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Have you read the chapter in Sommerfeld's book? What's the specific problem with it?
the specific problem is that I'm too stupid hahavanhees71 said:Have you read the chapter in Sommerfeld's book? What's the specific problem with it?
tade said:since the charge density is zero, there could be tangential components
tade said:for the field normals, is it possible that they are non-zero? but some stick in and others stick out such that the net amount is zero
what's the mathematical principle, because technically such fields won't affect equilibriumPeterDonis said:Not at the surface of a conductor. The tangential electric field there must be zero regardless of whether there is any charge density present.
do you know of more documentation on this theorem, thanksPeterDonis said:Not over the entire sphere at the inside surface of the conductor, because any field with the property you describe would have to have a component tangent to the sphere at at least one point (actually I think it would have to along at least one closed curve on the sphere), and we know there cannot be any nonzero tangential component anywhere on a sphere at the inside surface of the conductor.
tade said:what's the mathematical principle
tade said:do you know of more documentation on this theorem
however, tangential fields at regions of zero charge density would not affect equilibriumPeterDonis said:Physically, the reason for this definition is that if there were any tangential electric field at the surface of a conductor, charges would move freely on the surface of the conductor to neutralize the field. So the more precise way to specify this property of a conductor is that in equilibrium there cannot be any tangential electric field at the surface of a conductor. Which for this problem amounts to the same thing.
PeterDonis said:It's an application of the mean value theorem. For the surface integral of a vector field over a sphere to be zero, the normal component of the field must be negative (pointing inward) on one part of the sphere and positive (pointing outward) on another part (this is basically what you described). But if the vector field is continuous (which the electric field must be), then there must be some set of points on the sphere where the normal component is zero, in between the negative points and the positive points. I believe, as I said, that that set of points must be a closed curve on the sphere, separating the positive region from the negative region.
Now, at any of the points where the normal component of the field is zero, either (1) the field as a whole must be zero, or (2) the field must have a nonzero tangential component. But if the field as a whole were zero at any point on the sphere (which is the only way to avoid having a nonzero tangential component), it would have to be zero everywhere on the sphere, and thus zero everywhere inside the sphere.
tade said:tangential fields at regions of zero charge density would not affect equilibrium
tade said:if there must be a point where both normal and tangent are zero, all normals everywhere have to be zero as well
PeterDonis said:You keep stating this as if it's relevant. It's not. We're not talking about general "regions of zero charge density". We're talking specifically about the surface of a conductor.
nice, yeah, describing all the necessary conditionsPeterDonis said:No, if there must be a point on a sphere over which the surface integral is zero, where both normal and tangent are zero, the field as a whole everywhere on and inside the sphere has to be zero.
tade said:in conductors, charges can flow, and tangent fields make them do that
tade said:do you know of the info or its name
honestly, i don't think i have ignored any point you have posted, at least, i have not intended to, i have tried to assess all the points thoroughlyPeterDonis said:And in equilibrium, all such charge flow will already have taken place in order to neutralize the tangent fields. So in equilibrium, there must be zero tangent field at the surface of the conductor. And you specified you are looking at equilibrium.
How many times are we going to have to say this before you understand it?
no worriesPeterDonis said:Unfortunately no, I don't. I have tried a little bit of searching online but haven't found any useful reference.
tade said:tangent fields where there are charge densities will produce flow