$\oint \vec{E}.\vec{dA}=\frac{Q_{enc}}{\epsilon_0}$

$\oint \vec{B}.\vec{dA}=0$

$\oint \vec{E}.\vec{dl}=-\frac{d}{dt} \phi_B = -\frac{d}{dt}\int \vec{R}.\vec{dA}$

$\oint \vec{B}.\vec{dl}=\mu I+\mu_0 \epsilon_0 \frac{d}{dt} \int \vec{E}.\vec{dA}$

$\oint \vec{E}.d \vec{A}=\frac{Q_{enc}}{\epsilon_0}$

$\oint \vec{E}.d \vec{A}=0$

$\oint \vec{E}.d \vec{l}=-\frac{d \phi_B}{dt}$

$\oint \vec{E}.d \vec{l}=\mu_0 I +\mu_0 \epsilon_0 \frac{d}{dt} \phi_E$

Displacement Current

$\oint \vec{E}.d\vec{A}=0$

$\oint \vec{B}.d\vec{A}=0$

$\oint \vec{E}.\vec{dl}=-\frac{d}{dt} \int \vec{B}.\vec{dA}$

$\oint \vec{B}.\vec{dl}= \mu_0 \epsilon_0 \frac{d}{dt} \int \vec{E}.d\vec{A}$

$y(x, t)=A \sin (kx- \omega t)$

A : Amplitude of the wave $(kx- \omega t)$, phase of in wave

At A = 0

$y(x, t=0)= A \sin kx$

$t=0$

$k \lambda = 2 \pi$

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

K : wave number or propagation constant

At x = 0

$y(x, t)= A \sin (kx- \omega t)$

at x = 0

$y(o,t)=-A \sin \omega t$

$\omega T = 2 \pi$

$\omega = \frac{2 \pi}{T}$

Frequency $v=\frac{1}{T}$

$\omega = 2 \pi v$

$\omega :$ Angular frequency

$y(x_1 t)=A \sin[\frac{2 \pi}{\lambda}x-2 \pi v t]=A \sin [2 \pi (\frac{x}{\lambda}-vt)]$

$y(x_1 t)= A \sin (kx- \omega t)$

$y(x_1 t)= A \sin (kx- \omega t)$

At t = 0; y(x_1 t) = $A \sin kx$

At = t; y(x_1 t) = $A \sin (kx - \omega t)$

$kx_1 - \omega t_1=0$

$kx_1 = \omega t_1$

$\frac{x_1}{t_1}=\frac{\omega}{k}$

Velocity of the wave = $\frac{\omega}{k}$

$=\frac{2 \pi v}{2 \pi / \lambda}$

$=v/ \lambda$

$v=v \lambda$

Propagative Wave

$y(x_1 t)= A \sin (kx- \omega t)$

At t = 0; y(x_1 t) = $A \sin kx$

At = t; y(x_1 t) = $A \sin (kx - \omega t)$

$y(x_1 t)= A \sin (kx + \omega t)$

$\frac{dy}{dt}=-A \omega \cos(kx- \omega t)$

$\frac{dy}{dx}= Ak \cos(kx- \omega t)$

Transverse waves

Can propogate in force space

Variation of electric and magnetic field in space

Electric and magnetic fields are perpendicular to the propagation reaction

Electric field is perpendicular to the magnetic field

$(\vec{E} \times \vec{B})$ is along the direction of travel of the wave

Electrical field and magnetic field are in phase

Electromagnetic waves

$\vec{E}=\hat{i} E_0 \sin (kz- \omega t)$

$\vec{B}=\hat{j} B_0 \sin (kz- \omega t)$

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

$\lambda :$ Wavelength of the electromagnetic wave

$\omega = 2 \pi v$

$v :$ Frequency of the electromagnetic wave

$=\frac{\omega}{k}$

Frequencies of various electromagnetic waves

$\Rightarrow$ Increasing frequencies & decreasing wavelength