Question

A copper has a diameter 0.5mm and resistivity of $1.6 \times {10^{ - 8}}\Omega m$. What will be the length of this wire to make its resistance $10\Omega $? How much does the resistance change if the diameter is doubled?

Answer 1

Hint: Resistance of a conductor is the property by virtue of which it opposes the flow of charge through it. The resistance of a conductor is given by $R = \rho \dfrac{l}{A}$ . the area of cross-section of the conductor is given by $A = \dfrac{{\pi {d^2}}}{4}$. Using these formulas we can calculate the length of the given conductor.

Complete step-by-step solution -
We know that at a constant temperature, the resistance of a conductor depends on its length, areas of cross-section, and nature of the material.
Mathematically:
$R = \rho \dfrac{l}{A}$
Also, area of cross section A is given by:
$A = \dfrac{{\pi {d^2}}}{4}$
This implies resistivity of a conductor is also given by:
$\eqalign{
  & R = \dfrac{{\rho l}}{{\dfrac{{\pi {d^2}}}{4}}} \cr
  & \Rightarrow l = \dfrac{{R\pi {d^2}}}{{4\rho }} \cr} $
Given:
Diameter of copper, $d = 0.5mm$
Resistivity of copper, $\rho = 1.6 \times {10^{ - 8}}\Omega m$
Resistance of copper, $R = 10\Omega $
Substituting values in the above equation, we get:
$\eqalign{
  & l = \dfrac{{10 \times 3.14 \times {{\left( {0.5 \times {{10}^{ - 3}}} \right)}^2}}}{{4 \times 1.6 \times {{10}^{ - 8}}}} \cr
  & \Rightarrow l = 122m \cr} $
It is clear from the above equation of resistance that resistance R is inversely proportional to the square of the diameter of the conductor.
Mathematically:
 $R \propto \dfrac{1}{{{d^2}}}$
So, if resistance is doubled, then as a consequence, the resistance will become one-fourth of its original value.
Resistance of copper, $R = 10\Omega $
$\eqalign{
  & {R_ \circ } = 10\Omega \cr
  & R = \dfrac{1}{4}{R_ \circ } \cr
  & \Rightarrow R = \dfrac{1}{4} \times 10\Omega \cr
  & \Rightarrow R = 2.5\Omega \cr} $
Therefore, a copper has a diameter 0.5mm and resistivity of $1.6 \times {10^{ - 8}}\Omega m$, has a length of 122 meters to make its resistance $10\Omega $. Additionally, the resistance change if the diameter is doubled will be one-fourth its original value i.e., $R = 2.5\Omega $

Note: Resistance of a conductor also depends upon the temperature in which the system is established. For most conductors, their resistance increases with the increase in temperature. Additionally, if some conductors are cooled down to a certain degree of temperature, their resistance becomes zero. But reaching such temperatures is possible only theoretically, and they are unachievable in the real world.