Question

Assume that the volumetric charge density within an infinite cylindrical circular region of radius 'a' is rho_v = rho_0(1 + alpha * r²) * c/(m³), where rho_0 is a constant factor and r is the radial distance from the cylinder axis. Determine the value of the alpha parameter for which the electric field everywhere outside the cylinder (r > a) is zero. For this value of alpha, what is the electric field inside the cylinder (r < a)?

Answer 1

The question pertains to the determination of a parameter to ensure the electric field outside a cylindrical charge distribution is zero and the corresponding field inside it, using Gauss's Law and the integration of charge density.

Step-by-step explanation:

The student is asking about electromagnetism in the context of a cylindrical charge distribution and the determination of the electric field both inside and outside the cylindrical region. Using Gauss's Law, one can show that for the electric field outside the cylinder to be zero, the volume charge density must not result in a net charge enclosed by a Gaussian surface surrounding the cylinder. This result is obtained by setting up an integral of the charge density over the volume of the cylinder and equating it with the electric displacement flux. As for the electric field inside the cylinder, it can be determined using the same principle but applying the integral only up to the radial distance r less than the radius 'a'. The parameter α affects how the charge density varies with r, and hence the electric field distribution.