Question

what is the ratio of drift velocity and rms velocity of an electron in room temperature?

Answer 1

Answer on Question #43910-Physics-Electrodynamics

What is the ratio of drift velocity and rms velocity of an electron in room temperature?

Solution

RMS velocity of an electron at room temperature is

$$v_{RMS} = \sqrt{\frac{3kT}{m_e}},$$

where $T = 300\,K$, $k$ is the Boltzmann constant, $m_e$ is the mass of an electron.

Drift velocity of an electron is

$$v_{drift} = \frac{eE\tau}{2m_e},$$

where $e$ is a charge of an electron, $\tau$ is mean free time between collisions (at room temperature $\tau = 3 \cdot 10^{-14}\,s$), $E$ is an electric field.

The ratio of drift velocity and rms velocity of an electron in room temperature is

$$\frac{v_{drift}}{v_{RMS}} = \frac{\frac{eE\tau}{2m_e}}{\sqrt{\frac{3kT}{m_e}}}$$

For example in an electric field of $100\,\frac{V}{m}$ this ratio is equal

$$\frac{v_{drift}}{v_{RMS}} = \frac{\frac{1.60 \cdot 10^{-19}\,C \cdot 100\,\frac{V}{m} \cdot 3 \cdot 10^{-14}\,s}{2 \cdot 9.11 \cdot 10^{-31}\,kg}}{\sqrt{\frac{3 \cdot 1.38 \cdot 10^{-23}\,\frac{1}{K} \cdot 300\,K}{9.11 \cdot 10^{-31}\,kg}}} = \frac{0.26}{1.2 \cdot 10^5} = 2.1 \cdot 10^{-6}.$$

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