Notes from Toppers
Maxwell’s Equations and Electromagnetic Waves
Gauss’s Law
Reference: NCERT Class 12, Chapter 1 - Electric Charges and Fields
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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
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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
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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
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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
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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
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Magnetic field due to a current-carrying wire:
- The magnetic field due to a current-carrying wire is given by the Biot-Savart 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 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: $$\Phi_B = \oint \vec{B} \cdot d\vec{A}$$
- 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
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Applications of Gauss’s law for magnetism:
- Gauss’s law for magnetism can be used to determine the magnetic