C={\frac {q}{V}},}$

where

q is the charge held by the conductor,

${\displaystyle V={1 \over 4\pi \varepsilon _{0}}\int {\sigma \,dS \over r}}$ is the electric potential,

σ is the surface charge density.

dS is an infinitesimal element of area,

r is the length from dS to a fixed point M within the plate

${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity

Using this method, the self-capacitance of a conducting sphere of radius R is

${\displaystyle C=4\pi \varepsilon _{0}R\,}$

Example values of self-capacitance are:

• for the top "plate" of a van de Graaff generator, typically a sphere 20 cm in radius: 22.24 pF,
• the planet Earth: about 710 µF.

The inter-winding capacitance of a coil is sometimes called self-capacitance, but this is a different phenomenon. It is actually mutual capacitance between the individual turns of the coil and is a form of stray, or parasitic capacitance. This self-capacitance is an important consideration at high frequencies: It changes the impedance of the coil and gives rise to parallel resonance. In many applications this is an undesirable effect and sets an upper frequency limit for the correct operation of the circuit.

Mutual capacitance

A common form is a parallel-plate capacitor, which consists of two conductive plates insulated from each other, usually sandwiching a dielectric material. In a parallel plate capacitor, capacitance is very nearly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates.

If the charges on the plates are +q and −q, and V gives the voltage between the plates, then the capacitance C is given by

${\displaystyle C={\frac {q}{V}}.}$

which gives the voltage/current relationship

${\displaystyle i(t)=C{\frac {\mathrm {d} v(t)}{\mathrm {d} t}}.}$

The energy stored in a capacitor is found by integrating the work W

${\displaystyle W_{\text{charging}}={\frac {1}{2}}CV^{2}}$

Capacitance matrix

The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition ${\displaystyle C=Q/V}$ does not apply when there are more than two charged plates, or when the net charge on the two plates is non-zero. To handle this case, Maxwell introduced his coefficients of potential. If three (nearly ideal) conductors are given charges ${\displaystyle Q_{1},Q_{2},Q_{3}}$, then the voltage at conductor 1 is given by

${\displaystyle V_{1}=P_{11}Q_{1}+P_{12}Q_{2}+P_{13}Q_{3},}$

and similarly for the other voltages. Hermann von Helmholtz and Sir William Thomson showed that the coefficients of potential are symmetric, so that ${\displaystyle P_{12}=P_{21}}$, etc. Thus the system can be described by a collection of coefficients known as the elastance matrix or reciprocal capacitance matrix, which is defined as:

${\displaystyle P_{ij}={\frac {\partial V_{i}}{\partial Q_{j}}}}$

From this, the mutual capacitance ${\displaystyle C_{m}}$ between two objects can be defined by solving for the total charge Q and using ${\displaystyle C_{m}=Q/V}$.

${\displaystyle C_{m}={\frac {1}{(P_{11}+P_{22})-(P_{12}+P_{21})}}}$

Since no actual device holds perfectly equal and opposite charges on each of the two "plates", it is the mutual capacitance that is reported on capacitors.

The collection of coefficients ${\displaystyle C_{ij}={\frac {\partial Q_{i}}{\partial V_{j}}}}$ is known as the capacitance matrix, and is the inverse of the elastance matrix.

Capacitors

The capacitance of the majority of capacitors used in electronic circuits is generally several orders of magnitude smaller than the farad. The most common subunits of capacitance in use today are the microfarad (µF), nanofarad (nF), picofarad (pF), and, in microcircuits, femtofarad (fF). However, specially made supercapacitors can be much larger (as much as hundreds of farads), and parasitic capacitive elements can be less than a femtofarad. In the past, alternate subunits were used in historical electronic books; "mfd" and "mf" for microfarad (µF); "mmfd", "mmf", "µµF" for picofarad (pF); but are rarely used any more.

Capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. A qualitative explanation for this can be given as follows.

Once a positive charge is put unto a conductor, this charge creates an electrical field, repelling any other positive charge to be moved onto the conductor; i.e., increasing the necessary voltage. But if nearby there is another conductor with a negative charge on it, the electrical field of the positive conductor repelling the second positive charge is weakened (the second positive charge also feels the attracting force of the negative charge). So due to the second conductor with a negative charge, it becomes easier to put a positive charge on the already positive charged first conductor, and vice versa; i.e., the necessary voltage is lowered.

As a quantitative example consider the capacitance of a capacitor constructed of two parallel plates both of area A separated by a distance d. If d is sufficiently small with respect to the smallest chord of A, there holds, to a high level of accuracy:

${\displaystyle \ C=\varepsilon _{r}\varepsilon _{0}{\frac {A}{d}}}$

where

C is the capacitance, in farads;

A is the area of overlap of the two plates, in square meters;

ε_r is the relative static permittivity (sometimes called the dielectric constant) of the material between the plates (for a vacuum, ε_r = 1);

ε₀ is the electric constant (ε₀ ≈ 8.854×10⁻¹² F⋅m⁻¹); and

d is the separation between the plates, in meters.

Capacitance is proportional to the area of overlap and inversely proportional to the separation between conducting sheets. The closer the sheets are to each other, the greater the capacitance. The equation is a good approximation if d is small compared to the other dimensions of the plates so that the electric field in the capacitor area is uniform, and the so-called fringing field around the periphery provides only a small contribution to the capacitance. In CGS units the equation has the form

${\displaystyle C=\varepsilon _{r}{\frac {A}{4\pi d}}}$

where C in this case has the units of length. Combining the SI equation for capacitance with the above equation for the energy stored in a capacitance, for a flat-plate capacitor the energy stored is

${\displaystyle W_{\text{stored}}={\frac {1}{2}}CV^{2}={\frac {1}{2}}\varepsilon _{r}\varepsilon _{0}{\frac {A}{d}}V^{2}.}$

where W is the energy, in joules; C is the capacitance, in farads; and V is the voltage, in volts.

Stray capacitance

Any two adjacent conductors can function as a capacitor, though the capacitance is small unless the conductors are close together for long distances or over a large area. This (often unwanted) capacitance is called parasitic or "stray capacitance". Stray capacitance can allow signals to leak between otherwise isolated circuits (an effect called crosstalk), and it can be a limiting factor for proper functioning of circuits at high frequency.

Stray capacitance between the input and output in amplifier circuits can be troublesome because it can form a path for feedback, which can cause instability and parasitic oscillation in the amplifier. It is often convenient for analytical purposes to replace this capacitance with a combination of one input-to-ground capacitance and one output-to-ground capacitance; the original configuration — including the input-to-output capacitance — is often referred to as a pi-configuration. Miller's theorem can be used to effect this replacement: it states that, if the gain ratio of two nodes is 1/K, then an impedance of Z connecting the two nodes can be replaced with a Z/(1 − k) impedance between the first node and ground and a KZ/(K − 1) impedance between the second node and ground. Since impedance varies inversely with capacitance, the internode capacitance, C, is replaced by a capacitance of KC from input to ground and a capacitance of (K − 1)C/K from output to ground. When the input-to-output gain is very large, the equivalent input-to-ground impedance is very small while the output-to-ground impedance is essentially equal to the original (input-to-output) impedance.

Capacitance of conductors with simple shapes

Calculating the capacitance of a system amounts to solving the Laplace equation ∇²φ = 0 with a constant potential φ on the surface of the conductors. This is trivial in cases with high symmetry. There is no solution in terms of elementary functions in more complicated cases.

For two-dimensional situations analytic functions may be used to map different geometries to each other.

Capacitance of simple systems

Type	Capacitance	Comment
Parallel-plate capacitor	${\displaystyle \varepsilon A/d}$	ε: Permittivity
Coaxial cable	${\displaystyle {\frac {2\pi \varepsilon \ell }{\ln \left(R_{2}/R_{1}\right)}}}$	ε: Permittivity
Pair of parallel wires	${\displaystyle {\frac {\pi \varepsilon \ell }{\operatorname {arcosh} \left({\frac {d}{2a}}\right)}}={\frac {\pi \varepsilon \ell }{\ln \left({\frac {d}{2a}}+{\sqrt {{\frac {d^{2}}{4a^{2}}}-1}}\right)}}}$
Wire parallel to wall	${\displaystyle {\frac {2\pi \varepsilon \ell }{\operatorname {arcosh} \left({\frac {d}{a}}\right)}}={\frac {2\pi \varepsilon \ell }{\ln \left({\frac {d}{a}}+{\sqrt {{\frac {d^{2}}{a^{2}}}-1}}\right)}}}$	a: Wire radius d: Distance, d > a ℓ: Wire length
Two parallel coplanar strips	${\displaystyle \varepsilon \ell {\frac {K\left({\sqrt {1-k^{2}}}\right)}{K\left(k\right)}}}$	d: Distance w1, w2: Strip width km: d/(2wm+d) k2: k1k2 K: Elliptic integral l: Length
Concentric spheres	${\displaystyle {\frac {4\pi \varepsilon }{{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}}}}$	ε: Permittivity
Two spheres, equal radius	${\displaystyle 2\pi \varepsilon a\sum _{n=1}^{\infty }{\frac {\sinh \left(\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}{\sinh \left(n\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}}}$ ${\displaystyle =2\pi \varepsilon a\left\{1+{\frac {1}{2D}}+{\frac {1}{4D^{2}}}+{\frac {1}{8D^{3}}}+{\frac {1}{8D^{4}}}+{\frac {3}{32D^{5}}}+O\left({\frac {1}{D^{6}}}\right)\right\}}$ ${\displaystyle =2\pi \varepsilon a\left\{\ln 2+\gamma -{\frac {1}{2}}\ln \left(2D-2\right)+O\left(2D-2\right)\right\}}$	a: Radius d: Distance, d > 2a D = d/2a, D > 1 γ: Euler's constant
Sphere in front of wall	${\displaystyle 4\pi \varepsilon a\sum _{n=1}^{\infty }{\frac {\sinh \left(\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}{\sinh \left(n\ln \left(D+{\sqrt {D^{2}-1}}\right)\right)}}}$	a: Radius d: Distance, d > a D = d/a
Sphere	${\displaystyle 4\pi \varepsilon a}$	a: Radius
Circular disc	${\displaystyle 8\varepsilon a}$	a: Radius
Prolate ellipsoid	${\displaystyle {\frac {\varepsilon a}{\ln \left(2a/b\right)}}}$	half-axes a>b=c
Thin straight wire, finite length	${\displaystyle {\frac {2\pi \varepsilon \ell }{\Lambda }}\left\{1+{\frac {1}{\Lambda }}\left(1-\ln 2\right)+{\frac {1}{\Lambda ^{2}}}\left[1+\left(1-\ln 2\right)^{2}-{\frac {\pi ^{2}}{12}}\right]+O\left({\frac {1}{\Lambda ^{3}}}\right)\right\}}$	a: Wire radius ℓ: Length Λ: ln(ℓ/a)

Energy storage

The energy (measured in joules) stored in a capacitor is equal to the work required to push the charges into the capacitor, i.e. to charge it. Consider a capacitor of capacitance C, holding a charge +q on one plate and −q on the other. Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW:

${\displaystyle \mathrm {d} W={\frac {q}{C}}\,\mathrm {d} q}$

where W is the work measured in joules, q is the charge measured in coulombs and C is the capacitance, measured in farads.

The energy stored in a capacitor is found by integrating this equation. Starting with an uncharged capacitance (q = 0) and moving charge from one plate to the other until the plates have charge +Q and −Q requires the work W:

${\displaystyle W_{\text{charging}}=\int _{0}^{Q}{\frac {q}{C}}\,\mathrm {d} q={\frac {1}{2}}{\frac {Q^{2}}{C}}={\frac {1}{2}}QV={\frac {1}{2}}CV^{2}=W_{\text{stored}}.}$

Nanoscale systems

The capacitance of nanoscale dielectric capacitors such as quantum dots may differ from conventional formulations of larger capacitors. In particular, the electrostatic potential difference experienced by electrons in conventional capacitors is spatially well-defined and fixed by the shape and size of metallic electrodes in addition to the statistically large number of electrons present in conventional capacitors. In nanoscale capacitors, however, the electrostatic potentials experienced by electrons are determined by the number and locations of all electrons that contribute to the electronic properties of the device. In such devices, the number of electrons may be very small, however, the resulting spatial distribution of equipotential surfaces within the device are exceedingly complex.

Single-electron devices

The capacitance of a connected, or "closed", single-electron device is twice the capacitance of an unconnected, or "open", single-electron device. This fact may be traced more fundamentally to the energy stored in the single-electron device whose "direct polarization" interaction energy may be equally divided into the interaction of the electron with the polarized charge on the device itself due to the presence of the electron and the amount of potential energy required to form the polarized charge on the device (the interaction of charges in the device's dielectric material with the potential due to the electron).

Few-electron devices

The derivation of a "quantum capacitance" of a few-electron device involves the thermodynamic chemical potential of an N-particle system given by

${\displaystyle \mu (N)=U(N)-U(N-1)}$

whose energy terms may be obtained as solutions of the Schrödinger equation. The definition of capacitance,

${\displaystyle {1 \over C}\equiv {\Delta \,V \over \Delta \,Q}}$,

with the potential difference

${\displaystyle \Delta \,V={\Delta \,\mu \, \over e}={\mu (N+\Delta \,N)-\mu (N) \over e}}$

may be applied to the device with the addition or removal of individual electrons,

${\displaystyle \Delta \,N=1}$ and ${\displaystyle \Delta \,Q=e}$.

Then

${\displaystyle C_{Q}(N)={e^{2} \over \mu (N+1)-\mu (N)}={e^{2} \over E(N)}}$

is the "quantum capacitance" of the device.

This expression of "quantum capacitance" may be written as

${\displaystyle C_{Q}(N)={e^{2} \over U(N)}}$

which differs from the conventional expression described in the introduction where ${\displaystyle W_{\text{stored}}=U}$, the stored electrostatic potential energy,

${\displaystyle C={Q^{2} \over 2U}}$

by a factor of 1/2 with ${\displaystyle Q=Ne}$.

However, within the framework of purely classical electrostatic interactions, the appearance of the factor of 1/2 is the result of integration in the conventional formulation,

${\displaystyle W_{\text{charging}}=U=\int _{0}^{Q}{\frac {q}{C}}\,\mathrm {d} q}$

which is appropriate since ${\displaystyle \mathrm {d} q=0}$ for systems involving either many electrons or metallic electrodes, but in few-electron systems, ${\displaystyle \mathrm {d} q\to \Delta \,Q=e}$. The integral generally becomes a summation. One may trivially combine the expressions of capacitance and electrostatic interaction energy,

${\displaystyle Q=CV}$ and ${\displaystyle U=QV}$,

respectively, to obtain,

${\displaystyle C=Q{1 \over V}=Q{Q \over U}={Q^{2} \over U}}$

which is similar to the quantum capacitance. A more rigorous derivation is reported in the literature. In particular, to circumvent the mathematical challenges of the spatially complex equipotential surfaces within the device, an average electrostatic potential experiences by each electron is utilized in the derivation.

The reason for apparent mathematical differences is understood more fundamentally as the potential energy, ${\displaystyle U(N)}$, of an isolated device (self-capacitance) is twice that stored in a "connected" device in the lower limit N=1. As N grows large, ${\displaystyle U(N)\to U}$. Thus, the general expression of capacitance is

${\displaystyle C(N)={(Ne)^{2} \over U(N)}}$.

In nanoscale devices such as quantum dots, the "capacitor" is often an isolated, or partially isolated, component within the device. The primary differences between nanoscale capacitors and macroscopic (conventional) capacitors are the number of excess electrons (charge carriers, or electrons, that contribute to the device's electronic behavior) and the shape and size of metallic electrodes. In nanoscale devices, nanowires consisting of metal atoms typically do not exhibit the same conductive properties as their macroscopic, or bulk material, counterparts.

Capacitance in electronic and semiconductor devices

In electronic and semiconductor devices, transient or frequency-dependent current between terminals contains both conduction and displacement components. Conduction current is related to moving charge carriers (electrons, holes, ions, etc.), while displacement current is caused by time-varying electric field. Carrier transport is affected by electric field and by a number of physical phenomena - such as carrier drift and diffusion, trapping, injection, contact-related effects, impact ionization, etc. As a result, device admittance is frequency-dependent, and a simple electrostatic formula for capacitance ${\displaystyle C=q/V,}$ is not applicable. A more general definition of capacitance, encompassing electrostatic formula, is:

${\displaystyle C={\frac {\operatorname {Im} (Y(\omega ))}{\omega }},}$

where ${\displaystyle Y(\omega )}$ is the device admittance, and ${\displaystyle \omega }$ is the angular frequency.

In general case, capacitance is a function of frequency. At high frequencies, capacitance approached a constant value, equal to "geometric" capacitance, determined by the terminals' geometry and dielectric content in the device. A paper by Steven Laux presents a review of numerical techniques for capacitance calculation. In particular, capacitance can be calculated by a Fourier transform of a transient current in response to a step-like voltage excitation:

${\displaystyle C(\omega )=1/(\Delta V)\int _{0}^{\infty }[i(t)-i(\infty )]cos(\omega t)dt.}$

Negative capacitance in semiconductor devices

Usually, capacitance in semiconductor devices is positive. However, in some devices and under certain conditions (temperature, applied voltages, frequency, etc.), capacitance can become negative. Non-monotonic behavior of the transient current in response to a step-like excitation has been proposed as the mechanism of negative capacitance. Negative capacitance has been demonstrated and explored in many different types of semiconductor devices.