Resistivity (ρ) :
As from the equation R = ρ × l/A
It is clear that, resistance 'R' of a conductor depends on the following factors:
(i) Length 'l' of the conductor; R ∝ l
i.e., resistance of a conductor varies directly as its length.
As potential difference is applied across the two ends, free electrons move from the end at lower potential to the end at a higher potential. In this process they collide against each other and undergo retardation. A greater length of conductor results in greater number of collisions, thereby producing greater retardation and hence greater resistance.
(ii) Area of cross-section 'a'; R ∝ 1/a
Resistance of a conductor varies inversely as its area of cross-section.
For a conductor having greater area of cross-section more free electrons cross that section of conductor in one second, thereby giving large current. A large current means a lesser resistance.
Combining the two factors together,
R ∝ l/a or R = ρ l/a
Where ρ is the resistivity or the specific resistance of the material.
If l = 1, A = 1, R = ρ
Definition of Resistivity:
The resistivity of a material is defined as the resistance of a conductor made up of the material having unit length and area of cross section.
Alternatively, it is also defined as the resistance in between the two opposite faces of a unit cube made up of that material.
Unit of resistivity (ρ) :
in S.I= ohm × m²/m
= ohm × m
n C.G.S= stat ohm×c.m²/c.m
= stat ohm×c.m
Dimension of resistivity
[ρ] = [R] × [a]/[l]
[ρ] = [M1 L2 T-3 A-2] × [L2]/[L1]
[ρ] = [M1 L3 T-3 A-2]So, the dimensions of resistivity are 1, 3, -3 and -2 in mass, length, time and electric current respectively.
Variation of Resistivity with Temperature:
(i) Conductors. The resistance R of a material is given by
R = ρ. l/a
As from the equation R = ρ × l/A
R = m/ne²τ . l/a
Comparing these two equations
ρ = m/ne²τ
'τ' is the 'relaxation time' which depends upon the temperature of the conductor. An increase in the temperature results in an increase in amplitude of atomic vibrations which in turn decreases τ. Therefore, resistivity ρ, of the conductor, increases with an increase in temperature. Vibration of ρ with temperature for copper is shown [in Fig. 1]. The curve is a straight lines around room temperature. It deviates beyond 800 K and also in the low temperature region. At absolute zero the material possesses some minimum finite resistivity.
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| Fig. 1. Variation of resistivity with temperature for a conductor. |
If ρ₀ and ρt are resistivity of the conductor at 0⁰ at t⁰C respectively, then
ρt = ρ₀ [ 1 + ∝ (T - T₀)
or ∝ = ρt - ρ₀/ ρ₀(T - T₀)
'∝' is called the temperature coefficient of the resistivity.
ρ₀ = 1, T - T₀ = 1, ∝ = ρt - ρ₀
Temperature coefficient of resistivity of a conductor is defined as the change in resistivity per unit resistivity per degree celsius rise of temperature.
(ii) Insulators. In case of insulators the resistivity, almost, increases exponentially with a decrease in temperature and tends to infinity as the temperature approaches absolute zero. Resistivity ρ₀ and ρt at 0⁰C and t⁰C, respectively, are connected by the relation
ρt = ρ₀ e-Eg/kT ..... (1)
Where 'Eg' is called the positive energy of the insulator and 'k' is Boltzmann constant. Eg represents the difference between conductor band and valence band and has value ranging from zero to several electron volt. Depending upon the value of Eg, the substance behaves like an insulator or as a semi-conductor.
For Eg < 1 eV, value of 'ρ' at room temperature is not very high. Therefore, the substance acts as a semi-conductor. For Eg > 1 eV, the value of ρ at room temperature is high (≈ 103 ohm). So, the substance acts as an insulator.
Since number of electrons per cm³ of the material varies inversely as the resistivity,
n ∝ 1/ρ
∴ From equation (1),
nt = n₀ e-EgkT
Where n₀ has a value of the order of 10²⁸.
Values of resistivity and temperature coefficient of some conductors, semi-conductors and insulators are given in table.
Table:
|
Material
|
Resistivity at 0⁰ C (ohmmeter)
|
Temperature co-eff. (∝)
|
|
Conductors
Aluminium
Copper
Silver
Gold
Iron
Lead
Platinum
Semi-conductor
Germanium
Silicon
Carbon
Insulators
Glass
Rubber
|
2.7 × 10-8 1.7 × 10-8 1.6 × 10-8 2.42 × 10-8 10 × 10-8 22 × 10-8 11 × 10-8
0.46 2300 3.5 × 10-5
10¹⁰ - 10¹⁴ 10¹³ - 10¹⁶
|
3.9 × 10^-3 3.9 × 10^-3 3.8 × 10^-3 3.4 ×10^-3 5.0 × 10^-3 3.90 × 10^-3 3.92 × 10^-3
-48 × 10^-3 -75 × 10^-4 -0.5 × 10^-3
— —
|
Important Notes:
- Resistivity is a constant of material. Two wires having different lengths and thickness but made up of same material will have same resistivity.
- Resistivity of a conductor is much small while that of insulator is large.
- Resistivity of a conductor increases with an increase in temperature.
- Resistivity of a insulator increases with a decrease in temperature.
- Resistivity and conductivity are inverse of each other.
Example-
A hollow copper tube of 5 m length has got external diameter equal to 0.1 m and the walls are 5 mm thick. If the resistivity of copper is 1.7 × 10-8 ohm metre, calculate the resistance of the tube.
Solution. A hollow tube of radius r, thickness d and length l [Fig. 2(i)] can be considered to be equivalent to a metallic sheet of breadth 2πr, thickness d and length l [Fig. 2(ii)].
 |
| Fig. 2. |
Area of Cross-section of the sheet,
A = 2πr × d
∴ R = ρ × l/A = ρ × l/2πr × d
Here r = 5 × 10-2 m,
d = 5 mm = 5 × 10-3 m, l = 5 m,
ρ = 1.7 × 10-8 ohm metre
Substituting, we get
R = 1.7 × 10-8 × 5 / 2 × π × 5 × 10-2 × 5 × 10-3 ohm
R = 5.4 × 10-5 ohm.
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