Question

For ohmic conductor, the drift velocity ${v_d}$ and the electric field applied across it are related as:
A)${v_d} \propto \sqrt E $
B)${v_d} \propto {E^2}$
C)${v_d} \propto E$
D)${v_d} \propto \dfrac{1}{E}$

Answer 1

Hint Drift velocity is the average velocity attained by charged particles such as electrons, in a material due to an electric field.
Now, we are supposed to find the relation between drift velocity and electric field responsible for motion of the charged particle. The product of electric field and charge of the particle gives us the force by which electrons flow in a circuit. Using, $F = ma$, we can get the acceleration and then by using equations of motion, we can get the required velocity and hence the required relation.

 Complete step-by-step solution:
When a battery is connected to a conductor, an electric field is set up inside the magnitude of which is
$E = \dfrac{V}{L}$
Where $V$ is the voltage of battery and $L$ is the length of the conductor.
Electric field has a direction, so it is a vector. Current flows in the direction of the electric field. Current, conventionally, is taken to be a flow of positive charges. So the electrons begin to flow in a direction opposite to the direction of the electric field since the flow of positive charges is not possible in a conductor.
Force exerted by the electric field on the electrons is the product of magnitude of electric field and charge of electrons. So, $F = eE$
From Newton’s law, we know that $F = ma$.
Equating the two relations that we have got above, we get
$
  ma = eE \\
  a = \dfrac{{eE}}{m} \\
 $
This is the acceleration by which the electrons move. Here, $m$ is the mass of the electron.
Now, we will make use of equations of motion. We will use the relation
$v = u + at$
Since, before the application of potential difference, the electrons were at rest. So,
Initial velocity, $u = 0$
$v$ that is the final velocity will be taken as the drift velocity ${v_d}$. We have already derived acceleration above and time will be taken as $\tau $. The electrons do not flow smoothly but undergo a number of collisions with other electrons. $\tau $ is the average time between two collisions. It is also called the relaxation time.
Substituting all the values in relation $v = u + at$, we get
$
  {v_d} = 0 + \dfrac{{eE}}{m} \times \tau \\
  {v_d} = \dfrac{{eE\tau }}{m} \\
   \Rightarrow {v_d} \propto E \\
 $
This gives us that drift velocity is directly proportional to the electric field.
So, option C is correct.

Note:- There may be many alternate ways of solving this question but this is the quickest and also you just need basic knowledge of concept by following this approach. Basic questions of kinematics should be known for following this approach to the question.