2. (10 points) A coaxial cable is made from an inner wire of radius a, perfectly

Question

2. (10 points) A coaxial cable is made from an inner wire of radius a, perfectly conducting, straight and inlinitely long surrounded by an outer hollow cylinder, also made from perfectly conducting material, having inner radius b and outer radius e. The hollow cylinder functions as a return wire, and is coaxial with the inner wire. If the space between the wires is air or vacuum, the electric field between the inner and return wires is shown to be * where is the charge per unit length and the cylindrical distance between the wires. Suppose the E field cannot be allowed to reach values |E above the value it reaches in the middle of the arrangement where t. This can be achieved by shielding the inner cable with a dielectric of permittivity e. a What is the mininuin value of and the minimum thickness of dielectric coating that will ensure Lhat the fell is everywhere below its maximal value? b) What is the capacitance per unit length of the resulting arrangement

Explanation / Answer

In this solution, permitivity will be denoted by e :

(a) The dielectric constant is k = e/e0 , where e is the permitivity of the dielectric medium and e0 permitivity of free space, since the dielectric constant for the space between the coaxial filled is 1.00059, therefore,

k = e/e0

1.00059 = e/8.854 x 10-12 farads per meter

e = 1.00059 x 8.854 x 10-12 = 8.85922386 x 10-12 farads per meter, since 1 farad = 1 coulomb , therefore, e = 8.85 x 10-12 coulomb per meter.  

The thickness of the dielectric coating, d = E/k ,

d = L / 2pi e0 p k , where p is the cylindrical distance between the wires and L is the charge per unit length.  

(b) Capacitance, C = ke0 A/d for the given arrangement , therefore, capacitance per unit length

C/l = ke0 A/l .d .

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