What is Gauss Law : Theory & Its Significance

As the scope of science is extensively enhancing and included with various developments and technologies, the more we learn the more we gain knowledge. And the one crucial topic which we need to be aware of is Gauss law which analyses electrical charge in addition to the surface and the concept of electrical flux. The law was initially articulated by Lagrange in the year 1773 and then it was supported by Friedrich in 1813. This law is one of the Maxwell proposed four equations where this is a fundamental concept for classical electrodynamics. So, let’s dive more into the concept and know all the related concepts of Gauss law.

What is the Gauss Law?

Gauss law can be defined in both the concepts of magnetic and electric fluxes. In the view of electricity, this law defines that electric flux all through the enclosed surface has direct proportion to the total electrical charge which is enclosed by the surface. It indicates that the insular electrical charges do exist and such similar charges get repelled whereas dissimilar charges get attracted. And in the scenario of magnetism, this law states that magnetic flux all through the enclosed surface is null. And gauss law seems to be stable in the scrutiny that the separated magnetic poles do not exist. The Gauss law diagram is shown as below:

This law can be either defined as that the net electrical flux in the enclosed surface equals to the electrical charge in correspondence to permittivity.

Ф_electric = Q/є₀

Where ‘Q’ corresponds to the entire electrical charge inside the closed surface

‘є₀’ corresponds to the electric constant factor

This is the fundamental gauss law formula.

Gauss Law Derivation

Gauss law is considered as the related concept of Coulomb’s law which permits the evaluation of the electric field of multiple configurations. This law correlates the electric field lines that create space across the surface which encloses the electric charge ‘Q’ internal to the surface. Let us assume that the Gauss law as in the right of Coulomb’s law where it is represented as follows:

E = (1/(4∏є₀)). (Q/r²)

Where EA = Q/є₀

In the above Gauss law mathematical expression, ‘A’ corresponds to the net area which encloses the electric charge that is 4∏ r². Gauss law is more applicable and functions when the electric charge lines are aligned in a perpendicular position to the surface, where ‘Q’ corresponds to the electric charge internal to the enclosed surface.

When some portion of the surface is not aligned at the right-angled position to the closed surface, then a factor of cosϴ will get combined which moves to null when the electric field lines are in a parallel position to the surface. Here, the term enclosed signifies that the surface should be free from any kind of gaps or holes. The term ‘EA’ represents electric flux which can be related to the total electric lines that are apart from the surface. The above concept explains the gauss law derivation.

As Gauss law is applicable for many situations, it is mainly beneficial is doing hand calculations when there exists increased symmetry levels in the electric field. These instances include of cylindrical symmetry and of spherical symmetry. The Gauss law SI unit is newton meters squared per each coulomb which is N m² C^-1.

Gauss Law in Dielectrics

For a dielectric substance, the electrostatic field is varied because of the polarization as it differs in vacuum also. So, the gauss law is represented as

∇E = ρ/є₀

This is applicable even in the vacuum and is reconsidered for the dielectric substance. This can be portrayed in two approaches and those are differential and integral forms.

Gauss Law for Magnetostatics

The basic concept of magnetic fields where it gets varied from the electric fields is the field lines that produce the surrounded loops. The magnet will not be observed as half to separate the south and the north poles.

The other approach is that in the view of magnetic fields, it seems to be simple to observe that the total magnetic flux which passes through the enclosed (Gaussian) surface is null. The thing that moves internally to the surface needs to become out. This states the Gauss law for magnetostatics where it can be represented as

ʃB.dS = 0 = µʃHds cosϴ = 0

This is also termed as the principle of magnetic flux conservation.

µcosϴʃI =0 which implies that ʃI = 0

So, the net sum of the currents moving into the enclosed surface is null.

Importance

This section gives a clear explanation of the significance of Gauss law.

Gauss’s law statement is correct for any type of closed surface without having a dependency on the size or shape of the object.

The term ‘Q’ in the basic formula of the law consists of the consolidation of all the charges those are completely enclosed no matter of any position internal to the surface.

In the case, the selected surface there exists both the internal and external charges of the electric field (where the flux is present on the left position is because of the electrical charges in both the in and out of the ‘S’).

Whereas the factor ‘q’ on the right position of the Gauss law signifies that the complete electrical charge internal to the ‘S’.

The selected surface for the functionality of Gauss law is termed as Gaussian surface, but this surface should not be passed through any kind of isolated charges. This is due to the reason that isolated charges are not exactly defined in the electric charge position. When you reach nearer to the electrical charge, the field enhances without any boundary. While the Gaussian surface goes through the continuous charge allocation.

Gauss law is mainly employed for a more simplified analysis of the electrostatic field in the scenario that the system holds some equilibrium. This is only accelerated only by the selection of an appropriate Gaussian surface.

On the whole, this law is dependent on the inverse square based on the location that is in Coulomb’s law. Any kind of breach in the Gauss law will signify the deviation of the inverse law.

Examples

Let us consider a few gauss law examples:

1). An enclosed gaussian surface in the 3D space where the electrical flux is measured. Provided the gaussian surface is spherical in shape which is enclosed with 30 electrons and has a radius of 0.5 meters.

• Calculate the electric flux that passes through the surface
• Find the electrical flux having a distance of 0.6 meters to the field measured from the center of the surface.
• Know the relation that exists between the enclosed charge and the electric flux.

Answer a.

With the formula of electric flux, the net charge that is enclosed in the surface can be calculated. This can be achieved by charge multiplication for the electron with the entire electrons that appear on the surface. Using this, the free space permittivity and the electric flux can be known.

Ф = Q/є₀ = [30(1.60 * 10-¹⁹)/8.85 * 10^-12]

= 5.42 * 10^-12 Newton*meter/Coulomb

Answer b.

Rearranging the equation of electric flux and expressing the area as per radius can be used to calculate the electric field.

Ф = EA = 5.42 * 10^-12 Newton*meter/Coulomb

E = (5.42 * 10^–)/A

= (5.42 * 10^–)/4∏(0.6)²

As the electric flux has a direct proportion with the enclosed electric charge, this signifies that when the electric charge on the surface enhances, then the flux which passes through it also will be enhanced.

2). Consider a sphere having a radius of 0.12 meters that has similar charge distribution on the surface. This sphere holds an electric field placed at a distance of 0.20 meters which has a value of -10 Newtons/Coulomb. Calculate the

• Calculate the amount of electrical charge that is disseminated on the sphere?
• Define why or why not the electrical field which is internal to the sphere is null?

Answer a.

To know Q, the formula we use here is

E = Q/(4∏r²є₀E)

With this Q = 4∏(0.20)²(8.85 * 10^-12)(-100)

Q = 4.45 * 10^-10C

Answer b.

In the empty spherical space, there exists no electric charge internally having total charge living at the surface. As there is no internal charge, the electrical field which is internal to the sphere is also null.

Applications of Gauss Law

Few of the applications where this law is used are as explained as below:

• The electric field in between the two parallelly placed condenser plates is E = σ/є0, where ‘σ’ corresponds to the density of surface charge.
• The electrical field intensity which is placed near the plane sheet having charge is E = σ/2є₀K and σ corresponds density of the surface charge
• The electrical field intensity which is placed near the conductor is E = σ/є₀K and σ corresponds density of the surface charge, when the medium is chosen as dielectric then E_air= σ/є₀
• In the scenario of having an infinite electrical charge placed at a distance of radius ‘r’, then E = ƴ/2∏rє₀

To select the Gaussian surface, we need to consider the states where the proportion of dielectric constant and the electrical charge is provided by a 2d surface that is integral than the charge distribution’s electric field symmetry. Here, comes the three various situations:

• In the case when the charge allocation is in the shape of cylindrically symmetric
• In the case when the charge allocation is in the shape of spherically symmetric
• The other scenario is that the charge allocation has translational symmetry all through the plane

The gaussian surface size is selected based on the condition of whether we need to measure the field. This theorem is more useful in knowing the field when there exists corresponding symmetry because it addresses the direction of the field.

And this is all about the concept of Gauss Law. Here, we have gone through a detailed analysis of knowing what Gauss law is, its examples, significance, theory, formula, and applications. In addition, one is more recommended to also know about the advantages of Gauss law and disadvantages of gauss law, its diagram, and others.