Jawaharlal Nehru Technological University Hyderabad B.Tech II ELECTROMAGNETIC FIELDS Previous Question Paper (March - 2021 - R15/R13)

Consider an infinite line charge with a uniform linear charge density ρ_L (C/m) placed along the z-axis. To find the electric field intensity E at a point P at a radial distance r from the line charge, we use Gauss's Law in integral form:

Due to the cylindrical symmetry of the infinite line charge, the electric flux density D is directed radially outward in the radial direction (a_r). We choose a cylindrical Gaussian surface of radius r and length L centered on the z-axis. The total flux through the surface consists of flux through the top disk, the bottom disk, and the curved side surface.

Since D is purely radial, the flux through the top and bottom disks is zero because the normal vector to these surfaces is in the a_z direction, making D · dS = 0. The flux through the curved surface is:

The enclosed charge Q_encl within the Gaussian surface of length L is:

Equating the two expressions:

Since D = ε₀E, the electric field intensity is:

Poisson's and Laplace's equations are derived from Gauss's Law for electrostatics in differential form and the relationship between the electric field and the electric potential.

Starting with Gauss's Law:

Substituting D = εE and E = -∇V (where V is the electric potential), we get:

Assuming a homogeneous medium where ε is constant, this simplifies to:

This is Poisson's equation:

In a charge-free region, the volume charge density ρ_v is zero. Substituting ρ_v = 0 into Poisson's equation yields Laplace's equation:

These equations are fundamental in solving electrostatic problems involving potential distributions.