Question

Define the term 'mobility’ of charge carriers in a current carrying conductor. Obtain the relation for mobility in terms of relaxation time.

Answer 1

You can approach the solution to this question by first defining the term mobility of charge carriers, and then develop the relation using the equation of drift velocity, which we will also derive in the complete solution, if you don’t know it already.

Complete step by step answer:
We will be solving the question exactly as we told in the hint section of the solution to this question. Firstly, we will define the term mobility of charge carriers in a current carrying conductor. Then, we will find the relation between drift velocity and relaxation time, and will use it to develop a relation between mobility and relaxation time.
First, let us define the term “mobility” of charge carriers:
The mobility of charge carriers in a current carrying conductor can be defined as the net average velocity with which the free-electrons move towards the positive end of a conductor under the influence of an external electric field that is being applied.
Mathematically, we can define it as:
$\mu = \dfrac{{{V_d}}}{E}$
Where, $\mu$ is the mobility of charge carriers in a current carrying conductor
${V_d}$ is the drift velocity of the charge carriers or electrons
$E$ is the external electric field that is being applied.
To find a relation between mobility and relaxation time, we firstly need to find the relation between drift velocity and relaxation time, so let’s do it.
Let us assume that an electron with charge $- e$ takes an average time $\tau$ in between collisions, referred to as relaxation time.
We know that if the external electric field is $E$ then the force acting on the electron will be:
$F = - eE$
From this, we can find the acceleration of electron as:
$a = - \dfrac{{eE}}{{{m_e}}}$
Where, ${m_e}$ is the mass of an electron.
If we assume the initial velocities of each electron to be $0$ , that is, assume all electrons to be at rest initially, then we get the drift velocity using the formula:
$v = u + at$
Substituting the values, we get:
${V_d} = 0 + \left( { - \dfrac{{eE}}{{{m_e}}}} \right)\tau$
Or,
${V_d} = - \dfrac{{eE}}{{{m_e}}}\tau$
Now, let us substitute this in the equation of mobility:
$\mu = \dfrac{{{V_d}}}{E}$
After substituting, we get:
$\mu = \dfrac{{\left( { - \dfrac{{eE}}{{{m_e}}}} \right)\tau }}{E} \\\ \mu = - \dfrac{e}{{{m_e}}}\tau \\\$
Since velocity is a vector but mobility is a scalar, it can’t be negative, so:
$\mu = \dfrac{e}{{{m_e}}}\tau$

Note: Many students think of mobility as simply the ability of electrons or charge carriers to move without facing any hindrance, but that thing is conductivity and not mobility, so one should never confuse between these two terms.