Question

(1) Ohm’s law and its derivations
(2) Drift velocity of free electrons and its relation with current $I=neA{v}_d$

Answer 1

Hint Ohm’s law explains the relationship between the potential difference and current. We have to derive an expression for this relation. Free electrons have a drift velocity and we have to derive the relation between the drift velocity of the electrons and the current also.

Complete step by step answer:
It was George Simon Ohm who established a relationship between the potential difference across a conductor and the current flowing through the conductor. This law is called Ohm’s law.
According to Ohm’s law, at a constant temperature, the current flowing through a conductor will be directly proportional to the potential difference between the ends of the conductor.
Let us take the potential difference to be $V$
Let the current through the conductor be, $I$
According to Ohm’s law, $V$ and $I$ are directly proportional
I.e.
$V\propto I$
We can change the proportionality sign into an equal sign by multiplying with a proportionality constant.
Therefore, we can write,
$V=IR$
Where the proportionality constant$R$is known as the resistance of the conductor. The unit of resistance is Ohm $\left(\mathrm{\Omega }\right)$
Drift velocity of electrons.
The average velocity with which the free electrons are drifted under the influence of an electric field is called drift velocity.
Drift velocity can be denoted as $v_d$.
The relation between current and drift velocity can be derived is as follows,
Let us consider a conductor of length $l$ and an area of cross-section $A$. Let the number of free electrons per unit volume be $n$.$v_d$is the drift velocity.
The current flowing through the conductor in a second can be written as,
$I={\displaystyle \frac{q}{t}}$
Let $N$ be the total number of electrons, and $e$be the charge of an electron.
Then we can write the total charge $q$ as $q=Ne$
Then,$I={\displaystyle \frac{Ne}{t}}$
The volume of the conductor can be written as,
$V=Al$
Where $A$ is the area of the conductor and $l$ is the length of the conductor.
The number of electrons per unit volume will be,
$n={\displaystyle \frac{N}{Al}}$
From this, we can write,
$N=nAl$
Substituting this value in the equation of current, we get
$I={\displaystyle \frac{nAle}{t}}$
Let the number of charges crossing any cross-section in one-second will be the electrons contained in a length equal to the drift velocity of the conductor.
i.e.
$l=v_{d}t$
Substituting this value in the above equation for current, we get
$I={\displaystyle \frac{nA{v}_{d}te}{t}}=nAe{v}_d$

Note
Ohm’s law holds good only when the current flows through a conductor is steady. According to Ohm’s law, the graph connecting the voltage and current will be a straight line. Mobility can be defined as the drift velocity per unit electric field. It is a positive quantity.