Question

(a) Define resistivity. Write an expression for the resistivity of a substance. Give the meaning of each symbol which occurs in it.
(b) State the SI unit of resistivity.
(c) Distinguish between resistance and resistivity.
(d) Name two factors on which the resistivity of a substance depends and two factors on which it does not depend.
(e) The resistance of a metal wire of length 1 m is 26 Ω at 20°C. If the diameter of the wire is 0.3 mm, what will be the resistivity of the metal at that temperature?

Answer 1

(a) We know that resistance,
$R=\frac{\mathrm{\rho }l}{A}$

So resistivity,

$\mathrm{\rho }=\frac{RA}{l}$
When the area of cross-section, A = 1 m² and the length of the wire, l = 1 m, $\mathrm{\rho }=R$.

Resistivity is the resistance of the conductor whose area of cross-section, A is 1 m² and length, l is 1 m.

(b) The SI unit of resistivity is Ω m or ohm-metre.

1 ohm-metre is the resistance of a conductor that has a resistance of 1 ohm and whose area of cross-section, A is 1 m² and length, l is 1 m.

(c) Difference between resistance and resistivity:

Resistance	Resistivity
1. It depends on the dimensions of the conductor like radius and length. 2. Resistance of a given material may change at constant temperature.	1. It does not depend on the dimensions of the conductor. 2. Resistivity of a given material does not change at constant temperature.

(d) Resistivity of a substance depends on the material and the temperature of the substance. It does not depend on the dimensions of the substance like length, radius etc.

(e) Length of the wire, l = 1 m

Diameter of the wire, d = 0.3 mm

So the radius of the wire, r = 0.15 mm = 0.00015 m = 1.5 x 10^-4 m

Area of cross-section of the wire, $A={\mathrm{\pi r}}^2=3.14\times (1.5\times {10}^{-4}{)}^2=7.065\times {10}^{-8}{\mathrm{m}}^2$

Resistance of the wire, R = 26 Ω

So resistivity,
$\rho =\frac{RA}{l}$

$$\begin{gathered} p=\frac{26\times 7.065\times {10}^{-8}}{1} \\ p=1.84\times {10}^{-8}\mathrm{\Omega }\mathrm{m} \end{gathered}$$

Thus, the resistivity of the material is $1.84\times {10}^{-8}\mathrm{\Omega }\mathrm{m}$.