### 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.

### Numerical Aperture and Acceptance Angle of the Optical Fibre

Angle of Acceptance → If incident angle of light on the core for which the incident angle on the core-cladding interface equals the critical angle then incident angle of light on the core is called the "Angle of Acceptance. Transmission of light when the incident angle is equal to the acceptance angle If the incident angle is greater then the acceptance angle i.e. $\theta_{i}>\theta_{0}$ then the angle of incidence on the core-cladding interface will be less than the critical angle due to which part of incident light is transmitted into cladding as shown in the figure below Transmission of light when the incident angle is greater than the acceptance angle If the incident angle is less then the acceptance angle i.e. $\theta_{i}<\theta_{0}$ then the angle of incidence on the core-cladding interface will be greater than the critical angle for which total internal reflection takes place inside the core. As shown in the figure below Transmission of light w

### Fraunhofer diffraction due to a single slit

Let $S$ be a point monochromatic source of light of wavelength $\lambda$ placed at the focus of collimating lens $L_{1}$. The light beam is incident normally from $S$ on a narrow slit $AB$ of width $e$ and is diffracted from it. The diffracted beam is focused at the screen $XY$ by another converging lens $L_{2}$. The diffraction pattern having a central bright band followed by an alternative dark and bright band of decreasing intensity on both sides is obtained. Analytical Explanation: The light from the source $S$ is incident as a plane wavefront on the slit $AB$. According to Huygens's wave theory, every point in $AB$ sends out secondary waves in all directions. The undeviated ray from $AB$ is focused at $C$ on the screen by the lens $L_{2}$ while the rays diffracted through an angle $\theta$ are focussed at point $p$ on the screen. The rays from the ends $A$ and $B$ reach $C$ in the same phase and hence the intensity is maximum. Fraunhofer diffraction due to

### Particle in one dimensional box (Infinite Potential Well)

Let us consider a particle of mass $m$ that is confined to one-dimensional region $0 \leq x \leq L$ or the particle is restricted to move along the $x$-axis between $x=0$ and $x=L$. Let the particle can move freely in either direction, between $x=0$ and $x=L$. The endpoints of the region behave as ideally reflecting barriers so that the particle can not leave the region. A potential energy function $V(x)$ for this situation is shown in the figure below. Particle in One-Dimensional Box(Infinite Potential Well) The potential energy inside the one -dimensional box can be represented as $\begin{Bmatrix} V(x)=0 &for \: 0\leq x \leq L \\ V(x)=\infty & for \: 0> x > L \\ \end{Bmatrix}$ $\frac{d^{2} \psi(x)}{d x^{2}}+\frac{2m}{\hbar^{2}}(E-V)\psi(x)=0 \qquad(1)$ If the particle is free in a one-dimensional box, Schrodinger's wave equation can be written as: $\frac{d^{2} \psi(x)}{d x^{2}}+\frac{2mE}{\hbar^{2}}\psi(x)=0$ \$\frac{d^{2} \psi(x)}{d x