3. A cylindrical capacitor consist on a long wire or radius R1, length L, and ch

Question

3. A cylindrical capacitor consist on a long wire or radius R1, length L, and charge +Q and a thin coaxial shell of radius R2 (R2>R1). length L, and charge -Q. (a) Find the electric field as a function of R from the axis. (b) How much energy resides in the region that has a radius R, thickness dR, and volume 2 RLdR between the wire and shell. (c) Integrate the expression found in (b) to get the total energy between the wire and shell and show that this is the same as the formula UQ2/(20). dR C 21s LA 2tie 21pt 2. lt

Explanation / Answer

3. given

a cylindrical capacitor, long wire of radius R1

length L

charge Q

coaxial shell, radius R2 > R1

Length L

charge -Q

a. electric field as a fuction of r from the axis

for r < R1

E(r) = 0 ( as electric field inside conductors is 0)

for R1 < r < R2

from gauss law

E(r)*2*pi*r*L = Q/epsilon

E(r) = Q/2*pi*r*L*epsilon

for r > R2

from gauss law

E(r) = 0

b. between wire and shell

at radius R

E(R) = Q/2*pi*R*L*epsilon

dV = 2*pi*R*L*dR

energy

de = 0.5*epsilon*Q^2*2*pi*R*L*dR/4*pi^2*R^2*L^2*epsilon^2

de = Q^2*dR/4*pi*R*L*epsilon

c. total energy = integral de

TE = Q^2*ln(R2/R1)/4*pi*L*epsilon

now, consider

C = 2*pi*L*epsilon/ln(R2/R1)

hence

TE = Q^2/2C

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