- #1
Demon117
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First, this is not a homework question, just something I've been confused about for some time. I understand how to use Guass's law in many ways but one thing I have always stumbled with is whether the E-field of a charge distribution should involve little r or big R in such an example:
In finding the E-field of a long cylindrical charge distribution with radius R and a uniform charge density [itex]\rho[/itex]. If I want to find the E-field inside (r<R) such a distribution shouldn't the final answer contain a big R?
Here is my work. Since we have cylindrical symmetry we can consider a guassian cylinder of radius r inside about the cylindrical axis. The total charge Q is given by [itex]Q=\rho V[/itex] where V is the volume of the entire cylinder correct? (This is where I get confused) Or just the Guassian cylinder?
If the entire cylinder then the volume is [itex]V=\pi R^{2}L[/itex], where L is the length of the cylindrical volume. Then working out the rest of the problem we have:
[itex]\oint EdA=E(2\pi r)L = \frac{Q}{\epsilon_{0}} = \frac{\rho \pi R^{2}L}{\epsilon_{0}}[/itex]
[itex]E = \frac{\rho R^2}{2 \epsilon_{0} r}[/itex]
If the Guassian cylinder then the volume is [itex]V=\pi r^{2}L[/itex], then the E-field is just:
[itex]E = \frac{\rho r}{2 \epsilon_{0}}[/itex]
Do you see my confusion? I need to understand this and I think it is just geometrical issue. . .
In finding the E-field of a long cylindrical charge distribution with radius R and a uniform charge density [itex]\rho[/itex]. If I want to find the E-field inside (r<R) such a distribution shouldn't the final answer contain a big R?
Here is my work. Since we have cylindrical symmetry we can consider a guassian cylinder of radius r inside about the cylindrical axis. The total charge Q is given by [itex]Q=\rho V[/itex] where V is the volume of the entire cylinder correct? (This is where I get confused) Or just the Guassian cylinder?
If the entire cylinder then the volume is [itex]V=\pi R^{2}L[/itex], where L is the length of the cylindrical volume. Then working out the rest of the problem we have:
[itex]\oint EdA=E(2\pi r)L = \frac{Q}{\epsilon_{0}} = \frac{\rho \pi R^{2}L}{\epsilon_{0}}[/itex]
[itex]E = \frac{\rho R^2}{2 \epsilon_{0} r}[/itex]
If the Guassian cylinder then the volume is [itex]V=\pi r^{2}L[/itex], then the E-field is just:
[itex]E = \frac{\rho r}{2 \epsilon_{0}}[/itex]
Do you see my confusion? I need to understand this and I think it is just geometrical issue. . .