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r16
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I was thinking about gauss's law and ran into this contradiction.
Consider this situation in electrostatics. You have an infinite line of charge, uniform charge density [text]\lambda[/tex]. In cgs units, E at a distance r from the line of charge along the +y axis (assuming a left handed coordinate system) is [tex]\frac{2 \lambda}{r}[/tex].
Now consider a right circular cylinder in empty space as a gaussian shape. Intuitively, I will tell you that the net flux is 0 through the entire shape: all field lines that flow into the shape flow out.
But thinking about this analytically gives you a different answer:
Flux through top bottom (closest to the charge) = [tex]-\frac{2 \lambda}{r} (\pi r^2)[/tex]
Flux through sides = 0
Flux through top (furthest away from the charge) = [tex]\frac{2 \lambda}{r + l} (\pi r^2)[/tex]
so the net flux is tex]-\frac{2 \lambda}{r} (\pi r^2) + \frac{2 \lambda}{r + l} (\pi r^2)[/tex] which must be less than 0 because [tex] r > r+l [/tex], which would mean that there is a net negative flux, or some sort of negative charge enclosed in the gaussian surface which is contradictory to the intuitive evaluation of the shape.
Any resolution to this contradiction?
Consider this situation in electrostatics. You have an infinite line of charge, uniform charge density [text]\lambda[/tex]. In cgs units, E at a distance r from the line of charge along the +y axis (assuming a left handed coordinate system) is [tex]\frac{2 \lambda}{r}[/tex].
Now consider a right circular cylinder in empty space as a gaussian shape. Intuitively, I will tell you that the net flux is 0 through the entire shape: all field lines that flow into the shape flow out.
But thinking about this analytically gives you a different answer:
Flux through top bottom (closest to the charge) = [tex]-\frac{2 \lambda}{r} (\pi r^2)[/tex]
Flux through sides = 0
Flux through top (furthest away from the charge) = [tex]\frac{2 \lambda}{r + l} (\pi r^2)[/tex]
so the net flux is tex]-\frac{2 \lambda}{r} (\pi r^2) + \frac{2 \lambda}{r + l} (\pi r^2)[/tex] which must be less than 0 because [tex] r > r+l [/tex], which would mean that there is a net negative flux, or some sort of negative charge enclosed in the gaussian surface which is contradictory to the intuitive evaluation of the shape.
Any resolution to this contradiction?