Question

Draw the graph showing the variation of conductivity with temperature for a metallic conductor?

Answer 1

The conductivity for a metallic conductor decreases with the increase in temperature.

Explanation: The Physics Behind the Graph

To understand this behavior, we need to look at the microscopic level inside the metal.

1. What is Electrical Conduction in Metals?
Metals are good conductors because they have a large number of "free electrons." These electrons are not bound to any single atom and are free to move throughout the metal's crystal lattice. When a voltage is applied, these free electrons drift in a specific direction, creating an electric current.

2. The Role of the Crystal Lattice
The free electrons move through a lattice of fixed positive metal ions. At absolute zero temperature (0 Kelvin), these ions would be perfectly still, and electrons could move with very little opposition. This would result in very high conductivity.

3. The Effect of Temperature
When the temperature of the metal increases, the positive ions in the lattice gain thermal energy. This causes them to vibrate more vigorously and with a larger amplitude around their fixed positions.

4. Electron Collisions (Scattering)
As the free electrons drift through the metal, they collide with these vibrating ions. These collisions are the primary source of electrical resistance.
At Low Temperatures: The ions vibrate less, so collisions are infrequent. Electrons can travel for a longer time and distance between collisions. This leads to a smooth flow of charge and high conductivity.
At High Temperatures: The ions vibrate intensely. This increases the frequency of collisions between the electrons and the ions. Each collision scatters the electron, hindering its smooth, directional motion. This increased scattering makes it harder for the current to flow, which means the resistance increases, and consequently, the conductivity decreases.

5. The Mathematical Relationship

The conductivity ($\sigma$) of a metal is given by the formula:

$$ \sigma = \frac{ne^2\tau}{m} $$

Where:
$n$ = number density of free electrons (number of free electrons per unit volume). This is nearly constant for a given metal.
$e$ = charge of an electron (a constant).
$m$ = mass of an electron (a constant).
$\tau$ = average relaxation time, which is the average time between two successive collisions of an electron with the lattice ions.

From the formula, you can see that conductivity ($\sigma$) is directly proportional to the relaxation time ($\tau$).

As Temperature (T) increases:
Vibrations of lattice ions increase.
The frequency of electron collisions increases.
Therefore, the average time between collisions, $\tau$, decreases.
Since $\sigma \propto \tau$, the conductivity ($\sigma$) decreases.

Summary

| Temperature | Lattice Ion Vibration | Electron Collision Frequency | Relaxation Time ($\tau$) | Conductivity ($\sigma$) |
| :---------- | :-------------------- | :--------------------------- | :---------------------- | :-------------------------- |
| Low | Low | Low | High | High |
| High | Vigorous | High | Low | Low |