### Characteristic impedance of electromagnetic wave

We know that the electromagnetic wave propagates perpendicular to both electric field and magnetic field which can describe as

$\overrightarrow{k} \times \overrightarrow{E}= \omega \overrightarrow{B} \qquad(1)$

If $\hat{n}$ is a unit vector in the direction of the propagation then

$\overrightarrow{k}=k \hat{n}$

Substitute these values in equation$(1)$ then we get

$k(\hat{n} \times \overrightarrow{E})= \omega \overrightarrow{B}$

$\overrightarrow{B}= \frac{k}{\omega}(\hat{n} \times \overrightarrow{E}) \qquad(2)$

But the value of $k$ and $\omega$ is

$k=\frac{2\pi}{\lambda}$

$\omega=2 \pi \nu$

Then value of $\frac{k}{\omega}=\frac{1}{c}$

Now substitute the value of $\frac{k}{\omega}$ in equation$(2)$ then we get

$\overrightarrow{B}= \frac{1}{c}(\hat{n} \times \overrightarrow{E})$

The magnitude form of the above equation can be written as

$B=\frac{E}{c}$

$\frac{E}{B}=c$

$\frac{E}{\mu_{0}H}=c \qquad (\because B=\mu_{0} H)$

$\frac{E}{H}=\mu_{0}c$

$\frac{E}{H}=\frac{\mu_{0}}{\sqrt{\mu_{0} \epsilon_{0}}} \qquad(\because c=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}})$

$\frac{E}{H}= \sqrt{\frac{\mu_{0}}{\epsilon_{0}}}$

The term $\frac{E}{H}$ has dimensions of the impedance and is known as characteristic impedance or intrinsic impedance of free space. It is represented by $(Z_{0})$.

$Z_{0}=\frac{E}{H}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}$

Now substitute the value of $\mu_{0}$ and $\epsilon_{0}$ i.e.

$\mu_{0}=4\pi \times 10^{-7}$

$\epsilon_{0}=8.854 \times 10^{-12}$

$Z_{0}=\sqrt{\frac{4\pi \times 10^{-7}}{8.854 \times 10^{-12}}}$

$Z_{0}=376.73\: \Omega$

$Z_{0}=120\pi \: \Omega$

### 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 than 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 the 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 than 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 lig

### 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

### Electromagnetic wave equation in free space

Maxwell's Equations: Maxwell's equation of the electromagnetic wave is a collection of four equations i.e. Gauss's law of electrostatic, Gauss's law of magnetism, Faraday's law of electromotive force, and Ampere's Circuital law. Maxwell converted the integral form of these equations into the differential form of the equations. The differential form of these equations is known as Maxwell's equations. $\overrightarrow{\nabla}. \overrightarrow{E}= \frac{\rho}{\epsilon_{0}}$ $\overrightarrow{\nabla}. \overrightarrow{B}=0$ $\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$ $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \overrightarrow{J}$ Modified Form: $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \left(\overrightarrow{J}+ \epsilon \frac{ \partial \overrightarrow{E}}{\partial t} \right)$ For free space