Electric Field Due to Infinitely Long Straight Wire

Gauss’ law is a key notion in physics and electromagnetics. It is used to link the charge distribution to the charges resulting in an infinitely long straight wire leading to the electric field. 

This law was developed by Joseph Lagrange in 1773 and confirmed by Carl Gauss in 1813. It is included in one of Maxwell’s four equations, which serves as the basis for classical electrodynamics. In electrostatics, the ultimate purpose of Gauss’ law is used to find the electric flux from a closed surface. It is the most fundamental electrostatic law. According to Gauss’ law, “the sum of the electric flux out of a closed surface is  equal to the charge enclosed in that region divided by the permittivity of a vacuum.” 

The Concept of Electric Fields Due to an Infinitely Long Straight Wire

The first important step is to choose a Gaussian surface. The ratio of the charge distribution, followed by the portion of the field, depicts a Gaussian surface as a circular cylinder. Let us suppose we have an infinitely large plain sheet and on this, positive charges are dispersed equally. For example, take an infinitely long, uniformly charged wire with a constant linear charge density λ (charge per unit length). 

Let P be a point r that is away from the wire, and E denotes the electric field at point P. 

The Gaussian surface is a cylinder with a length and radius that is closed at both ends by plane caps that are typical to the axis. The electric field would be of a similar magnitude and would be directed radially outward at all points on the curving surface of the cylinder because of symmetry. Both E, as well as ds, are pointing in the same direction.

If the linear charge density is positive, the electric field will be completely outward. If the linear charge density is negative, however, it will be dramatically inward. The cylinder’s top and bottom surfaces are parallel to the electric field. As a result, the angle formed by the area vector and the electric field is 90 degrees, with cos θ = 0.

Electric Field Due to an Infinitely Long Straight Uniformly Charged Wire

The electric field intensity is interpreted as the force that is encountered by a unit positive test charge in the electric field at any point.

E = F/ qo 

Here E = electric field intensity

qo = charge on the particle

Electric Field Intensity Due to an Infinitely Long Straight Uniformly Charged Wire

The electric field direction at every point must be radial ( if λ > 0, then outward and λ < 0 if inward).

E = λ/2𝜋ϵor

Here λ = linear charge density 

ϵo = permittivity,

r = distance of the point from the wire. 

Application of Gauss’ Law 

Let us suppose a charge line that is infinitely long and has a charge per unit length. The cylindrical symmetry of the circumstance could be used to our advantage. Because of symmetry, all-electric fields point radially out from the line of charge. There is no component parallel to the line of charge.

We can use a cylinder with an arbitrary radius (r) and length (l) centred on the charge line as our Gaussian surface. Presume an electric field is perpendicular to the curved surface of the cylinder. As a consequence, the angle created between the electric field and the area vector is 0, and cos is equal to one.

Both the top and bottom surfaces of the cylinder are aligned to the electric field. As a consequence, the area vector and the electric field form a 90-degree angle, and cos = 0. As a result, the curved surface is the sole source of electric flux.

Conclusion

The Gaussian surface is a closed imaginary surface. According to Gauss’ law, the flow of E through a closed surface S is solely determined by the value of the net charge inside the surface, not by the location of the charges. It states that the electric field flux out of every other closed surface is proportional to the electric charge encompassed by the surface in its integral form, regardless of how that charge is distributed. To create the formulas for an electrical field, Gauss’ law applies to several charged forms. Consider an endlessly long wire carrying a charge. The charge for every unit length of such a wire is represented by lambda. The purpose is to come up with an equation for calculating the electric field produced by this wire.