Finding the energy density outside of an isolated charged sphere

In summary: Since the sphere is charging and the radius is infinite, Q approaches infinity as r approaches infinity.In summary, the energy density in the electric field near the surface of an isolated metal sphere is 2eVs.
  • #1
Coulombus
2
0

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?
 
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  • #2
Your solution looks OK at the surface of the sphere.
The problem stated "near" the surface of the sphere.
 
  • #3
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}##
 
  • #4
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?
 

1. What is energy density?

Energy density refers to the amount of energy per unit volume in a given space. It is typically measured in units of joules per cubic meter (J/m3).

2. How is energy density calculated?

To calculate energy density outside of an isolated charged sphere, you can use the equation:
energy density = electric field strength x permittivity of free space / 2

3. What is an isolated charged sphere?

An isolated charged sphere is a spherical object that has a net charge and is not influenced by any external electric fields.

4. Why is it important to find the energy density outside of an isolated charged sphere?

Understanding the energy density outside of an isolated charged sphere is important in studying the behavior of electric fields and their effects on surrounding objects. It also has practical applications in fields such as engineering and physics.

5. How does the energy density change as you move further away from the charged sphere?

The energy density decreases as you move further away from the charged sphere. This is because the electric field strength also decreases with distance, according to the inverse square law.

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