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What is Gauss’ Law? Explain with Examples!

Gauss’ law has two laws which are both laws of physics. They are:

  • Gauss’s Law for Electrostatics (stationery charges), known as Gauss’s Flux Theorem
  • Gauss’s Law for Magnetism

Let us understand Gauss’s Law for Electrostatics. It is one of the four equations known as Maxwell’s Equations.

In the year 1835, a man named Carl Friedrich Gauss came up with the idea that an electric charge that causes your hair to stand on end, or you get a shock when you scoot your feet against the carpet, has a relation to an electric field.

 

Now let us understand what an electric field is. The electric charges can apply force on uncharged objects over big or small distances. For example, when our hair stands on, there is a force which causes the hair to stand which is caused by electrostatic charges. There is a positive charge somewhere which attract a negative one somewhere else. When they get attracted towards each other, they exert a force on one another.

 

Force and charge can be related to each other by using an electric field. Mathematically, it can be represented as:

The electric field,   is derived by dividing the electric force,   by the charge, q. Thus, the electric field is the force per (divided by) unit charge.

 

Here is the actual law:

The net electric flux passing through any closed surface is equal to  times the net electric charge which is within that surface.

 

Now we have a closed surface, an electric field, charges inside the surface, electric force. Let us put all of them together to understand what an electric flux is.

The charge is inside another surface. Here, it is in a ball. It is called a gaussian or enclosed surface which means it is around something else. In the diagram given below, the blue lines are electric field lines of force which arise from the charge inside and intersects the surface. These are the electric flux lines.

 

Gauss’s law helps us in giving a mathematical way to know how much charge, or force, is there at a point where the flux lines intersect the gaussian surface. In the diagram given below, the flux is shown by the aqua half-circles at the base of the lines.

 

 

 

  • When there is no charge inside, then there will be no net (excess) flux (field lines of force).
  • When there is an enclosed charge, the flux will be proportional to that charge inside. In this example, each charge of coulomb will be equal to one line of flux. When the charge inside is +4 C, then the flux lines might be 4. When the charge is 8, there will be 8 flux lines.

 

Gauss’ Law Equation

 

Now let us understand Gauss’ law through an integral equation. Gauss’s law in integral form is shown as below:

where,

 

  • E stands for the electric field
  • Q stands for the enclosed electric charge
  •  is the electric permittivity of free space
  • A stand for the outward pointing normal area vector

 

Flux is a measure of the strength of a field which passes through a surface. Electric field can be defined as:

 

 

The field can be understood better as flux density. Gauss’s law states that the net electric flux passing through any given closed surface is equal to zero unless the volume occupied by that surface has a net charge.

 

Gauss’s law for electric fields can be easily understood by neglecting electric displacement (d).

The dielectric permittivity in matters, might not be equal to the permittivity of free-space. The density of electric charges in the matter can be separated into a “free” charge density and a “bounded” charge density.

 

Examples

Let us understand Gauss’ law with the help of examples.

 

According to Gauss’ law, the flux is equal to the charge Q, over the permittivity of free space, epsilon-zero. Flux is also equal to the electric field E which is multiplied by the area of the surface A. Hence, EA is equal to Q over epsilon-zero.

 

In a sphere, the surface area is represented by , so that we can plug in A. When the electric field E is rearranged, the electric field E is then equal to Q over

Another example is a conducting cylinder whose surface area is equal to       2pirL, where r stands for the radius of the cross-section of the cylinder and L is the length. When they are rearranged for E again, we get Q over    which is an expression of electric field created by a charged cylinder.

Can you answer this question: A charge q is placed at the center of the open end of a cylindrical vessel. The flux of the electric field through the surface of the vessel is?

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