Question

A negligibly small current passed through a wire of length 15m and uniform cross-section $6 \times {10^{ - 7}}{m^2}$ and its resistance is measured to be $5\Omega$ what is the resistivity of the material at this temperature of experiment?

Answer 1

Hint: The resistance is the obstruction offered to the flow of current through it. However, it should be known that the resistance depends on the dimensions of the material and an inherent property of the material known as resistivity.

Complete step by step answer:
The concept of resistance is given by the Ohm’s law, which states that the voltage and current are directly proportional to each other.
$V \propto I$
The constant in this equation is called the resistance, which represents the obstruction offered to the flow of current I, when a potential difference of V is maintained.
$R = \dfrac{V}{I}$
However, the resistance of a conductor is given by the formula –
$R = \dfrac{{\rho l}}{A}$
where l = length, A = area of cross-section and $\rho$= resistivity of the material.
Given that length $l = 15m$
Area of cross-section $A = 6 \times {10^{ - 7}}{m^2}$,
Resistance of material $R = 5\Omega$
Substitute all the values in above equation,
$5 = \dfrac{{\rho \times 15}}{{6 \times {{10}^{ - 7}}}}$
Simplifying,
$\rho = \dfrac{{5 \times 6 \times {{10}^{ - 7}}}}{{15}}$
$\Rightarrow \rho = 2 \times {10^{ - 7}}\Omega m$
Hence, the value of resistivity, $\rho = 2 \times {10^{ - 7}}\Omega m$

Note: There are two quantities, resistance and resistivity, both of which represent the obstruction offered by the material to flow of the current. However, the quantity resistivity is an independent quantity whose value is obtained from standards at a particular constant temperature and the quantity resistance is dependent on the physical dimensions of the material.