Uniformly Charged Straight Wire

If we assume a uniformly charged straight wire that carries a uniform charge per unit length λ. The expected electric field that the charge generates would have cylindrical symmetry. We will use a cylindrical Gaussian surface to find out the electric field of a uniformly charged straight wire. Gauss’s Law says that the total flux of an electric field in a closed surface and the enclosed electric charge will be directly proportional. There are four equations of electromagnetism and Gauss’s Law is one of them. This law is used to calculate electric fields due to uniformly charged straight wire, uniformly charged infinite plate sheet, and uniformly charged thin spherical shell.

Applications of Gauss’s Law

Gauss’s Law is an essential part of electromagnetism. It is helpful in relating the distribution of charge with the resultant electric field because of the charge. Joseph Lagrange formulated this law in 1773 and further by Carl Gauss in 1813. It has many applications, some of which are:

Electric field due to uniformly charged straight wire

We take a uniformly charged straight wire that has a linear charge density λ and Length (L). We would make an assumption of a cylindrical Gaussian surface to calculate the electric field. The electric flux across the end of the cylindrical surface would be zero because of the electric field E s circular in direction. The reason for this is that the area vector and the electric field are perpendicular. It is seen that the electric field is upright to each point of the curved surface; it can be said that it has a constant magnitude.

If the surface area of the cylindrical surface is given by 2πrl, then the electric flux through the curve would be:

E × 2πrl

According to Gauss’s Law:

Φ = q/εo

E × 2πrl = λl/εo

E = λ/2π ɛo r

It must be remembered that if the charge of linear density is positive, then the direction of the electric field would be radially outward.

Electric field because of uniformly charged infinite plate sheet:

We consider a uniformly charged infinite plate sheet that has a surface charge density σ with a cross-sectional area A. The infinite charged sheet will cause the direction of the electric field to be perpendicular to the plane of the sheet. If we take a cylindrical Gaussian surface, that has its axis normal to the plane of the sheet. So, by Gauss’s Law:

2EA = σA/εO

E = σ/2 εO

The unit vector, which is normal to surface 1 is in –x-direction, and the unit vector which is normal to surface 2 would be in +x direction. Hence, the fluxes from both the surfaces would be equal and add up. So, the total flux through the Gaussian surface would be 2 EA. The closed surface enclosing the charge is σ A.

Electric Field because of a uniformly charged thin spherical shell:

We assume a thin spherical shell with radius R and σ is its uniform surface charge density. At any point P, whether inside or outside, the field may depend only on r and it should be radial.

The field outside the shell:

If we consider a point P outside the shell that has a radius vector r. To calculate E, at point P, we would take the Gaussian surface that is a sphere with radius r and its centre is O, which would pass through P. Every point on this sphere is equally relative to the configuration of the charge that is given. At every point of the Gaussian surface, the electric field would have the same intensity E and is radial at each point.

So, E and ΔS would be parallel to each other at every point, and the flux through every element would be E ΔS. The flux across the Gaussian surface would be E × 4 π r2 and σ × 4 π R2 is the charge enclosed in it.

Conclusion

Gauss’s law is used for the derivation of the equations for electrical fields for a number of charged surfaces. We can take any object like a uniformly charged straight wire or a uniformly charged infinite plate sheet or a uniformly charged thin spherical shell. We can define the electric flux as the product of the electric field and the area is the surface that is perpendicular to the electric field. The law is very important, as it helps us in estimating the electrical charge that is enclosed inside a closed surface. The law is useful for geometries that are symmetrical.