According to Faraday's law of induction, $E=-\frac{d \Phi_B}{d }$

Magnetic flux through a closed loop, ${\Phi}_B=\int \vec{B} \cdot d \vec{A}$

$\vec{B}$ is uniform.

$\Phi_B=\vec{B} \cdot \vec{A}=B . A \cos \theta$

The magnetic flux passing through loop is proportional to the magnetic field, depends on the area of the loop, and the angle between the area vector, and the magnetic field.

Change in magnetic flux will induce any EMF.

We can rotate the coil with respect to the magnetic field as a function of time.

When the coil rotates, the area vector rotates, the rotation of the area vector implies that the change that is a change in angle

The magnetic flux passing through loop changes with time and the changing magnetic flux will induce an EMF which will develop a potential difference across two points in the outside circuit.

$\frac{d \Phi_B}{dt}<0$, $E>0$ and $\frac{d \Phi_B}{dt}>0$, $E<0$.

$\Phi_B=B.A \cos\theta$

$\theta= \omega{t}$

$\Phi_B=B.A\cos\omega t$

Induced emf $E=-\frac{d\Phi_B}{dt}$

$= - \text{B A} \omega (-\sin\omega t)$

$= \text{B A} \omega \sin \omega t$

Hydro Electric Generator

Mechanical energy from falling water eg. dams.

Thermal Generators

Water is converted to steam using coal or other sources.

Steam at high pressure is used for rotation

Nuclear Generators

Nuclear fuel instead of coal

DC generator works on the principle of electromagnetic induction.

When a conductor is placed in a changing magnetic field, an electromotive force (EMF) is induced within the conductor.

Using this principle you can convert a mechanical energy to electrical energy.

Ampere's Law

$\oint \vec{B} \cdot \vec{dl}=\mu_0 ~I_{enc}$

Charging the Capacitor

There is a current which is changing as a function of time.

You have a charge potential difference between these two plates there will be an electric field pointing between two plates.

$\text { Electric flux }$

${\Phi}_E=\int \vec{E} \cdot d \vec{A}$

$= E A$

$\text{Current I} = \frac{dQ}{dt}$

$= {\epsilon_0}\frac{d\Phi_E}{dt}$

$\Phi_E=\text{E A} = \frac{\sigma}{\epsilon_0}A=\frac{\theta}{\epsilon_0}$

${\sigma}$ : Surface Charge Density

$\frac{d\Phi_E}{dt}=\frac{1}{\epsilon_0}\frac{dQ}{dt}$

Current ${I_c}=\frac{dQ}{dt}$, Conduction Current.

$\frac{d\Phi_E}{dt}=\frac{1}{\epsilon_0}~{I_c}$

${I_c}=\epsilon_0\frac{d\Phi_E}{dt}$

$\oint \vec{B} \cdot \vec{d l}=\mu_0 I_c+\mu_0 ~\epsilon_0 \frac{d \Phi_E}{d t}$

Generalized form of Ampere's Law

James Clerk Maxwell (1831 - 1879) introduced modification to Ampere's Law in 1865.

Displacement Current ${I_D}={\epsilon_0}\frac{d\Phi_E}{dt}$

$\oint \vec{B} \cdot \vec{d l}=\mu_0 I_c+\mu_0 I_D$

Displacement Current Density (Free Space)

$\vec{J_D}=\epsilon_0\frac{d\vec{E}}{dt}$

A changing magnetic flux induces an electric field a changing electric flux induces a magnetic field.

$\oint \vec{B} \cdot \vec{d l}=\mu_0 \epsilon_0 \frac{d {\Phi}_E}{d t}$

$\oint \vec{E} \cdot \vec{d l}=-\frac{d \Phi_B}{d t}$

It predicts the existence of waves.

Parallel plate capacitor

Circular plates of radii r

Capacitor getting charged

$\oint \vec{B} \cdot \vec{dl}=\mu_0 \cdot I_c+\mu_0 ~\epsilon_0 \frac{d \Phi_E}{d t}$

$=\mu_0 ~\epsilon_0 \frac{d \Phi_E}{d t}$

$\Phi_E = \text{E. Area} = \frac{\sigma}{\epsilon_0}.\pi r^2$

$= \frac{Q}{\pi R^2\epsilon_0}\pi r^2 = \frac{Q r^2}{\epsilon_0 R^2}$

$\frac{d {\Phi}_E}{d t} = \frac{r^2}{\epsilon_0 R^2}\frac{dQ}{dt}$

$= \frac{r^2}{\epsilon_0 R^2} I$

$\oint \vec{B} \cdot \vec{d l}=2 \pi ~r B$

$\oint \vec{B} \cdot \vec{dl}=\mu_0 ~\epsilon_0 \frac{d \Phi_E}{d t}$

$2 \pi ~r B=\mu_0 ~\epsilon_0 \frac{r^2}{\epsilon _0R^2} I$

$B=\frac{\mu_0 I}{2 \pi R^2} r$

${0 < r < R}$

$\Phi_E=E.~\pi R^2$

$= \frac{\sigma} {\epsilon_0}\pi R^2 =\frac{Q}{\epsilon_0}$

$\frac{d\Phi_E}{dt} =\frac{1}{\epsilon_0} \frac{dQ}{dt}=\frac{1}{\epsilon_0} I$

$2\pi r ~B = \mu_0 ~\epsilon_0 \frac{I}{\epsilon_0}$

$B = \frac{\mu_0~I}{2\pi r}$

$r > R$

$\mu_0~I/2\pi R$