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Ohm's Law Explained

Topic Summary

Ohm's law states that the electric current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the three mathematical equations used to describe this relationship: where I is the current through the conductor, V is the voltage measured across the conductor and R is the resistance of the conductor. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.

If the resistance is not constant, the previous equation cannot be called Ohm's law, but it can still be used as a definition of static/DC resistance. Ohm's law is an empirical relation which accurately describes the conductivity of the vast majority of electrically conductive materials over many orders of magnitude of current. However some materials do not obey Ohm's law; these are called non-ohmic.

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Related references

Resistivity and Resistance

The resistance of a conductor depends on its material and geometry: R = ρL/A, where ρ is the resistivity (Ω·m), L is the length, and A is the cross-sectional area. Resistivity is a material property: copper ≈ 1.7×10⁻⁸ Ω·m; nichrome (heating wire) ≈ 1.1×10⁻⁶ Ω·m; silicon ≈ 10³ Ω·m.

Resistivity increases with temperature for metals (more lattice vibrations scatter electrons): ρ(T) = ρ₀[1 + α(T − T₀)], where α is the temperature coefficient. For semiconductors and thermistors, resistivity decreases with temperature, enabling temperature sensing applications.

Kirchhoff's Laws and Circuit Analysis

Ohm's law alone describes individual resistors; Kirchhoff's laws handle complete circuits. Kirchhoff's Voltage Law (KVL): the sum of voltages around any closed loop is zero. Kirchhoff's Current Law (KCL): the sum of currents at any node is zero (charge conservation). Together with Ohm's law, these three relationships allow the analysis of any resistive circuit.

For resistors in series: R_total = R₁ + R₂ + ... (voltages add, current is the same). For resistors in parallel: 1/R_total = 1/R₁ + 1/R₂ + ... (currents add, voltage is the same).

Power Dissipation

The power dissipated as heat in a resistor is P = IV = I²R = V²/R. Joule heating is the basis of electric heaters, incandescent bulbs, fuses, and toasters. In power transmission lines, low current (achieved with high voltage via transformers) minimises I²R losses — this is why the national grid operates at hundreds of kilovolts.

Worked Examples

Example 1: A 12 Ω resistor is connected to a 6 V supply. Find the current and power.
I = V/R = 6/12 = 0.5 A. P = IV = 0.5 × 6 = 3 W.

Example 2: Two resistors 4 Ω and 6 Ω are in parallel across 12 V. Find the total resistance and total current.
1/R = 1/4 + 1/6 = 3/12 + 2/12 = 5/12. R = 12/5 = 2.4 Ω. I = V/R = 12/2.4 = 5 A.

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Non-Ohmic Devices and Real-World Limitations

Ohm's law (V = IR with constant R) holds for many materials at fixed temperature but fails for others. A diode allows current in only one direction; its I-V curve is exponential, not linear. A transistor has current controlled by a gate voltage, enabling switching and amplification. Incandescent bulbs violate Ohm's law because their resistance rises with temperature — resistance at operating temperature (~2,700 K) is about 10× higher than at room temperature.

In practice, Ohm's law provides an excellent first approximation for metallic conductors at constant temperature. The microscopic basis is the Drude model: conduction electrons scatter off lattice ions with a mean collision time τ, giving resistivity ρ = m/(ne²τ), where n is the electron density. This derivation reveals why higher temperature (shorter τ due to more vigorous lattice vibrations) increases resistivity in metals.

Key Equations Summary

Ohm's Law: V = IR. Resistance from geometry: R = ρL/A. Series: R_total = R₁ + R₂ + ... Parallel: 1/R_total = 1/R₁ + 1/R₂ + ... Power dissipated: P = IV = I²R = V²/R. KVL: sum of voltages around any closed loop = 0. KCL: sum of currents at any node = 0. The SI unit of resistance, the ohm (Ω), is named after Georg Simon Ohm, who published his law in 1827 after systematic experimental study of current flow through wires. Modern understanding derives resistance from electron-phonon scattering as described by the Drude and Sommerfeld models of metallic conduction.

Ohm's Law underpins the design of every electronic circuit. From simple resistor voltage dividers used in sensor circuits, to the internal resistance of batteries (which reduces terminal voltage under load), to the skin-effect resistance of conductors at high frequencies — all are analysed using V = IR. Kirchhoff's laws extend this to arbitrary circuit topologies, forming the foundation of circuit analysis taught in every electrical engineering curriculum worldwide.

References and further reading