Gausss Law In Electrostatics

Concepts: 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,

(\overrightarrow{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 the unit vector pointing from the point charge to the observation location.

Electric flux:

• Electric flux is a measure of the amount of electric field passing through a given surface. It is given by the equation: $$\Phi_E=\oint\overrightarrow{E}\cdot\hat{n}dA$$ Where (\Phi_E) is the electric flux (\overrightarrow{E}) is the electric field vector. (\hat{n}) is the normal unit vector perpendicular to the surface. (dA) is the differential area of the surface.

Gauss’s law:

• Gauss’s law states that the total electric flux through any closed surface is equal to the total charge enclosed by that surface. It is given by the equation: $$\oint\overrightarrow{E}\cdot\hat{n}dA=\frac{Q_{enc}}{\epsilon_0}$$ Where, $$\oint\overrightarrow{E}\cdot\hat{n}dA$$ represents the total electric flux through the closed surface. (Q_{enc}) is the total charge enclosed by the closed surface. (\epsilon_0) is permittivity of free space.

Applications of Gauss’s law:

Electric field of a uniformly charged sphere: For a uniformly charged sphere, the electric field at a distance (r) from the center of the sphere is given by:

• Inside the sphere ((r<R)) $$\overrightarrow{E}=0$$

• Outside the sphere ((r>R ))

$$\overrightarrow{E}=\frac{Q}{4\pi\epsilon_0 r^2}\hat{r}$$

Electric field of a uniformly charged infinite plane: For a uniformly charged infinite plane with surface charge density (\sigma), the electric field at a distance (d) from the plane is given by: $$\overrightarrow{E}=\frac{\sigma}{2\epsilon_0}\hat{n}$$ Where (\hat{n}) is the normal unit vector perpendicular to the plane.

Electric field of a charged conducting sphere: For a charged conducting sphere, the electric field at a distance (r) from the center of the sphere is given by:

• Inside the sphere (r<R):

$$\overrightarrow{E}=0$$

• Outside the sphere (r>R): $$\overrightarrow{E}=\frac{Q}{4\pi\epsilon r^2}\hat{r}$$