Notes from Toppers
Maxwell’s Equations and Electromagnetic Waves
Gauss’s Law
Reference: NCERT Class 12, Chapter 1  Electric Charges and Fields

Electric field due to a point charge:
 The electric field due to a point charge is given by the equation: $$ \vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{r} $$
 Where:
 (\vec{E}) is the electric field vector
 (Q) is the magnitude of the point charge
 (r) is the distance from the point charge to the observation location
 (\hat{r}) is a unit vector pointing from the point charge to the observation location
 (\varepsilon_0) is the permittivity of free space

Electric flux:
 Electric flux is a measure of the amount of electric field passing through a given surface.
 It is defined as the dot product of the electric field vector and the area vector of the surface: $$\Phi_E = \oint \vec{E} \cdot d\vec{A}$$
 Where:
 (\Phi_E) is the electric flux
 (\vec{E}) is the electric field vector
 (d\vec{A}) is the area vector of the surface

Gauss’s law in integral form:
 Gauss’s law states that the total electric flux through a closed surface is equal to the total charge enclosed by the surface: $$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$$
 Where:
 (\oint \vec{E} \cdot d\vec{A}) is the total electric flux through the closed surface
 (Q_{enc}) is the total charge enclosed by the surface
 (\varepsilon_0) is the permittivity of free space

Gauss’s law in differential form:
 The differential form of Gauss’s law is: $$ \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0} $$
 Where:
 (\nabla \cdot \vec{E}) is the divergence of the electric field vector
 (\rho) is the charge density
 (\varepsilon_0) is the permittivity of free space

Applications of Gauss’s law:
 Gauss’s law can be used to calculate the electric field due to various charge distributions, such as point charges, charged spheres, and charged conductors.
 It can also be used to determine the electric flux through a given surface.
Gauss’s Law for Magnetism
Reference: NCERT Class 12, Chapter 4  Moving Charges and Magnetism

Magnetic field due to a currentcarrying wire:
 The magnetic field due to a currentcarrying wire is given by the BiotSavart law: $$ \vec{B} = \frac{\mu_0}{4\pi} \int \frac{I d\vec{l} \times \hat{r}}{r^2} $$
 Where:
 (\vec{B}) is the magnetic field vector
 (\mu_0) is the permeability of free space
 (I) is the current flowing through the wire
 (d\vec{l}) is a vector element of the currentcarrying wire
 (\hat{r}) is a unit vector pointing from the current element to the observation location
 (r) is the distance from the current element to the observation location

Magnetic flux:
 Magnetic flux is a measure of the amount of magnetic field passing through a given surface.
 It is defined as the dot product of the magnetic field vector and the area vector of the surface: $$\Phi_B = \oint \vec{B} \cdot d\vec{A}$$
 Where:
 (\Phi_B) is the magnetic flux
 (\vec{B}) is the magnetic field vector
 (d\vec{A}) is the area vector of the surface

Gauss’s law for magnetism in integral form:
 Gauss’s law for magnetism states that the total magnetic flux through a closed surface is equal to zero: $$\oint \vec{B} \cdot d\vec{A} = 0$$
 This means that there are no magnetic monopoles, which are isolated north or south poles.

Gauss’s law for magnetism in differential form:
 The differential form of Gauss’s law for magnetism is: $$ \nabla \cdot \vec{B} = 0 $$
 Where:
 (\nabla \cdot \vec{B}) is the divergence of the magnetic field vector

Applications of Gauss’s law for magnetism:
 Gauss’s law for magnetism can be used to determine the magnetic