An Expression for the Linear Charge Density of an Infinitely Long Cylinder

Linear Charge Density of an Infinitely Long Cylinder

An infinitely long cylinder carries a charge density throughout its volume given by ρ(r) = ρ0 (1 - r/R) where r is the radial distance from its central axis and ρ0, R, and λ are positive constants. The radius of the cylinder is R beyond which the charge density is zero. To obtain an expression for the linear charge density of the cylinder, we need to integrate the charge density over its volume.

Final Answer:

The linear charge density of an infinitely long cylinder with volume charge density ρ(r) = ρ0 (1 - r/R) is obtained by integrating the charge density over the volume of the cylinder. The expression for the linear charge density can be given as λ = ∫ ρ0 (1 - r/R) r dr dφ dz over appropriate limits.

Explanation:

The problem involves an application of Gauss' Law and integration within the field of electrostatics in Physics. The charge density is given by ρ(r) = ρ0 (1 - r/R) for r ≤ R and zero otherwise. The total linear charge density (λ) of the cylinder would be obtained by integrating the charge density over the volume of the cylinder. It can be expressed as:

λ = ∫ρ(r) dV, where dV represents a differential volume element inside the cylinder. Assuming the cylinder axis to be along the z-axis, the volume element can be expressed in cylindrical coordinate as dV = r dr dφ dz. Therefore, integrating ρ(r) over the volume of the cylinder, λ = ∫ ρ0 (1 - r/R) r dr dφ dz, over the limits 0 to R (for r), 0 to 2π (for φ), and -∞ to ∞ (for z), gives the total charge per unit length λ on the cylinder.