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Ohm's law applies to electrical circuits; it states that the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points, and inversely proportional to the resistance between them.

The mathematical equation that describes this relationship is:


 * $$V = IR \quad \text{or} \quad I = \frac{V}{R} \quad \text{or} \quad R = \frac{V}{I} $$

where:

I is the current in units of amperes (often shortened to "amps").

V is the potential difference measured across the resistance in units of volts. The potential difference is also known as the voltage drop, and is sometimes denoted by U, E or EMF (electromotive force) instead of V.

R is a circuit parameter called the resistance in units of ohms. The ohm is usually represented by the Greek letter capital omega (Ω) and is equivalent to "volts per ampere".

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wire. He presented a slightly more complex equation than the one above (see History section below) to explain his experimental results. The above equation is the modern form of Ohm's law.

Physics
Ohm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineering because it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly, as circuit elements in an electrical circuit. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, having variables that are closely related to the I, V and R scalar variables of Ohm's law, but are each functions of position within the conductor. Physicists often use this continuum form of Ohm's Law:



\mathbf{E} = \mathbf{J}\rho $$

where "E" is the electric field with units of volts per meter (analogous to "V" of Ohm's law which has units of volts), "J" is the current density with units of amperes per unit area (analogous to "I" of Ohm's law which has units of amperes), and "$$\rho$$" (Greek "rho") is the resistivity with units of ohm&middot;meters (analogous to "R" of Ohm's law which has units of ohms). The above equation is sometimes written as J = $$\sigma$$E where "$$\sigma$$" is the conductivity which is the reciprocal of $$\rho$$.

The potential difference between two points is defined as


 * $${\Delta V} = -\int {\mathbf E \cdot d \mathbf l} $$

with $$d \mathbf l$$ the element of path along the integration of electric field vector E. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage in the usual convention of being opposite in direction to the field:


 * $$V = {E}{l} \ \  \text{or} \ \ E = \frac{V}{l} \ . $$

Substituting the above in for E, and substituting $$I/a$$ (where I is the current in amperes and a is the cross sectional area of the conductor) for the current per unit area J, the continuum form shown at the beginning of this section becomes:


 * $$\frac{V}{l} = \frac{I}{a}\rho \quad \text{or} \quad V = I \rho \frac{l}{a}$$

The electrical resistance of a uniform conductor is given in terms of resistivity by:
 * $${R} = \rho \frac{l}{a} $$

where l is the length of the conductor in SI units of meters, a is the cross-sectional area (for a round wire a = &pi;r2 if r is radius) in units of meters squared, and ρ is the resistivity in units of ohm&middot;meters.

After substitution of R from the above equation into the equation preceding it, the continuum form of Ohm's law for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form:
 * $${V}={I}{R} \ $$

A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity, but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scatter from all of these, resulting in resistance to their flow.

Magnetic effects
The continuum form of the equation is only valid in the reference frame of the conducting material. If the material is moving at velocity v relative to a magnetic field B, a term must be added as follows:
 * $$\mathbf{E} + \mathbf{v}\times\mathbf{B} = \mathbf{J}{\rho} $$

See Lorentz force for more on this and Hall effect for some other implications of a magnetic field. This equation is not a modification to Ohm's law. Rather, it is analogous in circuit analysis terms to taking into account inductance as well as resistance.

Temperature effects
When the temperature of the conductor increases, the collisions between electrons and ions increase. Thus as a substance heats up because of electricity flowing through it (or by any heating process), the resistance will usually increase. The exception is semiconductors. The resistance of an ohmic substance depends on temperature in the following way:



R = R_0 [\alpha (T - T_0) + 1]\,\! $$

where T is its temperature, $$T_0$$ is a reference temperature (usually room temperature), $$R_0$$ is the resistance at $$T_0$$, and $$\alpha$$ is the percentage change in resistivity per unit temperature. The constant $$\alpha$$ depends only on the material being considered. The relationship stated is actually only an approximate one, the true physics being somewhat non-linear, or looking at it another way, $$\alpha$$ itself varies with temperature. For this reason it is usual to specify the temperature that $$\alpha$$ was measured at with a suffix, such as $$\alpha_{15}$$ and the relationship only holds in a range of temperatures around the reference.

Intrinsic semiconductors exhibit the opposite temperature behavior, becoming better conductors as the temperature increases. This occurs because the electrons are bumped to the conduction energy band by the thermal energy, where they can flow freely and in doing so they leave behind holes in the valence band which can also flow freely.

Extrinsic semiconductors have much more complex temperature behaviour. First the electrons (or holes) leave the donors (or acceptors) giving a decreasing resistance. Then there is a fairly flat phase in which the semiconductor is normally operated where almost all of the donors (or acceptors) have lost their electrons (or holes) but the number of electrons that have jumped right over the energy gap is negligible compared to the number of electrons (or holes) from the donors (or acceptors). Finally as the temperature increases further the carriers that jump the energy gap becomes the dominant figure and the material starts behaving like an intrinsic semiconductor.

Strain (mechanical) effects
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.

Resistive circuits
Resistors are circuit elements that impede the passage of electric charge in agreement with Ohm's law, and are designed to have a specific resistance value R. In a schematic diagram the resistor is shown as a zig-zag symbol. An element (resistor or conductor) that behaves according to Ohm's law over some operating range is referred to as an ohmic device (or an ohmic resistor) because Ohm's law and a single value for the resistance suffice to describe the behavior of the device over that range.

Ohm's law holds for circuits containing only resistive elements (no capacitances or inductances) for all forms of driving voltage or current, regardless of whether the driving voltage or current is constant (DC) or time-varying such as AC. At any instant of time Ohm's law is valid for such circuits.

Resistors which are in series or in parallel may be grouped together into a single "equivalent resistance" in order to apply Ohm's law in analyzing the circuit. This application of Ohm's law is illustrated with examples in "How To Analyze Resistive Circuits Using Ohm's Law" on wikiHow.

Reactive circuits with time-varying signals
When reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC or time-varying voltage or current is applied, the relationship between voltage and current becomes the solution to a differential equation, so Ohm's law (as defined above) does not directly apply since that form contains only resistances having value R, not complex impedances which may contain capacitance ("C") or inductance ("L").

Equations for time-invariant AC circuits take the same form as Ohm's law, however, if the variables are generalized to complex numbers and the current and voltage waveforms are complex exponentials.

In this approach, a voltage or current waveform takes the form $$Ae^{st}$$, where t is time, s is a complex parameter, and A is a complex scalar. In any linear time-invariant system, all of the currents and voltages can be expressed with the same s parameter as the input to the system, allowing the time-varying complex exponential term to be canceled out and the system described algebraically in terms of the complex scalars in the current and voltage waveforms.

The complex generalization of resistance is impedance, usually denoted Z; it can be shown that for an inductor,

$$Z = sL\,$$

and for a capacitor,

$$Z = \frac{1}{sC}.$$

We can now write,

$$\mathbf{V} = \mathbf{I} \cdot \mathbf{Z}$$

where V and I are the complex scalars in the voltage and current respectively and Z is the complex impedance.

While this has the form of Ohm's law, with Z taking the place of R, it is not the same as Ohm's law. When Z is complex, only the real part is responsible for dissipating heat.

In the general AC circuit, Z will vary strongly with the frequency parameter s, and so also will the relationship between voltage and current.

For the common case of a steady sinusoid, the s parameter is taken to be $$j\omega$$, corresponding to a complex sinusoid $$Ae^{j\omega t}$$. The real parts of such complex current and voltage waveforms describe the actual sinusoidal currents and voltages in a circuit, which can be in different phases due to the different complex scalars.

Linear approximations
Ohm's law is one of the basic equations used in the analysis of electrical circuits. It applies to both metal conductors and circuit components (resistors) specifically made for this behaviour. Both are ubiquitous in electrical engineering. Materials and components that obey Ohm's law are described as "ohmic" which means they produce the same value for resistance (R = V/I) regardless of the value of V or I which is applied and whether the applied voltage or current is DC (direct current) of either positive or negative polarity or AC (alternating current).

In a true ohmic device, the same value of resistance will be calculated from R = V/I regardless of the value of the applied voltage V. That is, the ratio of V/I is constant, and when current is plotted as a function of voltage the curve is linear (a straight line). If voltage is forced to some value V, then that voltage V divided by measured current I will equal R. Or if the current is forced to some value I, then the measured voltage V divided by that current I is also R.  Since the plot of I versus V is a straight line, then it is also true that for any set of two different voltages V1 and V2 applied across a given device of resistance R, producing currents I1 = V1/R and I2 = V2/R, that the ratio (V1-V2)/(I1-I2) is also a constant equal to R.  The operator "delta" (&Delta;} is used to represent a difference in a quantity, so we can write &Delta;V = V1-V2 and &Delta;I = I1-I2.  Summarizing, for any truly ohmic device having resistance R, V/I = &Delta;V/&Delta;I = R for any applied voltage or current or for the difference between any set of applied voltages or currents.



There are, however, components of electrical circuits which do not obey Ohm's law; that is, their relationship between current and voltage (their I–V curve) is nonlinear. An example is the p-n junction diode (curve at right). As seen in the figure, the current does not increase linearly with applied voltage for a diode. One can determine a value of current (I) for a given value of applied voltage (V) from the curve, but not from Ohm's law, since the value of "resistance" is not constant as a function of applied voltage. Further, the current only increases significantly if the applied voltage is positive, not negative. The ratio V/I for some point along the nonlinear curve is sometimes called the static, or chordal, or DC, resistance, but as seen in the figure the value of total V over total I varies depending on the particular point along the nonlinear curve which is chosen. This means the "DC resistance" V/I at some point on the curve is not the same as what would be determined by applying an AC signal having peak amplitude &Delta;V volts or &Delta;I amps centered at that same point along the curve and measuring &Delta;V/&Delta;I. However, in some diode applications, the AC signal applied to the device is small and it is possible to analyze the circuit in terms of the dynamic, small-signal, or incremental resistance, defined as the one over the slope of the V–I curve at the average value (DC operating point) of the voltage (that is, one over the derivative of current with respect to voltage). For sufficiently small signals, the dynamic resistance allows the Ohm's law small signal resistance to be calculated as approximately one over the slope of a line drawn tangentially to the V-I curve at the DC operating point.

Hydraulic analogies
A hydraulic analogy is sometimes used to describe Ohm's Law. Water pressure, measured by pascals (or PSI), is the analog of voltage because establishing a water pressure difference between two points along a (horizontal) pipe causes water to flow. Water flow rate, as in liters per second, is the analog of current, as in coulombs per second. Finally, flow restrictors — such as apertures placed in pipes between points where the water pressure is measured — are the analog of resistors. We say that the rate of water flow through an aperture restrictor is proportional to the difference in water pressure across the restrictor. Similarly, the rate of flow of electrical charge, that is, the electric current, through an electrical resistor is proportional to the difference in voltage measured across the resistor.

Flow and pressure variables can be calculated in fluid flow network with the use of the hydraulic ohm analogy. The method can be applied to both steady and transient flow situations.

Relation to heat conduction
Ohm's principle predicts the flow of electrical charge (i.e. current) in electrical conductors when subjected to the influence of voltage differences; Jean-Baptiste-Joseph Fourier's principle predicts the flow of heat in heat conductors when subjected to the influence of temperature differences.

The same equation describes both phenomena, the equation's variables taking on different meanings in the two cases. Specifically, solving a heat conduction (Fourier) problem with temperature (the driving "force") and flux of heat (the rate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous electrical conduction (Ohm) problem having electric potential (the driving "force") and electric current (the rate of flow of the driven "quantity", i.e. charge) variables.

The basis of Fourier's work was his clear conception and definition of thermal conductivity. He assumed that, all else being the same, the flux of heat is strictly proportional to the gradient of temperature. Although undoubtedly true for small temperature gradients, strictly proportional behavior will be lost when real materials (e.g. ones having a thermal conductivity that is a function of temperature) are subjected to large temperature gradients.

A similar assumption is made in the statement of Ohm's law: other things being alike, the strength of the current at each point is proportional to the gradient of electric potential. The accuracy of the assumption that flow is proportional to the gradient is more readily tested, using modern measurement methods, for the electrical case than for the heat case.

History
In January 1781, before Georg Ohm's work, Henry Cavendish experimented with Leyden jars and glass tubes of varying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt as he completed the circuit with his body. Cavendish wrote that the "velocity" (current) varied directly as the "degree of electrification" (voltage). He did not communicate his results to other scientists at the time, and his results were unknown until Maxwell published them in 1879.

Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827 as the book Die galvanische Kette, mathematisch bearbeite (The galvanic Circuit investigated mathematically). He drew considerable inspiration from Fourier's work on heat conduction in the theoretical explanation of his work. For experiments, he initially used voltaic piles, but later used a thermocouple as this provided a more stable voltage source in terms of internal resistance and constant potential difference. He used a galvanometer to measure current, and knew that the voltage between the thermocouple terminals was proportional to the junction temperature. He then added test wires of varying length, diameter, and material to complete the circuit. He found that his data could be modeled through the equation
 * $$\mathbf{X} = \frac{\mathbf{a}}{\mathbf{b} + \mathbf{l}},$$

where X was the reading from the galvanometer, l was the length of the test conductor, a depended only on the thermocouple junction temperature, and b was a constant of the entire setup. From this, Ohm determined his law of proportionality and published his results.

Ohm's law was probably the most important of the early quantitative descriptions of the physics of electricity. We consider it almost obvious today. When Ohm first published his work, this was not the case; critics reacted to his treatment of the subject with hostility. They called his work a "web of naked fancies" and the German Minister of Education proclaimed that Ohm was "a professor who preached such heresies was unworthy to teach science." The prevailing scientific philosophy in Germany at the time, led by Hegel, asserted that experiments need not be performed to develop an understanding of nature because nature is so well ordered, and that scientific truths may be deduced through reasoning alone. Also, Ohm's brother Martin, a mathematician, was battling the German educational system. These factors hindered the acceptance of Ohm's work, and his work did not become widely accepted until the 1840s. Fortunately, Ohm received recognition for his contributions to science well before he died.

In the 1850s, Ohm's law was known as such, and was widely considered proved, and alternatives such as "Barlow's law" discredited, in terms of real applications to telegraph system design, as discussed by Samuel F. B. Morse in 1855.

While the old term for electrical conductance, the mho (the inverse of the resistance unit ohm), is still used, a new name, the siemens, was adopted in 1971, honoring Ernst Werner von Siemens. The siemens is preferred in formal papers.

In the 1920s, it was discovered that the current through an ideal resistor actually has statistical fluctuations, which depend on temperature, even when voltage and resistance are exactly constant; this fluctuation, now known as Johnson–Nyquist noise, is due to the discrete nature of charge. This thermal effect implies that measurements of current and voltage that are taken over sufficiently short periods of time will yield ratios of V/I that fluctuate from the value of R implied by the time average or ensemble average of the measured current; Ohm's law remains correct for the average current, in the case of ordinary resistive materials.

Ohm's work long preceded Maxwell's equations and any understanding of frequency-dependent effects in AC circuits. Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when they are evaluated within the appropriate limits.