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Resistance

Level 1 - Physics topic page in Current and Resistance.

Principle

Resistance is the ratio of potential difference across a device to current through it.

Notation

\(R\)
resistance
\(\mathrm{\Omega}\)
\(V\)
potential difference across a device
\(\mathrm{V}\)
\(I\)
current through a device
\(\mathrm{A}\)
\(\rho\)
resistivity
\Omega\,m
\(L\)
conductor length
\(\mathrm{m}\)
\(A\)
cross-sectional area
\(\mathrm{m^{2}}\)

Method

Derivation 1: Define resistance from an operating point

Resistance compares the voltage across a two-terminal device with the current through it.

Resistance definition
\[R=\frac{V}{I}\]
Ohm
\[1\,\Omega=1\,\mathrm{V\,A^{-1}}\]
Ohmic device
\[V=IR\]
\(R\) is constant only for an ohmic device in its operating range.

Derivation 2: Derive resistance of a uniform wire

Combine the material relation with the geometry of a straight wire.

Material law
\[E=\rho J\]
Uniform field and current density
\[E=\frac{V}{L},\qquad J=\frac{I}{A}\]
Substitute
\[\frac{V}{L}=\rho\frac{I}{A}\]
Wire resistance
\[R=\frac{V}{I}=\rho\frac{L}{A}\]

Derivation 3: Read an \(I\)-\(V\) graph

For an ohmic resistor, \(I\) is proportional to \(V\). The graph slope depends on which quantity is on the vertical axis.

\(V\) versus \(I\)
\[\text{slope}=\frac{\Delta V}{\Delta I}=R\]
\(I\) versus \(V\)
\[\text{slope}=\frac{\Delta I}{\Delta V}=\frac{1}{R}\]

Rules

These are the compact resistance relations.

Resistance
\[R=\frac{V}{I}\]
Ohm's law
\[V=IR\quad\text{for an ohmic resistor}\]
Uniform wire
\[R=\rho\frac{L}{A}\]
Ohm
\[1\,\Omega=1\,\mathrm{V\,A^{-1}}\]

Examples

Question
A resistor has
\[12\,\mathrm{V}\]
across it and current
\[0.40\,\mathrm{A}\]
Find \(R\).
Answer
\[R=\frac{V}{I}=\frac{12}{0.40}=30\,\Omega\]

Checks

  • \(V=IR\) is a model for ohmic behavior, not every device.
  • Wire resistance increases with length and decreases with cross-sectional area.
  • On a \(V\)-against-\(I\) graph, slope is \(R\); on an \(I\)-against-\(V\) graph, slope is \(1/R\).
  • Resistance is positive for ordinary passive resistors.

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