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PPT No. 3 * Divergence Theorem * Gauss’s law- Integral form * Gauss’s law- Differential Form Gauss’ Law Gauss’s Law- Integral Form Statement of the Gauss’s Law Electric Flux through any closed surface enclosing a charge QS is proportional to the enclosed electric charge Gauss’ Law Quantitatively Gauss’s Law is given by for Charge QS inside the surface =0 for Charges outside the surface QS is the total charge enclosed by the surface (including both Free and Bound charge) and ε0 is the permittivity of free space or Electric constant Gauss’ Law LHS is a surface integral of E-Field through a closed surface S denoting the electric flux ΦE RHS is the total charge Qs enclosed by S divided by the Electric constant ε0 S is called as Gaussian surface Only the charge enclosed within a volume defined by a closed surface contributes to the net electric flux through the surface Gauss’ Law- Significance Gauss's law relates the distribution of electric charge Qs to the resulting electric field E. Gauss Law is One of the fundamental laws of physics Gauss’ Law is One of the four Maxwell’s Equations which form the basis of Classical Electrodynamics Gauss's law can be used to derive Coulomb’s Law & Coulomb’s Law can be used to derive Gauss' Law Gauss’ Law- Applications Gauss's law is very useful to find the distribution of electric charge when the electric field E is known By integrating the electric field, the flux can be found and the charge distribution in the region can be deduced by using Gauss Law. Gauss's law can be employed in the reverse problem also (i.e. The electric charge distribution is known and the electric field needs to be computed) under certain conditions Types of Charge Distributions •3-D Charge distributions are specified in terms of • Volume charge density ρ defined as •Total Charge / Volume •2-D Charge distributions are specified in •Surface charge density σ defined as •Total Charge / Area •1-D Charge distributions are specified in terms of • Linear charge density λ defined as •Total Charge / Length Gauss’s Law If the electric charge distribution has some kind symmetry, then the electric field E passes through the surface uniformly and E can be computed using Gauss Law e. g. The Gaussian surface is a hypothetical surface enclosing the charge constucted to calculate the flux or electric field Selection of the Gaussian surface Selection of the proper Gaussian surface is the Key to solve the problems using Gauss Law Usually the shape of the Gaussian surface is such that it mimics the symmetry of the charge distribution The Gaussian surface is constructed so that the electric flux density is normal to the surface and it has a constant value on the surface. Selection of the Gaussian surface Charge Distributions having Spherical Symmetry are * A point charge A charged sphere Charged spherical shell Their Gaussian Surface is Concentric sphere with centre at the charge Selection of the Gaussian surface Charge Distributions having Cylindrical Symmetry are Long straight line of charge Charged infinite cylinder Their Gaussian Surface is A Concentric cylinder with axis along the line of charge Selection of the Gaussian surface Charge Distributions having Planar Symmetry are Infinite layer of charge Infinite plane sheet Their Gaussian Surface is A Cylinder with sides ┴ to sheet i.e. Faces ║ to sheet Selection of the Gaussian surface A coordinate system is selected that most nearly matches the symmetry of the charge distribution Spherical co-ordinate system is used to determine the flux due to a point charge as its flux has spherical symmetry Cartesian co-ordinate system is used to determine the flux due to an Infinite plane sheet of charge as charge-sheet along x-y plane has E-field along z-axis which is independent of x or y axis & symmetric under reflection about the x-y plane due to its planar symmetry Gauss' Law If symmetry of any kind is not present in electric charge distribution, then it is difficult to compute electric field E from the given distribution of electric charge Q using Gauss’s law in Integral form In that case the differential form of Gauss's law is useful Gauss’ Law- Differential Form According to the integral form of Gauss's law, where charge Q is in terms of Volume charge density ρ, dτ is an infinitesimal element of volume. Using the Divergence theorem LHS is a volume integral over the volume V, RHS is the surface integral over the boundary of volume V. Gauss’ Law- Differential Form (contd) where · E is the Divergence of the electric field ρ is the Volume Charge density ε0 is the electric constant. This is the differential form of Gauss's law Equivalence of Two Forms of Gauss's law Integral form of Gauss's law Differential form of Gauss's law Equivalence of Two Forms of Gauss's law The Integral and Differential forms of Gauss's law are equivalent and are related by the Divergence theorem.