### Viscosity, Viscous force and Coefficient of Viscosity

Definition of Viscosity:

It is the property of a fluid that opposes the relative motion between its adjacent layers. This property of the fluid is known as viscosity. It is also called the resistance of fluid to flow or deformation or fluid thickness.

Effect of temperature on Viscosity:

The viscosity of the fluid decreases sharply with the temperature rise and becomes zero at boiling temperature. On the other hand, the viscosity of the gases increases with the temperature rise.
Definition of Viscous Force (Internal Frictional Force):

When a layer of fluid slide over another layer of the same fluid then an internal tangential frictional force act between them which opposes the relative motion between the layers. This tangential force is called viscous force or internal frictional force.

In the absence of external force, the viscous force would soon bring the fluid to rest.

Factor affecting the viscous force:

There are the following factors that affect the viscous force-

1.) The viscous force is directly proportional to the contact area of the layers. i.e.

$F \propto A \qquad(1)$

Where $A \rightarrow$ The contact area of the layers

2.) The viscous force is directly proportional to the velocity gradient between the layers. i.e.

$F \propto \frac{\Delta v_{x}}{\Delta z} \qquad(2)$

Where $\frac{\Delta v_{x}}{\Delta z} \rightarrow$ The velocity gradient that is shown in the figure below
From equation $(1)$ and equation $(2)$

$F \propto A \: \frac{\Delta v_{x}}{\Delta z}$

$F = \pm \: \eta \: A \: \frac{\Delta v_{x}}{\Delta z} \qquad(3)$

Where

$\eta \rightarrow$ It is a constant and also known as the "coefficient of viscosity" of the fluid.

$\pm \rightarrow$ It indicates that the viscous force $F$ between two layers of fluid is a mutually interacting force. As shown above in figure (B), A layer that is just above layer $A$ exerts a force in the forward direction while a layer that is just below layer $A$ exerts an equal and opposite force i.e. in the backward direction.

Definition of Coefficient of Viscosity:

If $A=1$ and $\frac{\Delta v_{x}}{\Delta z} = 1$, then from equation $(3)$

$\eta = 1$

So from the above equation,  If the contact area and velocity gradient are units then the viscous force between the fluid is known as the coefficient of velocity.

Dimension of coefficient of viscosity:

$\left[ M L^{-1} T^{-1} \right]$

Unit of coefficient of viscosity:

$Kg-m^{-1}s^{-1}$

Here

$1 \: Kg-m^{-1}s^{-1} = 10\: poise = 1 \: decapoise$

### 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