Question

Question:If electrical conductivity were determined by the mere static presence of positive ions rather than by their motion the collision time would be inversely proportional to the electron's average speed. If however, it were dominated by the motion of the ions, it should be inversely proportional to the “area" presented by a jiggling ion, which is in turn proportional to the square of its amplitude as an oscillator. Argue that only the latter view gives the correct temperature dependence in conductors of $\mathit{\sigma }\mathbf{\propto }{\mathit{T}}^{\mathbf{-}\mathbf{1}}$. Use the equipartition theorem (usually covered in introductory thermodynamics and also discussed in Section 9.9).

Short answer

Answer

It is shown that correct temperature dependence in conductor is $\sigma \propto T^{-1}$.

Step-by-step solution

Given data

The electrical conductivity is inversely proportional to the square of the amplitude of the ion so, .

$\sigma \propto \frac{1}{A^2}$

Definition of Electrical Conductivity

The current carrying capability of a material is called the electrical conductivity of that particular material.

It is an intrinsic property of the material. It is represented by the symbol$\mathit{\sigma }$ .

The electrical conductivity is inversely proportional to the square of the amplitude of the ion so, $\mathit{\sigma }\mathbf{\propto }\frac{\mathbf{1}}{{\mathbf{A}}^{\mathbf{2}}}$.

Derive the function for Correct Temperature Dependence in Conductors

The current carrying capability of a material is called the electrical conductivity of that particular material. It is an intrinsic property of the material.

It is represented by the symbol $\sigma$.

The electrical conductivity is inversely proportional to the square of the amplitude of the ion so,

$\sigma \propto \frac{1}{A^2}$ .

Now, since, oscillator energy is directly proportional to the square of the amplitude of the ion so,

$\sigma \propto \frac{1}{\text{oscillationenergy}}$.

By the equipartition theorem, the oscillation energy is directly proportional the temperature so:

$$\begin{gathered} \sigma \propto \frac{1}{T} \\ \propto T^{-1} \end{gathered}$$

Therefore, it is shown that correct temperature dependence in conductor is $\sigma \propto T^{-1}$.