Hi there!
a)
We can begin by using the equation for energy density.
U = Energy (J)
ε₀ = permittivity of free space
E = electric field (V/m)
First, derive the equation for the electric field using Gauss's Law:
Creating a Gaussian surface being the lateral surface area of a cylinder:
Now, we can calculate the energy density using the equation:
Plug in the expression for the electric field and solve.
b)
Now, we can integrate over the volume with respect to the radius.
Recall:
Now, we can take the integral of the above expression. Let: = inner cylinder radius
= outer cylindrical shell inner radius
Total energy-field energy:
Plug in the equation for the electric field energy density and solve.
Bring constants in front and integrate. Recall the following integration rule:
Now, we can solve!
To find the total electric field energy per unit length, we can simply divide by the length, 'L'.
And here's our equation!