3. A coaxial cable made of a solid cylindrical conducting wire or radius ? surro

Question

3. A coaxial cable made of a solid cylindrical conducting wire or radius ? surrounded by a coaxial conducting sheath of radius b.

a. Find the capacitance per unit length of a coaxial cable, assuming that there is a vacuum between the central wire and the sheath.

b. Consider that the cable has an equal but opposite charges per unit length ? on the two elements of the cable, which therefore acts as a capacitor. A plug of material of dielectric constant is inserted between the wire and the sheath to a depth of from the end. What is the change in potential energy of the charged cable? What electric force acts on the plug as it is inserted?

Please answer with clear and legible step-by-step solutions and explanations(So I can understand). Take your time, Thanks!

Explanation / Answer

condisder a cylinder of length L and radius r

so charge q for this is given by q = EeoA = eo E (2pirL)

so solving for E

E = q/(2pieorL)

but eelctric potneatil V = integral E.ds   ( limits from a to b)

so V = (q/2pieoL) int (dr/r) from a to b

V = q/(2pieoL){ln (b/a)

now appli it for capaciatcne C

as C = Q/V

C = q/(q/(2pieoL){ln (b/a)

C = 2pieo L/ln(b/a)-------answer

b. insertion of didlectric constant enchnaces the net capaciatcne by a factor of K, dieelctric consatnt

so

Cnet = Kco

also eenrgy U = 0.5CV^2   without dieelctric aqnd

U = 0.5 K CV^2

so change of energy = (K-1) the initial energy ( that is with out didelctric)

U = 0.5 * 2pieo L/ln(b/a)* V^2

force F = U/L = 0.5 * 2pieo* V^2/ln(b/a)

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