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Mathematics for Physics

53 Resistance

A property of a conductor that is used to relate the potential difference across it to the current that flows through it.

definition [d] (Resistance) From Knight: we define the resistance of a conductor to be

  • \(R = \dfrac{\rho L}{A}\) ,

where \(\rho\) is the resistivity, \(L\) is the length, and \(A\) is the cross-section area. Establishing a potential difference \(\Delta V\) between the ends of a conductor of resistance \(R\) creates a current. This simple relationship between potential difference and current is known as Ohm’s law:

  • \(I = \dfrac{\Delta V}{R}\) .

where

  • \(R\) is the resistance.
  • \(\rho\) is the resistivity of the material.
  • \(L\) is the length of the conductor.
  • \(A\) is the cross-sectional area.
  • \(I\) is the current.
  • \(\Delta V\) is the potential difference between the ends.

53.1 Elementary Example

53.1.1 Simple

A resistor with \(R = 10\,\Omega\) across \(\Delta V = 5\,\mathrm{V}\) carries

\[ I = \dfrac{\Delta V}{R} = 0.5\,\mathrm{A} \]

where

  • \(I\) is the current through the resistor.

53.1.2 General

For fixed \(\rho\) and \(A\), tripling the length triples \(R\).

\[ R_{i} = \dfrac{\rho L_{i}}{A},\quad L_{i} \in \{L,2L,3L\} \]

\[ I_{i} = \dfrac{\Delta V}{R_{i}} \]

where

  • longer conductors give smaller current at the same \(\Delta V\).