Question

Derive \[R = \rho \dfrac{l}{A}\].

Answer 1

Hint: Recall the factors on which the resistance of the conductor depends. Identify the proportionality of these factors and write the equation. Assume the proportionality constant as resistivity of the conductor.

Complete step by step answer:
We know that a resistance of a conductor is proportional to the length of the conductor at constant temperature and constant cross-sectional area. With increase in the length, the resistance also increases, therefore, we can write,
\[R \propto l\] …….. (1)
We also know that the resistance of the conductor varies inversely with the cross-sectional area of the conductor. This means that if the area of cross-section of the conductor is greater, the current flowing through the conductor will be greater and conversely the resistance will be smaller. Also, if the area of cross-section of the conductor is smaller, the path of the current gets narrowed and hence the resistance will be larger. Therefore, we can write,
\[R \propto \dfrac{1}{A}\], where, A is the area of cross-section of the conductor. …… (2)
From equation (1) and (2), we can write,
\[R \propto \dfrac{l}{A}\]
\[ \Rightarrow R = \rho \dfrac{l}{A}\]
Here, \[\rho \] is the proportionality constant and it is known as resistivity of the conductor.
The resistivity of a material is the property which measures how strongly it opposes the flow of current through the conductor. Therefore, the resistance should be proportional to the resistivity.
So, we have derived the formula for resistance of a conductor given by,
\[ \Rightarrow R = \rho \dfrac{l}{A}\]

Note:
Since the resistivity is inverse of conductivity, you can also write the formula as, \[R = \dfrac{l}{{\sigma A}}\], where, \[\sigma \] is the conductivity. The term A in the formula is the cross-sectional area of the conductor and not the total surface area. The cross-sectional area of the conductor of radius of cross-section r is expressed as, \[A = \pi {r^2}\].