- #1
wbrigg
- 16
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hey,
i have a question from an exam paper which isn't worded too nicely (most of the questions on the exam are worded in similar ways )
[tex]U = \frac{n dq^{2}}{4 \pi \epsilon _{0} R}[/tex]
where n is the number of shells already added.
so the total Electrostatic energy of the whole thing (with all the shells assembled) is going to be:
[tex]U=\Sigma^{n}_{i=1} \frac{i dq^{2}}{4 \pi \epsilon_{0} R}[/tex] Where [tex]n dq = Q[/tex]
I can turn that into an integral (Replacing one of the dqs with [tex]\frac{q}{dq}[/tex]) ;
[tex]U = \int ^{Q}_{0} \frac{q dq}{4 \pi \epsilon_{0} R} dq[/tex]
This goes to [tex]U=\frac{Q^{2}}{8 \pi \epsilon_{0} R}[/tex]
now i make it so it's shells rather than points;
[tex]U=\int^{2\pi}_{0}\int^{\pi}_{0}\frac{Q^{2}}{8 \pi \epsilon_{0} R} d\theta d\varphi[/tex]
[tex]U=\frac{\pi Q^{2}}{4 \epsilon_0 R}[/tex]Could someone confirm whether or not this is correct please?
i have a question from an exam paper which isn't worded too nicely (most of the questions on the exam are worded in similar ways )
The way I've done it is to first put in my first shell of infintesimal charge and then treat it as a point charge. Then i treat the next shell as just a point, and place it at a distance R from the point (electrostatic energy of this;Determine the electrostatic energy U of a thin spherical shell of radius R which carries chare Q uniformly distributed over it's surface [Hint: imagine to assemble the sperical shell by superimposing sheels of radius R and infintesimal charge dq. ].
[tex]U = \frac{n dq^{2}}{4 \pi \epsilon _{0} R}[/tex]
where n is the number of shells already added.
so the total Electrostatic energy of the whole thing (with all the shells assembled) is going to be:
[tex]U=\Sigma^{n}_{i=1} \frac{i dq^{2}}{4 \pi \epsilon_{0} R}[/tex] Where [tex]n dq = Q[/tex]
I can turn that into an integral (Replacing one of the dqs with [tex]\frac{q}{dq}[/tex]) ;
[tex]U = \int ^{Q}_{0} \frac{q dq}{4 \pi \epsilon_{0} R} dq[/tex]
This goes to [tex]U=\frac{Q^{2}}{8 \pi \epsilon_{0} R}[/tex]
now i make it so it's shells rather than points;
[tex]U=\int^{2\pi}_{0}\int^{\pi}_{0}\frac{Q^{2}}{8 \pi \epsilon_{0} R} d\theta d\varphi[/tex]
[tex]U=\frac{\pi Q^{2}}{4 \epsilon_0 R}[/tex]Could someone confirm whether or not this is correct please?
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