Specific Resistance

Resistivity or specific resistance of a material is a measure of the resistance, which it offers to the flow of current through it. It is an intrinsic property of a material. Specific resistance depends on the composition, temperature, pressure of the material. The reciprocal of resistivity is defined as specific conductance which amounts to the ability to conduct electricity. The value of resistivity is very low for conductors and very high for insulators. The resistivity of a material is a scalar quantity. Like any other physical quantity, the description of specific resistance requires a number (magnitude), associated with a unit.

Definition of Resistivity

The resistance R of a conductor depends on its length L, cross-section A and its composition. For a fixed cross-section, the resistance is proportional to the length of the conductor. Whereas the resistance is inversely proportional to the cross-section for a fixed length. These two dependencies can be written down combinedly as,

R \[\propto\] \[\frac{L}{A}\]

R = \[\frac{\rho L}{A}\]

Here, is a proportionality constant which is known as the specific resistance. The constant has different values for different materials. Specific resistance depends on the physical properties of the material e.g. density and composition. For unit length and unit cross-sectional area i.e. L=1 and A=1,

\[\rho\] = R

This condition can be used to define resistivity.

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Resistivity

Definition: The resistance of a homogeneous chunk of a material of unit length and unit cross-section is defined as the resistivity or specific resistance of the material. Quantitatively,

\[\rho\] = \[\frac{RA}{L}\]

The conductivity of material is defined as the inverse of resistivity,

\[\sigma\] = \[\frac{1}{\rho}\]

Unit of Resistivity

From the specific resistance formula,

Unit of \[\rho\] = \[\frac{Unit \; of\; R\; \times \; Unit \; of \; A}{Unit \;of \;L}\]

One useful unit is obtained when resistance R is expressed in Ohm (Ω) and distances are expressed in centimeters (cm). In this convention, the unit of resistivity is Ohmcm (Ω . cm) .

If the distances are expressed in meter (m), the SI unit of specific resistance is given by Ohmm (Ω . m) . The SI unit of conductivity is Ohm-1 m-1 or siemens m-1 (S . m-1).

Metals are good conductors of electricity whereas insulators can carry little to zero current. The specific resistance of metals is very low ( \[\sim 10^{-8}\]Ω . m )but for typical insulators, the values are very large ( \[\sim 10^{16}\]Ω . m ). The values of resistivity at 200C for some standard materials are listed below.

Resistivity of Copper

The specific resistance of copper is 1.68 x 10-8 Ω. m (200C) i.e. the resistance between two opposite surfaces of a copper cube of side 1 m is 1.68 x 10-8 Ω at temperature 200C. The conductivity of copper is5.96 x 107 Sm-1 . Due to its very low resistivity and high conductivity, the resistance of copper to the flow of current is negligible. In electrical circuits, copper wires are widely used to conduct electricity.

Solved Examples

l. Find the length and cross-sectional area of a copper wire made from a chunk of copper having a mass of 10 g, if the resistance of the copper wire is 2 Ω. The density and resistivity of copper wire are 9 g/cm3 and 1.8 x 10-6 Ω . cm respectively.

Solution: The density of copper is d = 9 g/cm3 and the mass of the chunk is m = 10 g such that the volume of the copper wire is

V = \[\frac{m}{d}\]

V = \[\frac{10}{9}\] cm3

If the length of the wire is L and the cross-section is A, the volume is given by V=LA. Therefore,

LA = \[\frac{10}{9}\] cm3 (1)

Substituting R = 2Ω and \[\rho\] = 1.8 x 10-6 Ω . cm in the formula of specific resistance,

1.8 x 10-6 = 2 x \[\frac{A}{L}\]

\[\frac{L}{A}\] = \[\frac{10}{9}\] x 106 cm-1 (2)

Multiplying equations (1) and (2),

LA x \[\frac{L}{A}\] = (\[\frac{10}{9}\])2 x 106 cm2

L = 11.1 m

Dividing equation (1) by (2),

LA x \[\frac{A}{L}\] = 10-6 cm4

A = 0.1 mm2

The length and cross-sectional area of the copper wire are 11.1 m and 0.1 mm2 respectively.

ll. A 5 mm diameter wire is produced from a chunk of metal. Another wire of diameter 1 cm is produced from an identical chunk. What is the ratio of the resistance of the two wires?

Solution: Resistance of a wire of length L and cross-section A is R = \[\frac{\rho L}{A}, where \[\rho\] is the resistivity of the material. The mass and density of the chunk are m and D respectively. If the diameter of a wire of volume V is d,

A = \[\frac{\pi d^{2}}{4}\]

V = \[\frac{m}{D}\]

L = \[\frac{V}{A}\]

Therefore, the resistance of a wire of diameter d is,

R = \[\frac{16 \rho m}{\pi D a^{4}}\]

According to the problem, mass and density of the two wires of diameters \[d_{1}\] = 5 mm=0.5cm and \[d_{2}\] = 1 cm are same such that the ratio of resistance is,

\[\frac{R_{1}}{R_{2}}\] = \[(\frac{d_{2}}{d_{1}})^{4}\]

\[\frac{R_{1}}{R_{2}}\] = \[(\frac{1 \; cm }{0.5\; cm})^{4}\]

\[\frac{R_{1}}{R_{2}}\] = 16

The ratio of the resistance of the wires is 16:1.

Did You Know?

Specific resistance depends on the ambient temperature. It increases with increasing temperature for metals. For glass, however, at very high temperatures, the resistivity decreases considerably.

Superconductors have zero resistivity in the superconducting state (at very low temperature).

The resistivity of semiconductors decreases with increasing temperature.

FAQ (Frequently Asked Questions)

1. What Is Meant By Resistance and Resistivity?

Both resistance and resistivity quantify the amount of difficulty provided by a material to the flow of electricity. The resistance between two ends of a wire is the ratio of the potential difference of the two points to the current flowing through the conductor. Resistivity or specific resistance of a material is defined as the resistance of unit length and unit cross-section of that material.

2. What is the Difference Between Resistance and Specific Resistance?

Specific resistance is an intrinsic property of a material that depends only on the composition of a material. But resistance is an extrinsic property that depends on the length and cross-section of a conductor as well as the applied potential difference. The SI unit of resistivity is Ohm (Ω . m) whereas the unit of resistance is Ohm.

3. What is the Specific Resistance Value of Copper Conductor?

The specific resistance of copper is very low, the value being 1.68 x 10^{-8} Ω . m at 20^{0}C. Due to such low resistivity, copper is a good conductor of electricity.