Electrostatic Potential Energy-related.

[Tarabas]

Homework Statement

E=(1/4πε0)(Q/r^2) for R<r<2R

Homework Equations

U= integral (2R,R) ( (ε0 E^2)/2*4πr^2 dr

The Attempt at a Solution

I have no idea where the U-formula comes from. Any help would be appreciated.
I added some pictures so that it could be easier to understand.

 

[collinsmark]

Hello Tarabas,

You might want to look through your textbook in the parts that talk about the energy stored in an electrostatic field. By that I mean the total energy it takes to create a given electrostatic field in the first place.

To point you in the right direction, the energy density (energy per unit volume) of an electrostatic field is

$$\frac{dU}{dV} = \frac{1}{2} \varepsilon_0 |E|^2$$

where $dU$ is the differential potential energy (the potential energy of the space enclosed within the differential volume), $dV$ here refers to the differential volume (where '$V$' here stands for volume, not to be confused with potential or voltage) and $E$ is the magnitude of the electric field at that point in space. Really though, you should check your textbook because it's likely there are at least a few pages dedicated to this idea.

Now do you see how the answer you posted in the image is integrating the energy density over the specified volume? [Edit: which gives you the energy stored in that region of space between $R$ and $2R$]

 

[Tarabas]

Hello Tarabas,

You might want to look through your textbook in the parts that talk about the energy stored in an electrostatic field. By that I mean the total energy it takes to create a given electrostatic field in the first place.

To point you in the right direction, the energy density (energy per unit volume) of an electrostatic field is

$$\frac{dU}{dV} = \frac{1}{2} \varepsilon_0 |E|^2$$

where $dU$ is the differential potential energy (the potential energy of the space enclosed within the differential volume), $dV$ here refers to the differential volume (where '$V$' here stands for volume, not to be confused with potential or voltage) and $E$ is the magnitude of the electric field at that point in space. Really though, you should check your textbook because it's likely there are at least a few pages dedicated to this idea.

Now do you see how the answer you posted in the image is integrating the energy density over the specified volume? [Edit: which gives you the energy stored in that region of space between $R$ and $2R$]

Thanks a lot. I actually checked my textbook and it was nowhere. :D