Conductivity Mobility Relationship Explained

The relationship between electrical conductivity, charge carrier concentration, and charge carrier mobility is a fundamental concept in solid-state physics and electronics. Conductivity measures how easily electric current flows through a material, while mobility describes how quickly charge carriers (like electrons or holes) can move through the material under the influence of an electric field.

Understanding the Key Terms

• Conductivity ($\sigma$): This property quantifies how well a material conducts electric current. It is the reciprocal of resistivity. Higher conductivity means current flows more easily.
• Mobility ($\mu$ or $U$ in this question): This represents the average drift velocity achieved by charge carriers per unit electric field. It indicates how mobile the charge carriers are within the material.
• Carrier Concentration ($N$): This is the number of charge carriers (electrons or holes) per unit volume in the material. A higher concentration generally leads to higher conductivity.
• Elementary Charge ($e$): This is the magnitude of the electric charge carried by a single electron or proton, approximately $1.602 \times 10^{-19}$ Coulombs.

The Fundamental Equation

The electrical conductivity ($\sigma$) of a material is directly related to the concentration of charge carriers ($N$), the charge of each carrier ($q$), and their mobility ($\mu$) by the following fundamental equation:

$$ \sigma = N \cdot q \cdot \mu $$

In this equation:

• $\sigma$ is the electrical conductivity.
• $N$ is the number of charge carriers per unit volume.
• $q$ is the magnitude of the charge on each carrier.
• $\mu$ is the mobility of the charge carriers.

Analyzing the Options for Conductivity-Mobility

Let's examine how the given options relate to this fundamental formula, assuming the symbols used are:

• $\sigma$ = Conductivity
• $N$ = Carrier Concentration
• $e$ = Elementary Charge (acting as $q$, the magnitude of charge)
• $U$ = Mobility (acting as $\mu$)

Substituting $e$ for $q$ and $U$ for $\mu$ into the fundamental equation $\sigma = N q \mu$, we get the expected relationship:

$$ \sigma = N e U $$

Now let's evaluate the provided options based on this derived relationship:

• Option 1: $d = NU\sigma$
  This equation uses a symbol '$d$' which is not standard for this relationship and rearranges the terms incorrectly. It does not represent the conductivity-mobility relationship.
• Option 2: $\sigma = NeU$
  This equation perfectly matches the fundamental relationship $\sigma = N q \mu$ when using the assumed symbols ($N$ for carrier concentration, $e$ for charge magnitude, and $U$ for mobility).
• Option 3: $N = \sigma U/e$
  This is an incorrect rearrangement of the correct formula. The correct rearrangement to solve for $N$ would be $N = \sigma / (eU)$.
• Option 4: $U = \sigma e/N$
  This is also an incorrect rearrangement. The correct rearrangement to solve for mobility ($U$) would be $U = \sigma / (Ne)$.

Conclusion on the Equation

Based on the standard physics principles governing electrical conductivity in materials, the equation that correctly relates conductivity ($\sigma$), carrier concentration ($N$), the elementary charge ($e$), and mobility ($U$) is $\sigma = NeU$. This highlights that conductivity increases with higher carrier concentration and greater mobility, which is intuitive as more mobile carriers contribute more effectively to current flow.