Displacement Current

Description of Displacement Current: The concept of displacement current was first introduced by Maxwell purely on the theoretical ground.

Maxwell postulates that "It is not only current in a conductor that produces a magnetic field but a changing electric field (or time varying electric field) in vacuum or in dielectric also produces the magnetic field. It means that a changing electric field is equivalent to a current which flows as long as the electric field is changing. This equivalent current in a vacuum or dielectric produces the same magnetic effect as an ordinary or conductor current in a conductor. This equivalent current is known as displacement current".

According to the Maxwell modified ampere's law.

$\oint \overrightarrow{B}. \overrightarrow{dl}= \mu_{\circ}i+\mu_{\circ}i_{d}$

Where $i_{d}$ = Displacement Current

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Mechanism of Flow of Charge in Metals: Free Electron Gas Theory

Theory of Free Electron Gas Model → According to the electron gas theory

1. The free electrons are continuous in motion inside the metal. The motion of free electrons are random inside the metal.

2. When the free electrons are collisied to each other then the direction of electrons are changed.

3. Mean free Path: The length covered by free electrons, between the two successive collisions is called the "Mean Free Path".

4. Relaxation Time: The time taken between the two successive collisions of free electrons is called the Relaxation time. It is represented by $\tau$.

5. Drift velocity: When a potential is applied across the metal then these electrons do not move own velocity but it move with an average velocity in the opposite direction of the electric field. This average velocity is called the "Drift Velocity". The drift velocity of electrons depends upon the applied potential.

6. Mobility of Electrons: When a potential $V$ is applied across the metal then electrostatic force $F$ acts on the electrons i.e

$F=qE $

$F=NeE \qquad(1)$

Where
$N$ →The number of free electrons inside the metal
$E$ → The electric field due to the applied potential

When this electrostatic force is applied to the electrons then these electrons are accelerated with accelerated $a$ i.e

$F=ma \qquad(2)$

From equation $(1)$ and equation $(2)$ we can write

$ma=N\:e\:E$

$a=\frac{N}{m}eE$

$\frac{v_{d}}{\tau}=\frac{N}{m}eE \qquad \left( \because a=\frac{v_{d}}{\tau} \right)$

$v_{d}=\frac{Ne\tau}{m} E$

$v_{d}=\mu E$

Where $\mu=\frac{Ne\tau}{m}$. It is known as the mobility of electrons.

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Derivation of Ohm's Law

Derivation→

Let us consider,

The length of the conductor = $l$
The cross-section area of the conductor = $A$
The potential difference across the conductor = $V$
The drift velocity of an electron in conductor = $v_{d}$

Now from the equation of the mobility of electron i.e.

$v_{d}= \mu E$

$v_{d}= \left( \frac{e\tau}{m} \right) E \qquad \left(\because \tau = \frac{e\tau}{m} \right)$

$v_{d}= \left( \frac{e\tau}{m} \right) \frac{V}{l}\qquad (1) \qquad \left(\because E = \frac{V}{l} \right)$

Now from the equation of drift velocity and electric current

$i=neAv_{d}$

Now substitute the value of $v_{d}$ from equation $(1)$ to above equation

$i=neA\left( \frac{e\tau}{m} \right) \frac{V}{l}$

$i=\left( \frac{ne^{2}A\tau}{ml} \right)V$

$\frac{V}{i}=\left( \frac{ml}{ne^{2}A\tau} \right)$

$\frac{V}{i}=R$

Where $R = \frac{ml}{ne^{2}A\tau} $ is known as electrical resistance of the conductor.

Thus

$V=iR$

This is Ohm's Law.

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Relation between electric current and drift velocity

Derivation→ Let us consider

The length of the conductor = $l$

The cross-section area of the conductor = $A$

The total number of free electrons inside the conductor = $N$

The current flow in the conductor = $i$

The flow of Electron in Conductor

The Relaxation time between the two successive collisions =$\tau$

According to the law of current density

$J=\frac{i} {A} $

$J=\frac{q} {A \tau} \qquad \left( \because i=\frac{q}{\tau} \right)$

$J=\frac{N \: e }{A \tau} \qquad \left( \because q=Ne \right)$

Now multiply by length of conductor $l$ in above equation. Therefore we get

$J=\frac{N \: e \: l}{A \tau\: l} $

$J=\frac{N \: e \: l}{V \tau\: }\qquad \left(\because V=A.l \right)$

Where $V$ is volume of the conductor.

$J=\frac{n \: e \: l}{ \tau }\qquad \left( \because n=\frac{N}{V} \right)$

Where $n$ is total number of electrons per unit .

$J=n \: e \: v_{d}\qquad \left( \because v_{d}=\frac{l}{\tau}\right)$

Where $v_{d}$ are known as drift velocity of charged particles.

Now substitute the value of current density $J$ from equation $(1)$ to above equation then above equation can be written as

$i=neAv_{d}$

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