Knowing more on Consider a cylindrical Gaussian surface

A surface on which the angle between the electric field and area vector is the same at every point is known as a Gaussian surface. Some common types of Gaussian surfaces are the cylindrical surface, spherical surface, and Gaussian pillbox. Generally, Gaussian surfaces are chosen carefully so that the calculation of the surface integral is simplified. If a Gaussian surface is chosen in a way that the electric field is constant at each point on the surface and along the regular vector, Gauss’s Law may be used to calculate the electric field (\vec E).

Cylindrical Gaussian Surface

We will consider a cylindrical Gaussian surface to calculate the electric flux or electric field produced by the given objects:

• A long infinite line of uniform charge
• An infinite plane with an unchanging charge
• An infinitely long cylinder with an unchanging charge

Imagine a point charge P, present at r distance with the charge density λ. If we take a closed surface in the shape of a cylinder, the rotational axis of this closed surface is the line charge. The length of the cylinder is ‘h’, then the charge inside the cylinder will be:

Q = λh,

Where the charge enclosed in the cylindrical Gaussian surface is denoted by q. 

There are three surfaces named a, b, and c. On each surface, dA is the differential vector area. The flux passing through the cylinder will be:

ΦE = ∫∫◯A E. dA = ∫∫◯a E. dA + ∫∫◯b E. dA + ∫∫◯c E. dA

E and dA will be perpendicular for surfaces a and b. However, E and dA will be parallel for surface c.

The cylinder’s surface area would be:

∫∫c dA = 2πrh

which suggests

ΦE = E2πrh

Gauss’s law states:

ΦE = q / ∈0

When we equate ΦE, we will find out that

E2πrh = λh / ∈0

E = λ / 2π∈0r

Electric Field Due to Infinite Plate Sheet

Consider an infinite plane sheet with a surface charge density σ with A as the cross-sectional area.

Infinite Charge Sheet

The electric field generated by the infinite charge sheet will be perpendicular to the sheet’s plane. Consider a cylindrical Gaussian surface that has its axis normal to the plane of the sheet. Gauss’s law will be used to evaluate the electric field.

According to the law

Φ = q / ∈0

Due to continuous charge distribution, the charge q would be σ A. If we talk about net flux, the electric flux would be considered only from the two ends of the Gaussian surface that we assumed. This is because the curved surface area and electric field are normal to each other. Thus, the electric flux produced is zero. Therefore, the total electric flux would be:

Φ = EA – (– EA) => Φ = EA + EA

Φ = 2EA

We can also write it as,

E = σ A/ ∈0

The term A in the equation cancels out, which means the electric field due to the infinite plane sheet is not dependent on cross-sectional area A and is equal to

E = σ / 2∈0

Gaussian Pillbox

The Gaussian pillbox is used when we need to determine the electric field due to an infinite sheet of charge with unchanging charge density or a finite thickness slab of charge. The shape of the pillbox is cylindrical, and it consists of three components. At one end of the cylinder, there is a disk that has an area of πR2. The disk at the other end of the cylinder has the same area as the cylinder’s side. The totality of the flux passing through each of its components is proportional to the charge enclosed in the pillbox. This is because the field close to the sheet could be estimated as constant. Moreover, the orientation of the pillbox is done in a manner that the field pierces the ends of the field at a 90-degree angle and the field lines are parallel to the cylinder’s side.

Conclusion

The types of surfaces that have the same angle between the electric field and area vector at every point are known as Gaussian surfaces. Gauss’s law may be used to calculate the electric field (\vec E). This law is extremely important in determining the relation between electric flux and electric charge. When we try to calculate the electric flux of an object, we use a Gaussian surface. There are commonly three types of Gaussian surfaces: cylindrical surface, spherical surface, and Gaussian pillbox.