A coaxial cable, see the figure, consists of an inner cylindrical conducting wir

Question

A coaxial cable, see the figure, consists of an inner cylindrical conducting wire of radius R_b surrounded by a dielectric insulator.(Figure 1) Surrounding the dielectric insulator is an outer conducting sheath of radius R_a, which is usually "grounded." Determine an expression for the capacitance per unit length of a cable whose insulator has dielectric constant K. Express your answer in terms of the variables R_a, R_b, K, and appropriate constants. For a given cable, R_b = 2.5 mm and R_a = 9.0 mm. The dielectric constant of the dielectric insulator is K = 2.6. Suppose that there is a potential of 1.0 kV between the inner conducting wire and the outer conducting sheath. Find the capacitance per meter of the cable. Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

a) C = kq/V (The dielectric constant k)
C/L = kq/(LV)

Electric field at a given radius via Gauss' Law:
E(2rL) = q/(o)
E = q/[2rL(o)]

V = int[E dr]
V = int{q/[2rL(o)]} dr
V = q ln(R) / [2L(o)]

Evaluate from outer to inner radius:
V = q ln[(Ra)/(Rb)] / [2L(o)]
q/(LV) = 2(o) / ln[(Ra)/(Rb)]

C/L = 2k(o) / ln[(Ra)/(Rb)]

b) C/L = 2(2.6)(8.85e-12 F/m) / ln[(9 mm)/(2.5 mm)]
C/L = 1.13e-10 F/m

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