Question

What is the resistance R given by in terms of length L, cross section A and the resistivity $ \rho $ of the wire?
(A) $ \dfrac{{\rho L}}{{{A^2}}} $
(B) $ \dfrac{{\rho L}}{A} $
(C) $ \dfrac{{\rho {L^2}}}{{{A^2}}} $
(D) $ \dfrac{{\rho {L^2}}}{A} $

Answer 1

Hint
The resistance of an object is dependent on its shape and the material of which it is made. It offers opposition to the current flow and is used widely in electrical circuits.

Complete step by step answer
The resistance of a body depends both on its dimensions and the material of which it is composed. As it is the opposition offered to current flow, its value will depend on factors that enhance or reduce the flow of current through an object.
As stated in the question, consider a cylindrical wire. For the whole length of the wire, the electrons will travel to the end and collide with one another. The longer the length, the more will be the collisions and hence, the resistance to current flow would increase. This means the resistance directly depends on the length of the wire.
Now, consider the cross section of the wire. For a large cross section, the amount of current that the wire will be able to carry will be more compared to when the diameter would be small. This is when all other parameters are considered constant. Hence, smaller diameter increases the resistance. So, resistance is inversely proportional to the cross section of the wire.
Lastly, consider the resistivity. Resistivity is a material property and does not depend on the shape or size of the wire. The difference in resistance offered by two wires of the same dimension would solely depend on the material which they are made of. Hence, the resistivity adds to the resistance and thus, the resistance is directly proportional to it as well.
Combining all of these results gives us:
 $ R = \dfrac{{\rho L}}{A} $
Hence, the correct answer is option (B).

Note
We can compare this property of resistance with the flow of water through a pipe. For a longer length of the pipe, the water will come out slowly. Similarly, for a larger cross section of the pipe, more water will be able to enter. Apart from the given parameters in the question, the resistance of a wire may also depend on the temperature. As the temperature increases, the collisions of electrons in the wire will increase and hinder the flow of current, adding to the resistance.