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What about the circles on the end of the cylinder? I’ve seen some examples where the value for E only depends on the radius of the cylinder. It doesn’t make sense for a point on the center of one of the circular ends of a cylinder to have the same electric field as the top for a situation like a long rod.Orodruin said:On the cylinder surface. Note that you have an integral, which depends on the value of the integrand at all points on the integration domain.
But the value of E solved for with Gauss’s law in some examples I’ve seen was some constants multiplied by r. If E represented the electric field at on the surface, and was only dependent on r, then wouldn’t the electric field have to be the same on every point on a cylinder of a specified radius, including the ends?Orodruin said:Of course it doesn't, the electric field is generally not constant on the surface.
In order to do this you need to be able to argue that the electric field component in the normal direction of the surface is constant on the surface due to symmetries. It is not a generally applicable method. The symmetries have to do with the source and not anything to do with the surface itself. In many cases you can choose a surface cleverly such that you can utilise symmetry arguments.FS98 said:But the value of E solved for with Gauss’s law in some examples I’ve seen was some constants multiplied by r.
Consider the significance of the dot product ##\mathbf{E} \cdot d \mathbf{A}##, particularly in the special cases where ##\mathbf{E}## is perpendicular and parallel to the surface at the location of ##d \mathbf{A}##. Which case applies at the ends of the cylindrical Gaussian surface, in the examples that you're looking at?FS98 said:including the ends
The example I’m thinking of is from this video.jtbell said:Consider the significance of the dot product ##\mathbf{E} \cdot d \mathbf{A}##, particularly in the special cases where ##\mathbf{E}## is perpendicular and parallel to the surface at the location of ##d \mathbf{A}##. Which case applies at the ends of the cylindrical Gaussian surface, in the examples that you're looking at?
It would help if you describe a specific example that confuses you, or point us to a page that describes it, preferably with a diagram. Otherwise we have to guess, and we might guess the wrong example, which would lead to confusion.
It does apply. But the contribution to the integral is zero due to the field being radial and therefore parallel to the surface. The reason you do a cylinder in the first place is the symmetry of the problem. You can argue that the field is radial everywhere and therefore ##\vec E \cdot d\vec S = E\, dS## on the mantle with ##E## constant. The contribution from the mantle is therefore ##EA##, where ##A## is the mantle area. The contribution to the integral from the end caps is zero because on the end caps ##\vec E \cdot d\vec S = 0##. The total integral is therefore ##EA##, which needs to be compared with the enclosed charge.FS98 said:I understand that in the example the flux on the ends is 0, but I don’t see how that would imply that the electric field calculated doesn’t apply at those points.
...and perpendicular to ##d\mathbf{A}##, which is what matters for the dot product ##\mathbf{E} \cdot d\mathbf{A}##. Note the diagrams in these examples. They use ##\Delta \mathbf{A}## instead of ##d\mathbf{A}##, but the idea is the same.Orodruin said:But the contribution to the integral [on the end caps] is zero due to the field being radial and therefore parallel to the surface.
I think my problem is the math. I’m familiar with calc 1 integrals but the math here seems to be different than what I’m used to.Orodruin said:It does apply. But the contribution to the integral is zero due to the field being radial and therefore parallel to the surface.
Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the total charge enclosed by that surface. It is named after the German mathematician and physicist Carl Friedrich Gauss.
Gauss's Law is used to calculate the electric field at a point in space due to a distribution of charges. It allows us to simplify complicated calculations by using symmetry and choosing an appropriate Gaussian surface.
The constant "E" in Gauss's Law represents the electric permittivity of free space, which is a measure of how easily electric fields can pass through a given medium. It is a fundamental constant in electromagnetism and has a value of approximately 8.85 x 10^-12 F/m.
Gauss's Law can be applied to any charge distribution, as long as the charge is enclosed by the chosen Gaussian surface. However, it is most useful for calculating the electric field for symmetrical charge distributions, such as a point charge, line of charge, or charged sphere.
Gauss's Law has many practical applications, such as in the design of capacitors, the calculation of the electric field inside a conductor, and the analysis of electric fields in lightning rods. It is also used in various technologies, including particle accelerators, MRI machines, and electronic devices.