Question

Show that \( \mathbf{E} = \rho \mathbf{J} \) leads to Ohm's law. Write a condition in which Ohm's law is not valid for a material.

Hint:

Ohm's law holds for linear, ohmic conductors, where the current density is proportional to the electric field. In materials with nonlinear responses, such as semiconductors or superconductors, Ohm's law is not valid.

Answer 1

Derivation of Ohm's Law and Condition for Its Breakdown 

1. Derivation of Ohm's Law from \( \mathbf{E} = \rho \mathbf{J} \)

The equation \( \mathbf{E} = \rho \mathbf{J} \) is known as the electrical conductivity equation. Here:

• \( \mathbf{E} \) is the electric field in the material,
• \( \mathbf{J} \) is the current density (current per unit area),
• \( \rho \) is the resistivity of the material.

From this equation, we can express the electric field in terms of current density:

\[ \mathbf{E} = \rho \mathbf{J} \]

Now, consider **Ohm's law**, which states that the current density \( \mathbf{J} \) is proportional to the electric field \( \mathbf{E} \) and the material's conductivity \( \sigma \) (the inverse of resistivity). So, we can write:

\[ \mathbf{J} = \sigma \mathbf{E} \]

Since \( \sigma = \frac{1}{\rho} \), we can substitute this into the above equation:

\[ \mathbf{J} = \frac{1}{\rho} \mathbf{E} \]

Rearranging the equation, we get:

\[ \mathbf{E} = \rho \mathbf{J} \]

This is exactly the form of the equation we started with, so we have derived Ohm's law from the equation \( \mathbf{E} = \rho \mathbf{J} \).

2. Condition Under Which Ohm's Law Is Not Valid

Ohm's law assumes that the material has a constant resistivity \( \rho \) and that the current is proportional to the applied voltage (i.e., linear response). However, there are conditions under which Ohm's law does not hold:

• Non-Linear Materials: If the resistivity \( \rho \) changes with the applied electric field, temperature, or current density, the relationship between \( \mathbf{E} \) and \( \mathbf{J} \) becomes non-linear, and Ohm's law does not apply.
• High Electric Fields: In materials with very high electric fields (such as in semiconductors at high bias voltages), the current may no longer be proportional to the electric field, leading to a breakdown of Ohm's law.
• Temperature Dependence: In certain materials, particularly at very high temperatures or low temperatures, resistivity \( \rho \) can vary significantly, leading to deviations from Ohm’s law.
• Superconductivity: In superconductors, the resistivity drops to zero below a certain temperature, and no electrical resistance is present, hence Ohm’s law does not hold.

Thus, Ohm’s law is not valid in situations where the material’s resistivity is not constant or when extreme conditions like high electric fields or temperatures cause a non-linear relationship between voltage and current.