Question

What is the relation between resistance R and resistivity of a wire?
A. \[R = \rho \dfrac{A}{l}\]
B. \[R = \rho \dfrac{l}{A}\]
C. \[R = \dfrac{A}{{\rho l}}\]
D. \[R = \dfrac{l}{{\rho A}}\]

Answer 1

Hint:Resistance is the property of a component to resist the flow of an electric current or electricity. Resistivity is the ability of a material which measures the resisting power of electric current. Because of this reason, the resistance is directly proportional to the resistivity.

Complete step by step answer:
We already know that resistance is directly proportional to the resistivity.
Mathematically, \[r\alpha \rho \] …… (1)
Practically it can be seen that, when the length of the wire (material) is increases then the resistance of the wire also increases and when the length of the wire (material) is decreases then the resistance of the wire also decreases Because of this the resistance is directly proportional to the length of the wire.
Mathematically, \[r\alpha l\] …… (2)
If we increase the cross sectional area of the material then due to the larger area, electrons get more space for travelling it means resistance to the electric current will decrease. Because of this the resistance is inversely proportional to the cross sectional area of the material or wire.
Mathematically, \[r\alpha \dfrac{1}{A}\] …… (3)
On comparing the equation (1),(2)and(3),we get the final relation as:
\[\begin{gathered}
  r\alpha \rho \times l \times \dfrac{1}{A} \\
  r = \rho \dfrac{l}{A} \\
\end{gathered} \]
Therefore, the correct relation between resistance R and resistivity \[{\text{\rho }}\]of a wire is, \[r = \rho \dfrac{l}{A}\]

Hence, the correct option for this problem is, B.

Note:The unit of resistance is ‘Ohm’ and the unit of resistivity is ‘Ohm-m’. Resistance is a macroscopic property while resistivity is a microscopic property of the material and resistivity is inversely proportional to the conductivity.