Gauss’S Law In Electrostatics

Slide 1 - Gauss’s Law in Electrostatics

Introduced by Carl Friedrich Gauss
Relates the electric flux through a closed surface to the charge enclosed within the surface
Helps in determining the strength of an electric field due to enclosed charge

Slide 2 - Electric Flux

Electric flux (Φ) is a measure of the electric field lines passing through a given surface
It depends on the magnitude and direction of the electric field and the area of the surface

Slide 3 - Gauss’s Law Equation

Gauss’s Law states that the electric flux (Φ) through any closed surface is equal to the enclosed charge (Q) divided by the permittivity of free space (ε₀): $\Phi = \frac{Q}{\varepsilon_0}$

Φ: Electric flux (in Nm²/C)
Q: Total charge enclosed by the surface (in C)
ε₀: Permittivity of free space (8.85 x 10⁻¹² C²/Nm²)

Slide 4 - Gauss’s Law and Closed Surfaces

Gauss’s Law is applied to closed surfaces
A closed surface is one that completely encloses a region of space
Examples of closed surfaces: sphere, cube, cylinder, etc.

Slide 5 - Sign Convention in Gauss’s Law

Positive flux: Electric field lines flowing outward from the surface
Negative flux: Electric field lines flowing inward towards the surface
The direction of the normal vector to the surface determines the sign of the flux

Slide 6 - Gaussian Surface

Gaussian surface is a hypothetical closed surface used to apply Gauss’s Law
The choice of surface depends on the symmetry of the charge distribution
Gauss’s Law simplifies the calculation of electric fields for highly symmetrical charge distributions

Slide 7 - Gauss’s Law for Spherical Symmetry

For spherically symmetrical charge distributions, we use a Gaussian surface in the form of a sphere
The electric field is constant on the Gaussian sphere’s surface radius

Slide 8 - Example: Electric Field due to a Point Charge

Consider a point charge Q located at the center of a Gaussian sphere of radius r
The electric field on the sphere’s surface is given by Gauss’s Law: $\Phi = \frac{Q}{\varepsilon_0}$

Slide 9 - Gauss’s Law for Planar Symmetry

For charge distributions with planar symmetry, we use a Gaussian surface in the form of a cylinder
The electric field is constant on the Gaussian cylinder’s side surface area

Slide 10 - Example: Electric Field due to an Infinite Plane of Charge

Consider an infinite plane of charge with charge density σ
The electric field on the Gaussian cylinder’s surface area is given by Gauss’s Law: $\Phi = \frac{\sigma \cdot A}{\varepsilon_0}$
σ: Charge density (in C/m²)
A: Area of the Gaussian cylinder’s side surface (in m²)

Gauss’s law in electrostatics - Gauss’s Law

Introduced by Carl Friedrich Gauss
Relates the electric flux through a closed surface to the charge enclosed within the surface
Helps in determining the strength of an electric field due to enclosed charge

Electric Flux

Electric flux (Φ) is a measure of the electric field lines passing through a given surface
It depends on the magnitude and direction of the electric field and the area of the surface

Gauss’s Law Equation

Gauss’s Law states that the electric flux (Φ) through any closed surface is equal to the enclosed charge (Q) divided by the permittivity of free space (ε₀): $\Phi = \frac{Q}{\varepsilon_0}$

Φ: Electric flux (in Nm²/C)
Q: Total charge enclosed by the surface (in C)
ε₀: Permittivity of free space (8.85 x 10⁻¹² C²/Nm²)

Gauss’s Law and Closed Surfaces

Gauss’s Law is applied to closed surfaces
A closed surface is one that completely encloses a region of space
Examples of closed surfaces: sphere, cube, cylinder, etc.

Sign Convention in Gauss’s Law

Positive flux: Electric field lines flowing outward from the surface
Negative flux: Electric field lines flowing inward towards the surface
The direction of the normal vector to the surface determines the sign of the flux

Gaussian Surface

Gaussian surface is a hypothetical closed surface used to apply Gauss’s Law
The choice of surface depends on the symmetry of the charge distribution
Gauss’s Law simplifies the calculation of electric fields for highly symmetrical charge distributions

Gauss’s Law for Spherical Symmetry

For spherically symmetrical charge distributions, we use a Gaussian surface in the form of a sphere
The electric field is constant on the Gaussian sphere’s surface radius

Example: Electric Field due to a Point Charge

Consider a point charge Q located at the center of a Gaussian sphere of radius r
The electric field on the sphere’s surface is given by Gauss’s Law: $\Phi = \frac{Q}{\varepsilon_0}$

Gauss’s Law for Planar Symmetry

For charge distributions with planar symmetry, we use a Gaussian surface in the form of a cylinder
The electric field is constant on the Gaussian cylinder’s side surface area

Example: Electric Field due to an Infinite Plane of Charge

Consider an infinite plane of charge with charge density σ
The electric field on the Gaussian cylinder’s surface area is given by Gauss’s Law: $\Phi = \frac{\sigma \cdot A}{\varepsilon_0}$
σ: Charge density (in C/m²)
A: Area of the Gaussian cylinder’s side surface (in m²)

Slide 21: Electric Field of a Charged Line

Charged line: Infinitely long line of charge with linear charge density (λ)
Gaussian surface: Cylinder with one end on the line and radius (r)
Electric field inside the cylinder: Zero (due to symmetry)
Electric field outside the cylinder: Given by Gauss’s Law: $E = \frac{\lambda}{2\pi\varepsilon_0 r}$

Slide 22: Electric Field of a Charged Disk

Charged disk: Flat disk of charge with surface charge density (σ)
Gaussian surface: Cylinder with one end on the disk and height (h)
Electric field inside the cylinder: Zero (due to symmetry)
Electric field outside the cylinder: Given by Gauss’s Law: $E = \frac{\sigma}{2\varepsilon_0}$

Slide 23: Electric Field of a Charged Sphere

Charged sphere: Sphere of charge with total charge (Q) and radius (R)
Gaussian surface: Sphere concentric with the charged sphere
Electric field inside the Gaussian sphere: Given by Gauss’s Law: $E = \frac{Q}{4\pi\varepsilon_0 R^2}$

Slide 24: Electric Field of Multiple Charges

Gauss’s Law can be used to find the electric field due to multiple charges
Calculate the net enclosed charge for each Gaussian surface
Sum the electric fields due to each enclosed charge

Slide 25: Flux Through a Closed Surface

The electric flux through a closed surface is given by Gauss’s Law: $\Phi = \frac{Q}{\varepsilon_0}$
Φ: Electric flux (in Nm²/C)
Q: Total charge enclosed by the surface (in C)
ε₀: Permittivity of free space (8.85 x 10⁻¹² C²/Nm²)

Slide 26: Gauss’s Law in Differential Form

Gauss’s Law can also be written in differential form using the divergence operator: $\vec{\Delta } \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$
∇: Del operator
E: Electric field (in N/C)
ρ: Charge density (in C/m³)

Slide 27: Gauss’s Law for Closed Surfaces

Gauss’s Law is applicable to any closed surface
The choice of Gaussian surface depends on the symmetry of the charge distribution
Gauss’s Law simplifies the calculation of electric fields for highly symmetrical charge distributions

Slide 28: Applications of Gauss’s Law

Calculating electric fields due to infinite line, plane, and spherical charge distributions
Determining the flux through closed surfaces
Understanding the distribution of electric charge in conductors

Slide 29: Limitations of Gauss’s Law

Gauss’s Law is based on the assumption of static charges
It does not hold for time-changing electric fields
It is only applicable in situations with sufficient symmetry

Slide 30: Summary

Gauss’s Law relates the electric flux through a closed surface to the enclosed charge
It simplifies the calculation of electric fields for highly symmetrical charge distributions
Gaussian surfaces are used to apply Gauss’s Law
Electric field inside closed surfaces can be determined using Gauss’s Law
Gauss’s Law can be written in differential form using the divergence operator