Gaussian Surface for different cases

Science is a fascinating subject that is full of amazing facts and things. The more we delve into the concepts of science and its interlinked subjects, the more knowledge and information we gain. The Gauss Law tells us about electric charge, Gaussian surface for different cases and the concept of electric flux. Let us learn more about this law and how it functions and how we can use the gaussian surface for different cases. In this article, we will read about Gaussian surfaces for different cases. 

Gauss law

Gauss law tells us that any enclosed object has an electric charge proportional to the net flow of an electric field of that closed surface. It’s one of Maxwell’s electromagnetic laws – and part of four equations. Carl Friedrich Gauss first proposed this law in 1835. This law connects the electric fields at points on a closed surface to the net charge encircled by that surface. The electric flux is then calculated by multiplying the electric field flowing through a specific area by the surface area in a plane perpendicular to the field. 

Or we can say that Gauss’s law states that the net flux of an electric field across a surface is divided by the enclosed charge, which should always be equal to a constant.

A positive electric charge produces a positive electric field in many cases. The law was published in 1867 as part of a collection of research work by Carl Friedrich Gauss, a notable German mathematician.

The Mathematical Equation for Gauss Law: 

Gauss law can be studied through the integral equation: 

∫E⋅dA=Q/ε0               …..(1)   

Here: 

• E is the electric field vector.

• Q is for enclosed electric charge.

• ε0 is the electric permittivity in free space.

• A tells us about outward pointing normal area vectors.

Here the flux can be measured by the strength of a field that passes through a surface. 

Electric flux can be defined as: 

Φ=∫E⋅dA                     ….(2)

The electric field can also be assumed as a flux density. Gauss law is defined as that unless the volume enclosed by a closed surface has a net charge, the net electric current flowing through that surface will be taken as zero.

The simplest way to understand Gauss’s law for electric fields is to overlook electric displacement (d). There are many examples where the dielectric permittivity may not equal the free-space permittivity (i.e. 0). Therefore the density of electric charges will be divided into two categories: free charge density (f) and bounded density (b).  

Then the formula would be: 

P = ρf + ρb. 

Question: Let’s assume that there are three charges q1, q2 and q3. All these have charge 6c, 5c and 3c respectively in an enclosed surface. Now find the total flux enclosed in that surface.

Solution: Let’s total charge be Q,

Q = q1 + q2 + q3

    = 6c + 5c + 3c

     = 14 C

Therefore the total flux, ϕ will be = Q/ϵ0

ϕ = 14C / (8.854×10−12 F/m)

ϕ = 1.584 Nm2/C

The solution would be the total flux enclosed by the surface equals 1.584 Nm2/C.

Now that we understand how the gauss law works and the electric charges and field. But, we do not know where to apply the Gauss law and on what Gaussian surface? 

What is the Gaussian surface for different cases? 

A Gaussian surface is where the angle (ө) between the electric field (E) and the area vector (A) is always constant at every point. As a vector quantity always has a magnitude and a direction, area vectors can be defined as the vectors of plane surfaces with a magnitude equal to the surface area that is directly perpendicular to it. So, how can you find a Gaussian surface when you see one? 

How can you tell that a surface is Gaussian?

We have learned that all Gaussian surfaces follow Gauss law. In Gauss law, the angle (ө) is between the electric field (E) and the area vector (A), which are constant at every point of the surface. To tell which surface is a Gaussian surface, we must check whether the angle (ө) in the plane is constant at every point. 

The steps to find out the Gaussian surface are as follows:

• Select a surface that can be any object or human body. 

You may be confused about where to start and what angle would best determine the Gaussian surface or whether that angle gives you the same points. You must understand that if you want to determine whether a surface on a body is Gaussian or not, you need to check the angle (ө). This angle must be constant at every point on the surface, not at every position on the entire body. 

You can shade the part where you want to find the gaussian surface. There can be objects that have many Gaussian surfaces, but you don’t have to worry about that. 

• Now, find the electric field’s direction (E). 

The next step after determining the surface would be finding the direction of the electric field emanating from the surface. On this surface, the electric field lines are emitted.

• Now check the area vector (A) of the surface. 

After that, you can find the direction of the electric field. Now you need to find the surface’s area vector. A vector whose direction is always perpendicular to the surface is referred to as an area vector.

• Find the angle formed by the area vector (A) and the electric field (E)

After finding the direction of the electric field and the area vector, you need the angle (ө) between them. Now, check to determine whether the angle between them is constant at some points. If all points are constant, then it is a Gaussian surface.

Conclusion

A Gaussian surface is a 3d enclosed surface where the flux of a vector field is determined (gravitational field, the electric field, or magnetic field.) As there is a symmetric charge distribution, the Gaussian surface will help determine the electric field intensity.