# Specific Resistance

## What is known as Specific Resistance?

Specific resistance of a material is defined as the electric resistance of that material of unit length and unit cross sectional area.

• Specific Resistance of a material is also known as Resistivity.

Let, a conductor of length ( l ) and cross sectional area ( A ) has a resistance ( R ) .

Then, the resistance of conductor will depend as –

1. R \propto l
2. R \propto \left ( \frac { 1 }{ A } \right )

Therefore, \quad R = \rho \left ( \frac { l }{ A } \right ) ………. (1)

Here, ( \rho ) is a constant of proportionality which depends upon the material of the conductor. It is known as resistivity or specific resistance of that material.

If a conductor is taken such that its length is unit i.e. ( l = 1 ) and area of cross section is also unit i.e. ( A = 1 ) then from equation (1) –

R = \rho \left ( \frac { 1 }{ 1 } \right )

Therefore, \quad \rho = R .

Hence, resistivity is the resistance of a material body of unit length and unit area i.e. of unit volume.

• SI unit of specific resistance or resistivity is ohm-metre Ωm )

### Temperature coefficient of Specific Resistance

Resistivity or specific resistance of a material depends upon temperature of the conductor. It increases with the increase in temperature as shown in figure.

At low temperatures, specific resistance increases at a higher rate as compared with rate of increase at higher temperatures. The relation between specific resistance and temperature is expressed as –

\rho = \rho_0 [ 1 + \alpha ( T - T_0 ) ]

So, \quad ( \rho - \rho_0 ) = [ \rho_0 \alpha ( T - T_0 ) ]

Or, \quad \alpha = \left [ \frac {\left ( \rho - \rho_0 \right )} {\rho_0 \left ( T - T_0 \right )} \right ] …….. (2)

Where ( \rho_0 ) is the resistivity at temperature \left ( T_0 = 0 \degree C \right ) and ( \rho ) is the resistivity at temperature ( T = t \degree C ) and ( \alpha ) is the temperature coefficient of specific resistance.

Therefore, temperature coefficient of specific resistance or resistivity is defined as the rate of change in resistivity by unit temperature change per unit original resistivity.

Thus, \quad \alpha = \left [ \frac { \left ( \rho - \rho_0 \right )}{ \rho_0 \left ( T - T_0 \right )} \right ]

= \left ( \frac { \Delta \rho }{ \rho_0 \Delta T } \right )

Temperature coefficient of specific resistance ( \alpha ) is different for different materials.

### Specific Resistance for Conductors

For good conducting metals, the resistivity is directly proportional to the temperature.

Therefore, \quad \rho \propto T .

• For good conductors the value of temperature coefficient of resistivity ( \alpha ) is positive.
• Their resistivity increases linearly with increase in temperature as shown in figure (A).
• For most of the metals, resistivity has linear relationship with temperature at normal temperature ranges.

### Specific Resistance for Semiconductors

For semiconductors, the resistivity decreases with increase in temperature. The variation of resistivity with temperature is given by the relation –

\rho = \rho_0 \ e^{E / 2 k T}

Where ( E ) is energy gap between conduction band and valence band ( k ) is Boltzmann’s constant and ( T ) is the absolute temperature.

• For semiconductors the value of temperature coefficient of resistivity ( \alpha ) is negative.
• Their resistivity decreases more rapidly with increase in temperature as shown in figure (B).
• At low temperatures, semiconductors behave as insulators but at room temperature they behave as conductors.

Examples of semiconductors are Germanium and Silicon etc.

### Specific resistance for Insulators

• Resistivity of insulators is similar to the resistivity of semiconductors.
• It decreases exponentially with increase in temperature as shown in figure (B).

The variation of resistivity of insulators with temperature is given by the same relation similar to semiconductors –

\rho = \rho_0 \ e^{E / 2 k T} .

• Energy gap ( E ) for insulators is much more than that of semiconductors.

## Electrical Conductance

The reciprocal of electric resistance is called electric conductance.

Therefore, electric conductance of a conductor is given by –

\quad G = \left ( \frac { 1 }{ R } \right ) .

• SI unit of conductance is {\text {ohm}}^{ - 1 } ( \omega^{ - 1 } ) or mho or siemen (S).

### Electrical Conductivity

The reciprocal of electrical resistivity is called electrical conductivity.

Therefore, electrical conductivity of a material is given by –

\sigma = \left ( \frac { 1 }{ \rho } \right )

• SI unit of conductivity is {\text {ohm}}^{ - 1 } m^{ - 1 } ( \omega^{ - 1 } m^{ -1 } ) or mho per mtr or siemen per mtr ( Sm -1).

### Effect of temperature on Electrical Conductivity

We know that, resistivity of a conductor is directly proportional to the temperature.

Therefore, \quad \rho \propto T .

Also, conductivity is the reciprocal of resistivity. Hence, conductivity is inversely proportional to temperature.

Therefore, \quad \sigma \propto \left ( \frac { 1 }{ T } \right ) .

As the temperature of conductor increases, the conductivity decreases exponentially as shown in figure.

## Super Conductivity

We know that the resistance of conductors increases with increase in temperature. Therefore, materials offer low resistance at low temperatures.

The property of a material which shows almost zero resistance at a very low temperature is called super-conductivity.

Materials like mercury, lead, alloys and oxides of certain metals etc. shows the property of super-conductivity at very low temperatures. These materials are called super-conductors.

Figure shows the behavior of mercury at low temperatures. Resistance of mercury (Hg) drops abruptly to zero below ( 4.2 \ K ) .

The temperature below which a material becomes super conductor is called Transition temperature or Critical temperature ( Tc ).

• Critical temperature ( T_c ) for mercury is ( 4.2 \ K ) .

### Use of Super Conductors

Electrical power consumption of electrical circuit solely depends upon magnitude of flow of current and resistance of circuit. Thus, if an element has zero resistance it means no power loss will occur in that element. Therefore, super-conductors have a number of useful applications because of negligible resistance.

Uses of super-conductors are –

1. Coils made of super conducting materials are used for making powerful magnets.
2. Transmission wires are made of super conducting materials which results for long transmission of electricity without power loss.
3. Super conductors are also used in high speed computing machines.
4. It is used in thin film devices.