Electrostatics

Concept:
Electric flux ( ) is defined as the surface integral of the electric field over a surface: From Gauss’s Law: Thus, the unit of electric flux can be determined from either definition.
Step 1:
The SI unit of electric field is:
Step 2:
The SI unit of area is:
Step 3:
Therefore, electric flux unit: Alternatively, Both forms are equivalent SI units.

Concept:
The dielectric constant (also called relative permittivity) measures how much a medium reduces the electric field compared to vacuum. It indicates how well a material can store electric energy in an electric field. Mathematically, it is defined as: where: = permittivity of the medium
= permittivity of free space

Step 1:
When a dielectric is placed in an electric field, it becomes polarized. This reduces the effective electric field inside the material.
Step 2:
The extent of this reduction is measured by the dielectric constant . A higher value of means stronger polarization and greater ability to store electric energy.
Step 3:
Thus, dielectric constant is a dimensionless quantity that compares electrical behavior of a medium with vacuum.

Concept:
Gauss’s law states that the total electric flux through a closed surface is equal to the charge enclosed divided by permittivity of free space. For an infinite plane sheet, the electric field is: - Perpendicular to the sheet - Same magnitude everywhere (symmetry)
Step 1: Choose a Gaussian surface
Take a cylindrical Gaussian surface (pillbox) with: - One face above the sheet - One face below the sheet - Axis perpendicular to the sheet
Step 2: Electric flux through the surface
Electric field is perpendicular to the plane and parallel to the curved surface. So flux through curved surface = 0. Flux only passes through the two flat faces:
Step 3: Charge enclosed
If surface charge density = , then charge enclosed:
Step 4: Apply Gauss’s law Cancel :
Step 5: Direction of electric field
- Away from sheet if charge is positive - Towards sheet if charge is negative

Concept:
Gauss’s law relates electric flux through a closed surface to the charge enclosed: For highly symmetric charge distributions (spherical, cylindrical, planar), Gauss’s law helps directly calculate electric field.
Step 1: Statement of Gauss’s law
The total electric flux through any closed surface is equal to times the total charge enclosed inside the surface.
Step 2: Symmetry of an infinite line charge
For an infinitely long straight wire:
- Electric field is radial (perpendicular to wire)
- Same magnitude at equal distance
- Cylindrical symmetry exists
Step 3: Choose Gaussian surface
Take a cylindrical Gaussian surface:
- Radius =
- Length =
- Wire along axis of cylinder
Step 4: Electric flux through surface
Electric field is:
- Perpendicular to curved surface
- Parallel to flat ends
So flux through flat ends = 0.
Flux only through curved surface:
Step 5: Charge enclosed
If linear charge density = , then:
Step 6: Apply Gauss’s law Cancel :
Step 7: Nature of electric field
- Decreases as - Radially outward for positive charge - Radially inward for negative charge