# Finding the energy density outside of an isolated charged sphere

Coulombus

## Homework Statement

A charged isolated metal sphere of diameter d has a potential V relative to V = 0 at infinity. Calculate the energy density in the electric field near the surface of the sphere. State your answer in terms of the given variables, using ε0 if necessary.

## Homework Equations

Since the chapter's homework is focused on is predominantly focused around capacitance, I believe that the equation for capacitance given by ##C = 4\pi \epsilon_0 R##, where R is the radius of the isolated sphere, will be useful. The core of this problem revolves around the equation for energy density given by $$u = \frac 1 2 \kappa \epsilon_0 E^2$$
Along with the Voltage equation ##V = \frac {Kq} R##
and Electric field magnitude ##E = \frac {Kq} R^2##
In both cases, ##K = \frac 1 {4 \pi \epsilon_0 }##

## The Attempt at a Solution

So my first attempt at finding the energy density involved a lot of solving and replacement of variables.
First, I solved the voltage equation for the charge and got ##q = \frac {RV} K##, then I substituted the result into the Electrical force magnitude equation and got this after simplifying: $$E = \frac V R$$
After substituting in that into the equation for the equation for energy density and replacing in R=d/2
My final equation looks something like this $$u = \frac {2V^2 \epsilon_0} {d^2}$$

That solution got rejected, but I think I'm in the ballpark at least. Any suggestions?

J Hann
Your solution looks OK at the surface of the sphere.
The problem stated "near" the surface of the sphere.

Coulombus
Alright, I've modified my formula to take into account the radius outside the sphere. by replacing placeholder R with ##\frac V {{\frac d 2} + r}## where "r" is the radius approaching infinity. plugging that back into the density equation, I eventually get this $$u = \frac {2 \epsilon_0 V^2} {d^2 + 4dr + 4r^2}$$ but when I tried plugging in that, it was also rejected.

Should I have written it as a limit instead? ##u = \lim_{ r\to\infty} \frac {2 \epsilon_0 V^2} {d^2 + 4dr + 4r^2}##

J Hann
You solved for Q using D/2 and V.
Why can't you just use E = k Q / r^2 where r is the distance from the
center of the sphere to a point external to the sphere?