Factor Affecting the Resistance: Electrical Resistivity or Specific Resistance
State the factors on which the resistance of a conductor depends at constant temperature and hence define electrical resistivity or specific resistance. Give SI unit and dimensional formula of specific resistance.
Resistance of a conductor at constant temperature depends upon:
(a) Length of the conductor: The resistance R of a conductor is directly proportional to its length $l$.
That is,
$R \propto l \quad \ldots (i)$
More the length of a conductor, more is its resistance.
(b) Area of cross-section of the conductor: The resistance of a conductor is inversely proportional to its area of cross-section A.
That is,
$R \propto \frac{1}{A} \quad \ldots (ii)$
Thus, thin wire has more resistance than the thick wire of same length.
Combining eqns. (i) and (ii), we get
$R \propto \frac{l}{A}$
or
$R = \rho \left( \frac{l}{A} \right) \quad \ldots (iii)$
where, $\rho$ is known as specific resistance or resistivity of the conductor of a given material.
Thus,
$\rho = R \left( \frac{A}{l} \right)$
If $A = 1$, $l = 1$, then $\rho = R$
Thus, resistivity of a conductor of a given material is defined as the resistance of the conductor of unit length and unit area of cross-section.
S.I. Unit of Resistivity :
Since, $\rho = R \left( \frac{A}{l} \right)$, therefore, S.I. unit of $\rho$ is $\frac{\text{ohm metre}^2}{\text{metre}}$ or ohm-metre ($\Omega$ m)
Dimensional formula of resistivity :
Resistivity, $[\rho] = \frac{[R] \times [A]}{[L]} = \frac{[ML^2T^{-3}A^{-2}] [L^2]}{[L]} = [ML^3T^{-3}A^{-2}]$
Important facts:
1. Resistivity of a conductor depends upon the temperature of the conductor also. If temperature of a conductor increases, its resistivity also increases. If the temperature of a conductor decreases, its resistivity also decreases.
2. Resistivity of an object does not depend upon the dimensions of the objects. It depends upon the nature of the material of the conductor.