Introduction to Gauss law:
Gauss's law represents the relation between an electric charge and the electric field which it sets up. It is an effect of Coulomb's law. However it includes no additional information, its mathematical form let us to resolve numerous problems of electric field computation far more expediently than via the use of Coulomb's law.
Electric field can be explained quantitatively by employing the concept of electric flux. Flux signifies flow. The rate of flow of a fluid (that is, the volume of the fluid crossing an area held perpendicular to the surface per unit time) is termed as the flux of the fluid and is equivalent to vds (that is, v is the velocity of fluid and ds is the small surface).
Analogous to the flux of fluid, the flux of the electric field is termed as the electric flux. Electric flux is proportional to the number of electric field lines penetrating a surface or passing via a virtual surface. Electric flux is stated as the electric field, E, multiplied through the component of area perpendicular to the field.
Assume that an 'E' field whose field lines cut via or pierce a loop. Define 'θ' as the angle between E and the normal or perpendicular direction to the loop. We will now state a new quantity, the electric flux via the loop, as Flux, or Φ = E⊥ A = E A cosθ
E⊥ is the component of E perpendicular to the loop: E⊥ = E cosθ.
The SI unit of Electrical flux is Newton meters squared per coulomb (N m2C-1), or, equivalently, volt meters (V m).
Calculation of Electric Flux: For uniform and non uniform field
Case I: Uniform Field
The figure illustrates field lines passing via a rectangular surface of area 'A' perpendicular to the field lines.
The electric flux passing via this surface is given by the product of electric intensity and the surface area perpendicular to the field lines.
f = EA where 'f' represents the electric flux and 'A' denotes the surface area.
Assume that the surface is not perpendicular to the field lines then the electric flux is provided by the equation:
f = EA cos q
Here 'q' is the angle between the direction of electric field 'E' and the normal drawn to the surface in the outward direction.
If the normal to surface is parallel to the electric field as represented in the figure then the electric flux is:
f = EA Cos q, (q = 0)
f = EA Cos 0
f = EA
The electric flux becomes zero when the normal to the surface is perpendicular to the electric field as represented in the figure.
That is, f = EA Cos q, (q = 90o)
f = EA Cos 90
f = EA x 0
f = 0
Case II: Non-Uniform field
Let us now compute the electric flux passing via a surface if the applied field is not uniform. The surface is generally divided into a large number of small area dA in such a way that the electric field remains constant over that surface as illustrated in the figure.
The electric flux passing via dA(df) = EdA cosq
Total electric flux = s∫E→.dA→
The electric flux per unit area is stated as the electric flux density.
Electric Flux via any closed surface enclosing a charge QS is proportional to the enclosed electric charge. Quantitatively it is represented by:
Φ = s E. dA = QS/εo (for Charge QS within the surface)
Φ = 0 (for Charges outside the surface)
QS is the total charge enclosed through the surface (comprising both free and bound charge), and εo is the permittivity of free space or electric constant. The left-hand side of the equation is a surface integral representing the electric flux via a closed surface S, and the right hand side of the equation is the net charge enclosed through S divided by the electric constant. This relation is termed as Gauss' law or Gauss's flux theorem, for electric field in its integral form. S is termed as Gaussian surface. Gauss's law associates the distribution of electric charge to the resultant electric field. Gauss' law is one of the four Maxwell's equations forming the base of classical electrodynamics.
Gauss's law can be employed to derive Coulomb's law and vice-versa. Gauss's law is much helpful to determine the distribution of electric charge if the electric field is acknowledged. By means of integrating the electric field, the flux can be found out and the charge distribution in the area can be deduced through by employing Gauss Law.
Gauss's law can be used in the reverse problem as well (that is, the electric charge distribution is acknowledged and the electric field requires to be computed) beneath certain conditions. When the electric charge distribution has several kinds of symmetry, then the electric field passes via the surface uniformly and E can be calculated by using Gauss Law example: electric fields due to infinite layer of charge, isolated charged sphere and charged infinite cylinder can be found through considering the Gaussian surfaces having the planar symmetry, spherical symmetry and cylindrical symmetry correspondingly and applying Gauss law.
Though, when symmetry of any type is not present in the electric charge distribution, then it is hard to calculate electric field from the given distribution of electric charge.
The gauss's law can be written in term of divergence theorem as:
∇. E = ρ/ε0
∇. E = electric field divergence
ρ = total electric charge density.
Gauss's Law for Magnetism:
The Gauss Law for Magnetism illustrates that there are no magnetic charges analogous to the electric charges.
The total magnetic flux out of any closed surface is zero.
Φ = B→.dA→ = 0
Application of Gauss's Law:
Gauss's law is applicable to any hypothetical closed surface (termed as a Gaussian surface) and enclosing a charge distribution. Though, the evaluation of the surface integral becomes simple only if the charge distribution has adequate symmetry. In such condition, Gauss's law permits us to compute the electric field far more simply than we could use Coulomb's law. As gauss's law is applicable for an arbitrary closed surface, we will make use of this freedom to select a surface having the similar symmetry as that of the charge distribution to calculate the surface integral.
1) Spherical Symmetry:
A charge distribution is spherically symmetric when the charge density (which is, the charge per unit volume) at any point based only on the distance of the point from a central point (termed as centre of symmetry) and not on the direction.
Figure above points out a spherically symmetric distribution of charge in such a way that the charge density is high at the centre and zero beyond r. Spherical symmetry of charge distribution means that the magnitude of electric field as well based on the distance 'r' from the centre of symmetry.
Φ = (q/4πεor2) x (4πr2) = q/εo
2) Electric Field of a Spherical Charge Distribution:
Charge q is spread out uniformly over the surface of the spherical shell of radius r.
The surface area of the shell is 4πr2 and the charge density is thus q/4πr2. By symmetry the field is perpendicular to the shell at each and every point; therefore the electric flux on any concentric sphere is EA.
For r > R
E = q/4πεor2
For r < R
E = qr/4πεoR3
3) Plane Symmetry:
By using a cylindrical Gaussian surface, one can represent that for a line of charge having a (positive) linear charge density 'λ', the electric field E at a distance 'r' from the points radially outward and consists of magnitude
E = λ/2πεor
E = σ/2εo
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