The inner cylinder of a long, cylindrical capacitor has radius r and linear charge density +λ. It is surrounded by a coaxial cylindrical conducting shell with inner radius r and linear charge density -λ. (a) What is the energy density in the region between the conductors at a distance r from the axis?b) Integrate the energy density calculated in part (a) over the volume between the conductors in a length L of the capacitor to obtain the total electric-field energy per unit length

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!

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