Gauss’s Law

According to Gauss Law, “The total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity. When it comes to calculating the electric flux, simply multiply the area of the field’s surface projected”. 

What is Gauss Law? 

The law was included in a compilation of work by the great German mathematician Carl Friedrich Gauss that was released later in 1867. The static electric field created by a distribution of electric charges is described by Gauss’s law in electrostatics for the electric field. It claims that the total electric charge enclosed by any closed surface is proportional to the electric flux through that surface. A positive electric charge, by convention, produces a positive electric field.

Applications of Gauss Law

Gauss’s Law can be utilised to address complex electrostatic issues with unusual symmetry, such as cylindrical, spherical, or planar. In other circumstances, calculating the electric field is also fairly difficult and requires a lot of integration. Gauss’s Law can be used to evaluate an electric field in a straightforward manner.

The following is how we use Gauss’s Law:

• Choose a Gaussian surface that makes evaluating the electric field simple.
• To make tasks easier, apply symmetry.
• Remember that the Gaussian surface does not have to overlap with the real surface; it might be inside or outside the Gaussian surface.

Electric Field due to Infinite Wire

Imagine a wire that is infinitely long and has a linear charge density. Due to the symmetry of wire, we assume a cylindrical Gaussian surface to compute the electric field. Because the electric field E is radial in direction, flux through the end of the cylindrical surface will be 0 because the electric field and the area vector are perpendicular. The curved Gaussian surface will be the only source of electric flux.

Electric Field due to Infinite Plate Sheet

Gauss’ law can be used to calculate the electric field of an infinite line charge with a uniform linear charge density. The electric field has the same amplitude at every point of the cylinder and is directed outward when considering a Gaussian surface in the form of a cylinder with radius r.

Electric Field due thin Spherical Shell

Assume a thin spherical shell with a radius “R” and a surface charge density of. Shell has spherical symmetry, as may be seen by looking at it. The electric field produced by the spherical shell can be measured in two ways:

• Electric Field Outside the Spherical Shell
• Electric Field Inside the Spherical Shell

Types of Symmetry Used to Deduce the Electric Field 

Gauss’s law in electrostatics can only be used to calculate the electric field if there are three types of symmetry. These include: 

1. Distribution of Charge with Spherical Symmetry

The density of charge varies exclusively with distance from a point in space and not with direction. In other words, when the system is rotated, it does not change its look. 

The sphere does not exhibit spherical symmetry if the sphere of radius R is charged such that the top half has a uniform charge density of 1 and the bottom half has a uniform charge density of 21 since the charge density is direction-dependent. Thus, whether or not a system exhibits spherical symmetry is determined by the geometry of the charge distribution rather than the shape of the object.

A sphere with separate shells, each with its uniform charge density, as depicted in the diagram below. Although charge density in the entire sphere is not uniform, the charge density function is only affected by distance from the centre, not by direction. 

Using Gauss’s law

1. Gauss’s law to find the electric field inside and outside a spherical shell. 

The flux via a closed surface is equal to the total charge enclosed within the closed surface divided by the permittivity of vacuum 0 as per Gauss’s law.

2. Distribution of Charge with Cylindrical Symmetry

Cylindrical symmetry can be defined as the condition if the charge density of a charge distribution depends only on the distance r from the cylinder’s axis and does not vary along the axis. In other words, you don’t have cylindrical symmetry if your system differs when you rotate it around the axis or shift it along the axis. In a cylindrically symmetrical condition, the electric field is simply affected by the distance from the axis. For positive charges, the electric field is directed away from the axis, while for negative charges, it is directed toward the axis.

Using Gauss’s law

The flux must match the quantity of charge within the volume encompassed by this surface, divided by the permittivity of open space, according to Gauss’s law. When you calculate qenc of Gauss’s law for a cylinder of length L, you’ll see that it’s directly proportional to L. Let’s put it this way: charge per unit length (enc) multiplied by length L:

qenc=λencL

Consequently, Gauss’s law produces the following magnitude of the electric field at a distance s from the axis for any cylindrically symmetrical charge distribution:

3. Charge Distribution with Planar Symmetry

When charges are evenly scattered across a broad flat surface, they generate a planar symmetry of charge density. All points in a plane parallel to the plane of charge are identical with respect to the charges in planar symmetry.

The two charges situated symmetrically from the field point P cancel out the components of the electric field parallel to a plane of charges. As a result, the field at any location is pointing vertically away from the charged plane. We can always obtain a q2 with this effect for any point P and charge q1.

Conclusion 

The relationship between the charge distribution that produces the electrostatic field and the behaviour of the electrostatic field in space is known as Gauss’s law. According to Gauss’s law, the excess charge would be totally on the surface of the conductor material, not in the interior. Remember, this is only true when the solid conductor is electrostatic and the ions and valence electrons that make up the conductor do not move in a net electric charge motion (current).