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!

r=rho l/a where a is the area, l is the length and rho is resistivity so the greater the area the lesser the resistance thats why the earth pin is thicker bcos current follow the shortest path, so less resistive .

the answer is a. at the equator