Electric field lines between charged plates and magnetic field patterns

Ohm's law and resistivity

PHYS 301 · Current and DC Circuits

Ohm's law relates voltage, current, and resistance for ohmic materials. This lesson explains resistance, resistivity, geometry, temperature effects, and non-ohmic devices.

Key equations

V=IRR=\rho\frac{L}{A}\sigma=\frac{1}{\rho}\vec{J}=\sigma\vec{E}\vec{E}=\rho\vec{J}\rho=\rho_0[1+\alpha(T-T_0)]P=IV=I^2R=\frac{V^2}{R}

Learning objectives

State Ohm's law and identify ohmic behavior.
Relate resistance to resistivity and geometry.
Use conductivity and microscopic Ohm's law.
Describe temperature effects in metals and semiconductors.
Recognize non-ohmic devices.

Ohm's law

Ohm's law states that, for many materials and devices under suitable conditions, current is proportional to voltage:

$V=IR$

Here $R$ is resistance. A device that obeys this linear relationship is called ohmic. Its current-voltage graph is a straight line through the origin with slope related to resistance.

Ohm's law is not a universal law of nature like charge conservation. It is a material behavior that holds approximately for many conductors over certain temperature and voltage ranges.

Resistance and geometry

For a uniform wire of length $L$ and cross-sectional area $A$ , resistance is

ho rac{L}{A}$$ where $ ho$ is resistivity. Longer wires have greater resistance because charges scatter over a longer path. Wider wires have lower resistance because more charge can flow in parallel. Resistivity is a property of the material. Resistance depends on both material and geometry. ## Conductivity Conductivity is the reciprocal of resistivity: $$sigma= rac{1}{ ho}$$ The microscopic form of Ohm's law is often written $$ ec{J}=sigma ec{E}$$ This says current density is proportional to electric field in an ohmic material. Equivalently, $$ ec{E}= ho ec{J}$$ ## Temperature dependence For many metals, resistivity increases with temperature. As temperature rises, lattice vibrations increase, causing more electron scattering. A common approximation is

ho= ho_0[1+alpha(T-T_0)]$$

where $alpha$ is the temperature coefficient of resistivity.

For semiconductors, resistivity often decreases with temperature because more charge carriers become available. This difference is central to electronics.

Superconductors

Some materials become superconductors below a critical temperature. In the superconducting state, electrical resistance drops to zero. A current can persist without continuous energy loss.

Superconductivity is a quantum phenomenon and cannot be explained by simple classical resistivity. It has applications in MRI machines, particle accelerators, sensitive magnetometers, and research magnets.

Non-ohmic devices

Many devices do not obey $V=IR$ with constant resistance. Diodes, transistors, filament bulbs, gas discharge tubes, and electrolytes can have nonlinear current-voltage relationships.

A light bulb filament heats up as current increases, changing its resistance. A diode allows current much more easily in one direction than the other. For such devices, resistance may depend on voltage, current, temperature, or direction.

Power and Ohmic heating

For an ohmic resistor,

$P=IV=I^2R= rac{V^2}{R}$

This energy transfer appears as thermal energy. Electric heaters intentionally use resistive heating. In power transmission, resistive heating in wires is usually an unwanted loss.

Reducing current for a given transmitted power reduces $I^2R$ losses, which is why high-voltage transmission is useful.

The big idea

Ohm's law describes linear voltage-current behavior in ohmic materials. Resistance depends on resistivity and geometry through $R= ho L/A$ . Resistivity depends on material and temperature, and many important devices are non-ohmic. Understanding Ohm's law and its limits is essential for circuit analysis and electrical technology.

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