Applications Of Gausss Law

Concepts

Visualize the wire as a source of electric fields radiating outward in all directions.
The field lines are parallel to the wire and their density decreases with the square of the distance from the wire.
Use the formula $$E = \frac{2 k_e \lambda}{r}$$, where $$k_e$$ is the Coulomb constant, (\lambda) is the linear charge density, and $$r$$ is the distance from the wire.

Imagine concentric field lines emanating radially from the center of the spherical shell.
The field outside the shell is the same as that of a point charge of the same magnitude located at the center.
Inside the shell, the electric field is zero.
Use the formula $$E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}$$, where $$\epsilon_0$$ is the vacuum permittivity, $$Q$$ is the total charge enclosed by the shell, and $$r$$ is the distance from the center of the shell.

The field has a non-zero value both inside and outside the sphere.
The field outside the sphere is the same as that of a point charge of the same magnitude located at the center of the sphere.
The field inside the sphere increases linearly with the distance from the center.
Use the formulas $$E_{outside} = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}$$ for points outside the sphere and $$E_{inside} = \frac{1}{4\pi\epsilon_0}\frac{Qr}{R^3}$$ for points inside the sphere, where $$R$$ is the radius of the sphere.

Gauss’s law states that the net flux of the electric field through any closed surface is proportional to the net charge enclosed by that surface.
Mathematically, $$\oint\overrightarrow{E}\cdot\hat{n}\text{d}A=\frac{Q_{enc}}{\epsilon_0}$$, where $$\hat{n}$$ is the unit normal vector perpendicular to the surface, $$\text{d}A$$ is the differential area element, and $$Q_{enc}$$ is the net charge enclosed by the surface.

Calculate the electric field of various charge distributions by choosing appropriate Gaussian surfaces.
Simplify complex charge distributions by enclosing them with symmetric Gaussian surfaces that can simplify the calculation.
Determine the net charge enclosed by a surface by calculating the flux of the electric field through that surface.

Flux measures the amount of electric field passing through a surface.
It is defined as the dot product of the electric field and the surface area vector $$(\overrightarrow{E}\cdot\hat{n})\text{d}A$$.
The net flux of the electric field through a closed surface is equal to the net charge enclosed by that surface divided by the vacuum permittivity.

Divergence measures the net outward flow of the electric field from a point in space.
It is calculated as the limit of the net flux through a small surface divided by the volume of the enclosed region as the volume approaches zero.
Divergence is a source term and is related to the charge density by the equation $$\overrightarrow{\nabla}\cdot\overrightarrow{E}=\frac{\rho}{\epsilon_0}$$

Divergence of the electric field at a point is directly proportional to the charge density at that point.
A positive divergence indicates a source of electric field lines (positive charge), while a negative divergence indicates a sink (negative charge).
Zero divergence implies the absence of any net charge or a balanced distribution of positive and negative charges.

Concept To Remember