Electric resistance of a constantan and a copper wire

Task number: 2119

How long is a constantan wire with a diameter of 0.6 mm that has a resistance of 10 Ω? What length of a copper wire of the same diameter would have the same resistance?

  • Hint

    Remember of look up the formula of the resistance of a wire with a uniform cross-section.

  • Solution

    The electrical resistance of a wire with uniform cross-section is given by the formula

    \[R=\rho\frac{l}{A}.\]

    We rearrange the formula to express the length of th wire:

    \[l =\frac{RA}{\rho}.\]

    We need to calculate the cross-sectional area A of the wire from its diameter d (assuming a cylindrical shape of the wire)

    \[A=\pi\left(\frac{d}{2}\right)^2.\]

    Substituting into the previous formula, we get the length of the constantan wire:

    \[l_{\mathrm{Co}}=\frac{R\pi \left(\frac{d}{2}\right)^2}{\rho_{\mathrm{Co}}}= \frac{R\pi d^2}{4\rho_\mathrm{Co}}.\]

    where ρCo is the constantat resistivity.

    Analogically, we get the length of the copper wire with resistivity ρCu

    \[l_{\mathrm{Cu}}=\frac{R\pi \left(\frac{d}{2} \right)^2}{\rho_{\mathrm{Cu}}}=\frac{R\pi d^2}{4\rho_\mathrm{Cu}}.\]
  • Notation and numerical calculation

    \(R = 10\,\mathrm\Omega\) Wire resistance
    \(d = 0.6\,\mathrm{mm}=0.6{\cdot}10^{-3}\, \mathrm m\) Wire diameter
    \(l_{\mathrm{Co}}= ?\,\mathrm{ (m)}\)Constantan wire length
    \(l_{\mathrm{Cu}}= ?\,\mathrm{ (m)}\)Copper wire length
    From the tables:
    \(\rho_{\mathrm{Co}}=0.5\,\mathrm{\mu \Omega m}=0.5{\cdot}10^{-6}\,\mathrm{\Omega m}\)Constantan electrical resistivity
    \(\rho_{\mathrm{Cu}}=0.018\,\mathrm{\mu \Omega m}=0.018{\cdot}10^{-6}\,\mathrm{\Omega m}\)Copper electrical resistivity

    \[l_{\mathrm{Co}}=\frac{R\pi d^2}{4\rho_{Co}}=\frac{10\,\pi \left(0.6{\cdot} 10^{-3}\right)^2}{4{\cdot} 0.5{\cdot} 10^{-6}}\,\mathrm m = 5.7\, \mathrm m\] \[l_{\mathrm{Cu}}=\frac{R\pi d^2}{4\rho_{Cu}}=\frac{10\,\pi \left(0.6{\cdot} 10^{-3}\right)^2}{4{\cdot} 0.018{\cdot} 10^{-6}} \,\mathrm m = 157\, \mathrm m\]
  • Odpověď

    The length of the constantan wire is 5.7 m. The copper wire would have to be 157 m long.

Difficulty level: Level 2 – Upper secondary level
Task requires extra constants
Tasks requiring routine calculations
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