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: $$\overrightarrow{E}=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}\hat{r}$$
  • 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: $${\mathrm{\Phi }}_E=\oint \overrightarrow{E}\cdot d\overrightarrow{A}$$
  • 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: $$\oint \overrightarrow{E}\cdot d\overrightarrow{A}=\frac{Q_{enc}}{{\epsilon }_0}$$
  • 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: $$\mathrm{\nabla }\cdot \overrightarrow{E}=\frac{\rho }{{\epsilon }_0}$$
  • 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 current-carrying wire:

  • The magnetic field due to a current-carrying wire is given by the Biot-Savart law: $$\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\int \frac{Id\overrightarrow{l}\times \hat{r}}{r^2}$$
  • 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 current-carrying 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: $${\mathrm{\Phi }}_B=\oint \overrightarrow{B}\cdot d\overrightarrow{A}$$
  • 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: $$\oint \overrightarrow{B}\cdot d\overrightarrow{A}=0$$
  • 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: $$\mathrm{\nabla }\cdot \overrightarrow{B}=0$$
  • 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