Derivation : Drift of Electrons and the Origin of Resistivity and Mobility

Derivation : Drift of Electrons and the Origin of Resistivity and Mobility

Drift of Electrons and the Origin of Resistivity : 

Drift Velocity : 

Consider a conductor with free electrons moving randomly. When an electric field $\vec{E}$ is applied across the conductor, the electrons experience a force $\vec{F} = -e\vec{E}$, where $-e$ is the charge of an electron.

This force causes the electrons to accelerate. According to Newton's second law, $\vec{F} = m\vec{a}$, where $m$ is the mass of the electron and $\vec{a}$ is its acceleration. Therefore,

$m\vec{a} = -e\vec{E}$

$\vec{a} = -\frac{e\vec{E}}{m}$

The electrons collide with the ions in the conductor. Let $\tau$ be the average time between successive collisions, known as the relaxation time. The average velocity acquired by the electrons due to the electric field is called the drift velocity $\vec{v}_d$.

$\vec{v}_d = \vec{a}\tau = -\frac{e\vec{E}\tau}{m}$

The negative sign indicates that the drift velocity is opposite to the direction of the electric field.

Origin of Resistivity : 

Let $n$ be the number density of free electrons in the conductor. The current $I$ flowing through the conductor is given by:

$ I = nAv_d e $

where $A$ is the cross-sectional area of the conductor.

Substituting the expression for $v_d$, we get:

$ I = nA \left( \frac{eE\tau}{m} \right) e$

$I = \frac{nA e^2 \tau E}{m}$

The current density $J$ is given by $J = I/A$. Therefore,

$J = \frac{ne^2\tau E}{m}$

Comparing this with Ohm's law in microscopic form, $J = \sigma E$, where $\sigma$ is the conductivity, we get:

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

The resistivity $\rho$ is the inverse of conductivity, $\rho = 1/\sigma$. Therefore,

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

This expression shows that the resistivity of a conductor depends on the number density of free electrons $n$, the relaxation time $\tau$, and the mass of the electron $m$.

Mobility : 

conductivity arises from mobile charge carriers. In metals, these mobile charge carriers are electrons; in an ionised gas, they are electrons and positive charged ions; in an electrolyte, these can be both positive and negative ions.

mobility $\mu$ defined as the magnitude of the drift velocity per unit electric field:

$\mu = \frac{|\vec{v}_d|}{E}$

The SI unit of mobility is m$^2$/Vs and is $10^4$ of the mobility in practical units (cm$^2$/Vs). Mobility is positive. 

$v_d = \frac{e\tau E}{m}$

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