- #1
NullSpace0
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I recently had a problem set with two questions that seemed to give very similar answers. I'm not asking how to do this, so I don't think this post belongs in the homework section. Rather, I'm asking if the similarity I think I see has any deeper meaning in the physics of electric fields.
Let's say I want to find the electric field due to a disk of uniform charge density along the disk's axis. I would integrate and I end up getting something like:
E=∫(2*pi*sigma*z*r*dr)/(r^2+z^2)^3/2... note that z/sqrt(r^2+z^2) comes in from multiplying by the cosine of the angle to get only the portion along the axis. In this integral, z is a constant.
For a hollow cylinder, you get essentially the same integral: E=∫(2*pi*sigma*R*z*dz)/(R^2+z^2)^(3/2)... again, note that R/sqrt(R^2+z^2) come from the cosine of the angle for similar reasons. In this case, R is a constant.
So they seem to be the same integral with R and z swapped out. Other than the fact that sigma is different in each case, does the similarity mean anything? It's almost like it's saying that a cylinder and a circle have basically the same electric field along the axis.
Let's say I want to find the electric field due to a disk of uniform charge density along the disk's axis. I would integrate and I end up getting something like:
E=∫(2*pi*sigma*z*r*dr)/(r^2+z^2)^3/2... note that z/sqrt(r^2+z^2) comes in from multiplying by the cosine of the angle to get only the portion along the axis. In this integral, z is a constant.
For a hollow cylinder, you get essentially the same integral: E=∫(2*pi*sigma*R*z*dz)/(R^2+z^2)^(3/2)... again, note that R/sqrt(R^2+z^2) come from the cosine of the angle for similar reasons. In this case, R is a constant.
So they seem to be the same integral with R and z swapped out. Other than the fact that sigma is different in each case, does the similarity mean anything? It's almost like it's saying that a cylinder and a circle have basically the same electric field along the axis.
