Conductivity Formula Derivation for Metals

The electrical conductivity ($\sigma$) of a metal describes its ability to conduct electric current. It depends on the properties of the charge carriers within the metal.

Key Formula for Electrical Conductivity

The standard formula for the electrical conductivity ($\sigma$) of a metal, often derived from the Drude model, is:

Where:

• $n$ represents the number density of charge carriers (electrons per unit volume).
• $e$ is the elementary charge (the magnitude of the charge of an electron).
• $τ$ (tau) is the mean free time or relaxation time between collisions of the charge carriers.
• $m$ is the mass of the charge carrier (electron mass).

Step-by-Step Reasoning

1. Relate Conductivity to Mobility: Electrical conductivity ($\sigma$) is directly proportional to the mobility ($\mu$) of the charge carriers:

   $ \sigma = n e \mu $

2. Define Mobility: Mobility ($\mu$) is defined as the drift velocity ($v_d$) per unit electric field ($E$):

   $ \mu = \frac{v_d}{E} $

3. Calculate Drift Velocity: Under an electric field $E$, the force on a charge carrier is $F = eE$. The acceleration is $a = F/m = \frac{eE}{m}$. Assuming the carrier accelerates for a time $\tau$ before a collision, the average drift velocity is:

   $ v_d = a \tau = \frac{eE\tau}{m} $

4. Determine Mobility Value: Substituting the drift velocity back into the mobility definition:

   $ \mu = \frac{(eE\tau / m)}{E} = \frac{e\tau}{m} $

5. Substitute Mobility into Conductivity Formula: Now substitute the expression for mobility back into the conductivity equation:

   $ \sigma = n e \mu = n e \left( \frac{e\tau}{m} \right) $

   This simplifies to the final formula:

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

Therefore, the correct expression for the electrical conductivity of a metal is $\sigma = \frac{ne^2\tau}{m}$. This matches Option B.