Ohm's Law · aiwiki.page

Ohm’s law is an empirical relationship between electric current, voltage, and electrical resistance. It states that the current through an ohmic conductor is proportional to the voltage across it, provided physical conditions, especially temperature, remain unchanged. Usually written $V=IR$ , it describes many metallic conductors and circuit components, but is not a universal rule for all electrical devices. (openstax.org)

Mathematical statement

The familiar circuit form is

V=IR,

where $V$ is the electric potential difference, or voltage, across a component; $I$ is the current through it; and $R$ is its resistance. Equivalent expressions are $I=V/R$ and $R=V/I$ , with the latter requiring nonzero current. In the International System of Units, voltage is measured in volts, current in amperes, and resistance in ohms, symbol $\Omega$ ; one ohm equals one volt per ampere. (web.mit.edu)

The essential assertion is that $R$ remains constant over the stated operating range. Merely calculating $V/I$ at one operating point does not establish ohmic behavior. An ohmic component has a straight-line voltage–current characteristic through the origin: on a graph with voltage vertically and current horizontally, its slope is $R$ . Reversing the voltage reverses the current without changing this slope under the same conditions. (openstax.org)

For example, an ideal $100\,\Omega$ resistor with $5\,\mathrm{V}$ across it carries $0.05\,\mathrm{A}$ , or $50\,\mathrm{mA}$ . Doubling the applied voltage doubles the current if the resistance remains unchanged. These values follow directly from substitution into the equation.

Historical development and experimental testing

The relationship is named after Georg Simon Ohm, who investigated electrical conduction experimentally and presented his mathematical treatment in the 1827 book The Galvanic Circuit Investigated Mathematically. His work established the proportional relationship between voltage and current in conducting wires. (physicstoday.aip.org)

A modern test uses a variable voltage source, an ammeter in series with the specimen, and a voltmeter connected across it. Measurements at several positive and negative voltages reveal whether the characteristic is linear. A single measurement establishes only one operating point, not proportionality over a range. (openstax.org)

Local form and material properties

For an isotropic material in its linear conduction regime, the local form relates current density $\mathbf J$ to the electric field $\mathbf E$ :

\mathbf J=\sigma\mathbf E.

Here $\sigma$ is electrical conductivity. Its reciprocal, electrical resistivity $\rho=1/\sigma$ , gives the equivalent expression $\mathbf E=\rho\mathbf J$ . Unlike the resistance of a particular object, conductivity and resistivity characterize the material under specified conditions. Resistivity is measured in ohm-metres. (web.mit.edu)

For a uniform conductor of length $L$ , cross-sectional area $A$ , and constant resistivity,

R=\rho\frac{L}{A}.

Using $E=V/L$ and $J=I/A$ connects the local relation to $V=IR$ . Thus a longer wire has greater resistance, whereas a larger cross-sectional area reduces resistance, other conditions being equal. The formula assumes uniform material properties and the simple geometry represented by these expressions. (openstax.org)

Microscopic interpretation

The Drude model provides a classical explanation of linear conduction. It treats conduction electrons as mobile particles whose velocities are randomized by collisions. An applied field produces an average drift superimposed on their otherwise disordered motion; collisions prevent indefinite acceleration of the average motion. (web.mit.edu)

In this model,

\sigma=\frac{ne^2\tau}{m},

where $n$ is the carrier number density, $e$ is the magnitude of the elementary charge, $m$ is the electron mass, and $\tau$ is a characteristic relaxation time. If these parameters remain effectively independent of the applied field, current density is proportional to field strength. This interpretation connects macroscopic resistance with carrier density and scattering processes rather than treating resistance as geometry alone. (web.mit.edu)

Temperature dependence and non-ohmic behavior

Ohm’s law applies within specified conditions, not necessarily across every voltage or temperature. For many metallic conductors, resistance near a reference temperature $T_0$ can be approximated by

R(T)\approx R_0[1+\alpha(T-T_0)],

where $\alpha$ is the temperature coefficient. The approximation has a limited range. Many metals have positive coefficients, so heating increases their resistance. (openstax.org)

A lamp filament illustrates the distinction: increasing voltage raises its temperature and changes its resistance, producing a nonlinear characteristic even though the filament material may behave approximately ohmically at a fixed temperature. A diode, commonly constructed from semiconductor materials, also exhibits a nonlinear voltage–current relationship, with markedly different behavior for opposite voltage polarities. Such devices cannot be represented by one constant resistance over their full operating range. (openstax.org)

Circuit analysis and power dissipation

For resistors in series, the same current passes through each and their resistances add. For parallel resistors, each branch has the same voltage and the reciprocals of their resistances add:

R_{\mathrm{series}}=\sum_k R_k, \qquad \frac{1}{R_{\mathrm{parallel}}}=\sum_k\frac{1}{R_k}.

These relations allow resistor networks to be reduced to equivalent resistances and analyzed using Ohm’s law. (openstax.org)

The electrical power absorbed by a resistor is $P=VI$ . Substitution gives

P=I^2R=\frac{V^2}{R}.

This conversion of energy into internal energy is associated with Joule heating. For the illustrative $100\,\Omega$ resistor at $5\,\mathrm{V}$ , the dissipated power is $0.25\,\mathrm{W}$ . At fixed resistance, doubling either voltage or current quadruples the power, showing why thermal effects can limit the operating range of a constant-resistance model. (openstax.org)