Electric energy density in the dielectric of a coaxial cable

[adamaero]

Homework Statement

A co-axial cable has an inner conductor of radius r_i = 0.00045 m and a thin outer conductor or radius r_o = 0.0018 m. The dielectric between the conductors has a relative permittivity εr = 2.25. The dielectric's radius is 0.00148 m.

Find the electric energy density in the dielectric.

Relevant Equations

V(ρ) = V_o*ln(ρ/ρ_o)/ln(ρ_i/p_o)

Work = 0.5∫∫∫D•E dv [J]
Work = 0.5D•E = 0.5*p_v*V

Maybe relevant:
-∇V = E

V(ρ) = V_o*ln(ρ/0.0018)/ln(45/180)

(Attached picture is where the unit vector of r is really ρ.)
In cylindrical coordinates
∇V = ρ*dV/dρ + 0 + 0
∇V =derivative[V_o*ln(ρ/0.0018)/1.386]dρ
∇V = V_o*0.0018/(1.386*ρ)
E = V_o*0.0012987/ρ

Work = 0.5∫∫∫εE•E dv
Bounds: 0.0018 to 0.00045 m

D = εE = 2.25*8.854e-12*E

Work = 0.5*2.25*8.854e-12*0.0012987^2*∫(V_o^2/ρ^2)dρ

...or can I just stop, before the integral, and do 0.5*D•E:
0.5*εE•E
0.5*2.25*8.854e-12*0.0012987^2
= 1.68e-17 J/m^3

 

[rude man]

Homework Statement: A co-axial cable has an inner conductor of radius r_i = 0.00045 m and a thin outer conductor or radius r_o = 0.0018 m. The dielectric between the conductors has a relative permittivity εr = 2.25. The dielectric's radius is 0.00148 m.

Find the electric energy density in the dielectric.

The problem is poorly phrased.
1. I assume the applied voltage is V_o?
2. I assume by "energy density" is meant the linear energy density, i.e. energy per unit length of cable?
If so, you might proceed by finding the charge q per unit length on what you will recognize is a cylindrical capacitor.
To do this, consider -V_o = $\int_a^c E \, dr$
and $\iint_S D \cdot dA$ = q.
That would give you E(r) and D(r) so you could volume-integrate 1/2 E(r)D(r) over the distance a to b and again b to c, the differential volume being $2\pi r dr$ per unit length.

 

[rude man]

Alternatively, if you previously solved for, or otherwise obtained, the capacitance of a cylindrical capacitor, it would be easier to compute the total capacitance per unit length, then use the familiar expression relating energy to C and V.