The discussion centers on the derivation of the potential energy formula \( U \) in the context of electrostatic fields, specifically using the equation \( E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2} \) for the region \( R < r < 2R \). The energy density of an electrostatic field is defined as \( \frac{dU}{dV} = \frac{1}{2} \varepsilon_0 |E|^2 \), which leads to the integration of energy density over the specified volume to find the total energy stored in that region. The discussion emphasizes the importance of consulting textbooks for a deeper understanding of energy stored in electrostatic fields.
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[Edit: which gives you the energy stored in that region of space between [itex]R[/itex] and [itex]2R[/itex]]collinsmark said: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
[tex]\frac{dU}{dV} = \frac{1}{2} \varepsilon_0 |E|^2[/tex]
where [itex]dU[/itex] is the differential potential energy (the potential energy of the space enclosed within the differential volume), [itex]dV[/itex] here refers to the differential volume (where '[itex]V[/itex]' here stands for volume, not to be confused with potential or voltage) and [itex]E[/itex] 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?