## Variation of Resistivity with Temperature:

We know,

ρ = m/ne^{2}τ …………………..(I)⇒ ρ ∝ 1/τ |

**In Most Metals-** When the temperature of the metal conductor is raised, the atoms/ions of the metal vibrate with greater frequency and hence relaxation time ‘τ’ decreases. We can conclude that the resistivity of the metal increases.

Hence when the temperature of the metal conductor is raised, its resistivity increases.

At 0°C, R_{o }= ρ_{o} x l/Aand At t°C, R _{t} = ρ_{t} x l/A We know, R _{t} = R_{o} (1 + α_{r}t)⇒ ρ _{t} x l/A = ρ_{o} x l/A (1 + α_{r}t) ⇒ ρ _{t} = ρ_{o} (1 + α_{r}t) Here α _{r} is called the temperature coefficient of resistivity.ρ _{t} = ρ_{o} + ρ_{o} α_{r}t ⇒ ρ _{t} – ρ_{o} = ρ_{o} α_{r}t ⇒ α _{r} = ρ_{t} – ρ_{o}/ ρ_{o}t ⇒ α _{r} = 1/ρ_{o} x ρ_{t} – ρ_{o}/tIf dρ be the change in resistivity when temperature changes by dt, then ⇒ α _{r} = 1/ρ_{o} x dρ/dt |

Hence temperature coefficient of resistivity is defined as the change in resistivity per unit resistivity of the material per degree change in temperature.

**For Conductors-** The value of α_{r} is positive i.e. resistivity of the conductors increases with an increase in temperature.

**For Semiconductors-** In the case of semiconductors, the value of α_{r} is negative. It means that the resistivity of semiconductors decreases as the temperature increases.

**For Insulators-** In the case of insulators, resistivity decreases exponentially with the rise in temperature. It becomes infinite at 0 k.

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