Question

A copper wire has a diameter of 0.5 mm and a resistivity of. How much of this wire would be necessary to make a resistance of 10 ohm ?

Answer 1

Resistance is the property of a material or a conductor to resist the flow of current. When an electric current is made to pass through a conductor or current carrying wire, then the conductor offers some blockage to the flow of current or the movement of electrons. This is known as resistance. It is denoted by R. The unit of resistance is ohm.

Complete step by step solution:
Step I: Resistivity is defined as the resistance offered to a conductor of unit length and unit cross sectional area. It is represented by the symbol rho $\rho$. The unit of resistivity is ohm-meter.
It is known that resistivity is given by formula
$R = \rho \dfrac{l}{A}$
$l = \dfrac{{RA}}{\rho }$

Step II:
Given diameter of the wire is, $d = 0.5mm = 0.0005m$
Resistivity of the wire is, $\rho = 1.6 \times {10^{ - 6}}ohm - cm$
Resistance of the wire is, $R = 10\Omega$
Area of cross section of the wire is $A = \pi {r^2} = \pi {(\dfrac{d}{2})^2}$

Step III: Substituting the given values in the above formula and find the length,
$l = \dfrac{{10 \times 3.14 \times {{\left( {\dfrac{{0.0005}}{2}} \right)}^2}}}{{1.6 \times {{10}^{ - 8}}}}$
On solving, the above equation, the length of the conducting wire is
$l = 122.7m$

Step IV: Therefore, the length of the wire required to make a resistance of $10\Omega$ is $122.7m$.

Note: It is to be noted that the terms resistance and resistivity are completely different. The resistance of a conductor varies directly with the length and temperature, and inversely proportional to the cross sectional area. But the resistivity depends only on the temperature and nature of the material of the conductor. The resistivity of a material increases as the temperature increases. Also resistance is directly proportional to the resistivity of the material of the conductor.