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A long metal cylinder with radius a is coaxial with, and entirely inside, an equally long metal tube with internal radius 2a and external radius 3a. The space between the cylinder and the tube is filled with an LIH dielectric material with relative permittivity r. This space is also permeated by a uniform free charge distribution with density pr. Outside the tube, the charge density is zero.

(a) Write down Poisson's equation for the electrostatic potential in the dielectric material in the region between the cylinder and the tube. (b)Find the general solution of Poisson's equation in the dielectric material. You should justify the choice of the coordinate system that you use.

(c)Using appropriate boundary conditions, show that the potential within the dielectric at distance r from the axis of symmetry is given by

(d) Determine the electrostatic field E in the dielectric material and Obtain an expression for the free charge per unit length, A, on the inner cylinder.'

(a) Write down Poisson's equation for the electrostatic potential in the dielectric material in the region between the cylinder and the tube. (b)Find the general solution of Poisson's equation in the dielectric material. You should justify the choice of the coordinate system that you use.

(c)Using appropriate boundary conditions, show that the potential within the dielectric at distance r from the axis of symmetry is given by

(d) Determine the electrostatic field E in the dielectric material and Obtain an expression for the free charge per unit length, A, on the inner cylinder.'

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