Question

The electrical resistance of a wire is directly proportional to its length. If a $48$-foot-long wire has a resistance of $16\,ohms$, how do you find the resistance of a $36$-foot-long wire?

Answer 1

Resistance is the hindrance offered to the flow of current by a conductor is dependent upon the geometry of the conductor. In this question, we are given the lengths of conductors for which we have to find the resistance. We shall first calculate the ratio of the resistivity and the cross-sectional area from the first set of parameters and then plug in this value to find the resistance of the 36-foot-long wire.

Complete step by step answer:
Resistance is the inherent property of conductors. It is the opposition caused to the flow of current through the conductor due to collisions happening within the conductor. The value is dependent on the geometry as:
-Directly proportional to the length of the conductor
-Inversely proportional to the cross-sectional area
This is given as $R \propto \dfrac{l}{A}$
Removing the proportionality sign, we have
$R = \rho \dfrac{l}{A}$ where $\rho$ is the resistivity of the conductor.

Given that a 48-foot-long wire has a resistance of 16 ohms,
Substituting the values we have,
$16 = \rho \dfrac{{48}}{A}$
This can be rewritten as
$\dfrac{{16}}{{48}} = \dfrac{\rho }{A}$
$\Rightarrow \dfrac{\rho }{A} = \dfrac{1}{3}\,\,\,\,\,\,\,\,\,\,\,\,.......(1)$
Now we are required to find the resistance of a 36-foot-long wire.
Substituting the known values in the formula for resistance we have
$R = 36\dfrac{\rho }{A}$
From (1) we can say that
$R = 36 \times \dfrac{1}{3}$
Further solving this we get,
$\therefore R = 12\,\Omega$

Hence, the resistance of a 36-foot-long wire is $R = 12\,\Omega$.

Note: The resistance of a conductor increases with temperature since the intermolecular collisions increase due to increased energies of the electrons. The resistivity also increases with temperature. However, at a constant temperature and for a given material, the resistivity is a constant whereas resistance can vary. This can be done by increasing or decreasing the length or the cross sectional area.