A long cylinder is composed of two parts. The core has a radius R 1 and charge d

Question

A long cylinder is composed of two parts. The core has a radius R 1 and charge distribution that varies linearly with the radius from zero at the center so that the total charge per unit length is Lamda.

Outside the core is a conductor of inner radius of R 1 and an outer radius of R 2 .

As a function of r, the distance from the center of the cylinder:

(a) Find the magnitude of the electric field E(r) outside of the cylinder (r > R 2 ).

(b) Find the surface charge density, sigma1 , at R 1 .

(c) Find the surface charge density, sigma2 , at R 2 .

(d) Find the varying volume charge density Q(r) , for r < R 1

(e) Find the magnitude of the electric field E(r) inside the core of the cylinder (r < R 1 ).

Please show work

Explanation / Answer

(a) Find the magnitude of the electric field E(r) outside of the cylinder (r > R 2 ).

Gauss's law: E = Lamda/(2 pi r epsilon0)

(b) Find the surface charge density, sigma1 , at R 1 .

inside the conductor the electrical field is zero, therefore total charge inside a guassian surface must be zero:

sigma_1 = -Lamda/(2 pi R1)

(c) Find the surface charge density, sigma2 , at R 2 .

sigma_2 = Lamda/(2 pi R2)

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