Q1. A uniformly charged solid cylinder of radius a and length L carries a total charge Q. Assuming the observation point lies at a distance x from the centre of the cylinder along its axis and that L >> a, derive an expression for the electric field on the axis of the cylinder. Discuss the limiting case when x >> L.
- Electric field on cylinder axis is found by integrating fields of elemental charged disks.
- Volume charge density ρ = Q / (πa²L) defines charge for each infinitesimal disk element.
- Field contribution from elemental disk dE = (ρ dy / 2ε₀) [1 - (x - y) / √((x - y)² + a²)].
- Total field E is obtained by integrating dE from -L/2 to L/2 along the cylinder's axis.
Answer: The electric field on the axis of a uniformly charged solid cylinder can be determined by integrating the electric field contributions from infinitesimally thin charged disks that constitute the cylinder. Let the cylinder have a total charge Q, radius 'a', and length 'L'. The uniform volume charge density is given by ρ = Q / (πa²L). Consider an elemental disk of thickness dy located at a distance 'y' from the cylinder's center along its axis. The charge of this elemental disk is dQ = ρ * (πa²) ...