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## 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

*ε*0 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?