Resistivity of Materials

What is Resistivity?

Resistivity is the measure of how much an electrical conductor opposes the flow of current through it.

Resistance has an application in protecting the circuit from high current flow.

When a potential difference (acceleration) is applied across the conductor (to car), the electrons start moving from negative to positive electrode).

The current flow increases, the resistance acts as a speed breaker to the accelerated car (high current flow).

The magnitude of the resistance is called the resistivity.  

Hence it is a magnitude of the resistance of a given size of a specific material or a conductor to electrical conduction.

Resistivity Formula

Resistivity of a material  is defined in terms of measurement of the electric field (E) across it that generates current density (J) . 

The formula for resistivity is given by,                   

ρ  = E /J 


and 


R = ρ L/A


Where ρ is the proportionality constant known as the resistivity of the material which  is the characteristic property of each material.

A = Area of cross-section

L = Length of the material of a conductor

Derive Resistivity

Resistivity of a material depends upon the following factors:

1. Length

Consider two conductors each of length ‘L’ and the area of cross-section ‘A’

Let V be the same potential difference applied across the ends of two slabs. 

Current ‘I’  flowing across each slab will be I/2. 

Then resistance via each slab is,

R = V/I (Ohm’s law)

Rs = V/ I/2 = 2 R

So, R increases with the increase in length 

R α L …(1)

2. Area of Cross-section 

Each slab of length ‘L’ has a cross-sectional area of A/2.

Similarly, on halving the area of the conductor, the resistance through each of the half slab will be

R’  =   V/ I/2 = 2 R

R increases with the decrease in the area of each half slab.

R α 1/A…(2)

Combining (1) and (2) we get

R   α L/A    

Removing the proportionality sign we get         

R = ρ L/A


Here, ρ is called the electrical resistivity or specific resistance of the material.

Resistivity definition

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The formula for the resistivity is given by,

R = ρ L/A…(a)

If L =1, A =1, then  R = ρ 

Thus, electrical resistivity of a material of a conductor is defined as the resistance offered by unit length and unit cross-sectional area of a wire of the given material.

Unit of Resistivity

The unit of resistivity is derived from eq(a)

If  R = ρ L/A

Then   ρ = R.A/L ….(b)

Given unit of R = Ohm (Ω),  A = m² and L = m 

Putting in eq(b)  we get

S.I. Unit of  ρ   = ohm . m²/ m 

= ohm . m = Ω . m   

In CGS system =  ohm.cm 


Define Resistivity of a Material

The resistivity is an attribute of each material which is useful in comparing various materials on the basis of their ability to conduct electric currents. 

Let’s Discuss the Resistivity of Some Materials are Discussed Below:

Name of the Material

Resistivity at 0°C

Name of the Material

Resistivity at 0°C

A. Conductors

3. Semiconductors


1. Metals

Carbon (Graphite)

3.5 x 10-8

Silver 

1.6 x 10-8

Germanium

0.46

Copper 

1.7 x 10-8

Silicon

2300

Aluminium

2.7 x 10-8

4. Insulators


Tungsten

5.8 x 10-8

Glass

1010 - 1014

Iron

10 x 10-8

Hard rubber

1013 - 1016

Platinum

11 x 10-8

Mica

1011 - 1015

Mercury

98 x 10-8

Wood

108 - 1011

Palladium

1.0 x 10-7

Paper (dry)

1012

2. Alloys

Amber

5 x 1014

Nichrome (Alloy - Iron, Nickel, Chromium)

100 x 10-8

Quartz

(fused)

7.5 x 1017

Manganin

44 x 10-8

Diamond

1012 - 1013

Constantin

49 x 10-8

Ebonite

1015 - 1017


Relation Between Conductivity and Resistivity 

The relation between conductivity and resistivity can be understood through an example.

You water a lot to the plants during the summer seasons.

If you just sprinkle a few drops of water and won’t supply enough water, after some time, they will get dried and hence die.

Therefore, the more is the resistance to sufficient supply of water to the plants, lesser will be their growth (conductivity).

Therefore, high resistivity signifies poor conductors.

 Resistivity is symbolized by a Greek letter  ‘ρ’ pronounced as ‘rho’ and the conductivity   as σ.

So, 

  σ =  1/ ρ  or ρ = 1/ σ

 

Since, conductivity is the inverse of resistivity.

Therefore, its unit is mho .m-1(Ω m-1)

Another unit:  Siemens per meter (S m-1)

FAQ (Frequently Asked Questions)

Q1: Two Wires of Length 3m and 5m Respectively are Made Up of the Same Material and have the Same Area of Cross Section, Which Wire has Higher Resistance?

Ans: With given relation  R α L, wire of length 5m will offer higher resistance.

Q2: Is Resistivity Directly Proportional to Resistance?

Ans: Resistivity is an innate attribute of a material directly proportional to the resistance while resistance is an external attribute that depends on the length and cross-sectional area of the resistor.

Q3: A Wire in the Circuit has Some Resistance as Shown in the Figure. Calculate its Resistivity if the Length of the Wire is 20m and its Area of Cross Section is 5m2.     

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Ans: Given:  A = 5 m2, L = 20 m, I = 5 A and V =200 V

             Calculating R by Ohm’s law

             R = V/ I  = 200 / 5 = 40 Ω

Since   ρ = R.A/L

Putting values of R, A and L we get,

  ρ   = 40 x 5/ 20 = 10 Ωm-1 

Q4: What is the Order of Resistivity of a Semiconductor?

Ans: Semiconductors have resistivity in the range from

heavily-doped i.e., 10-4 ohm-cm to undoped i.e., 103 ohm-cm.