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(II) A long cylindrical shell of radius $R_{0}$ and length $\ell$ $\left(R_{0}<\ell\right)$ possesses a uniform surface charge density (charge per unit area) $\sigma$ (Fig. 33$)$ . Determine the electric field at points $(a)$ outside the cylinder $\left(R>R_{0}\right)$ and $(b)$ inside the cylinder $\left(0<R<R_{0}\right) ;$ assume the points are far from the ends and not too far from the shell $(R<\ell) .$ (c) Compare to the result for a long line of charge, Example 6 of "Gauss's Law." Neglect the thickness of shell.

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a) $\frac{\sigma R_{0}}{\varepsilon_{0} R},$ radially outwardb) 0c)$\lambda=2 \pi R_{0} \sigma$

Physics 102 Electricity and Magnetism

Chapter 22

Gauss's Law

Electric Charge and Electric Field

Electric Potential

Rutgers, The State University of New Jersey

University of Washington

Simon Fraser University

University of Sheffield

Lectures

13:02

In physics, potential energy is the energy possessed by a body or a system due to its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units (SI) is the joule (J). One joule is the energy expended (or work done) in applying a force of one newton through a distance of one metre (1 newton metre). The term potential energy was introduced by the 19th century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concepts of potentiality. Potential energy is associated with forces that act on a body in a way that the work done by these forces on the body depends only on the initial and final positions of the body, and not on the specific path between them. These forces, that are called potential forces, can be represented at every point in space by vectors expressed as gradients of a scalar function called potential. Potential energy is the energy of an object. It is the energy by virtue of a position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force that works against the force field of the potential. This work is stored in the field, which is said to be stored as potential energy.

18:38

In physics, electric flux is a measure of the quantity of electric charge passing through a surface. It is used in the study of electromagnetic radiation. The SI unit of electric flux is the weber (symbol: Wb). The electric flux through a surface is calculated by dividing the electric charge passing through the surface by the area of the surface, and multiplying by the permittivity of free space (the permittivity of vacuum is used in the case of a vacuum). The electric flux through a closed surface is zero, by Gauss's law.

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(II) A long cylindrical sh…

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(II) A very long solid non…

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2. (23.34) A cylindrical s…

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(a) Consider a uniformly c…

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(II) Two thin concentric s…

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An infinitely long, cylind…

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okay, is there were doing Chapter 22 problems 33 here. So it says a long cylindrical show of radius are big are not and think a little low with our not much smaller than Elsa's very long rod. And it possesses a uniform's surface charged Density Sigma. So the charges the uniform across the surface of the cylindrical show. So for part A, we want to determine the electric field at points are greater than our not so outside of the cylinder. So if we're follow, exactly are similarly how example. 20 to 6 goes. We can draw a galaxy and cylinder around our why air here around this electric cylindrical show of radius. Big are, and we should see that the electric field is going to be distributed perpendicular to the surface at all points. And that's what we need for Galaxies law to apply. So it's right out gases little here, so we need the electric field. If it's perpendicular to the surface at all points, we then can pull it out and evaluate just the surface area, and this then becomes e the surface area of our cylindrical show. Then at this point is given by two pi are times the length l and this equals the charge Enclosed over Absolutely not. From that we can rearrange and solve for what he is. And this is given as the surface density times the area enclosed surface area enclosed here over to pie. Absolutely not big R l awesome. So now for are greater than are not. We can see that the area enclosed is just the surface complete surface area. Just two pi r. Not so. If we plug that in for our area enclosed E then becomes Sigma or not over Absalon, Not our as our two pies and medals cancel out. And since this is positive, we already know it's gonna be perpendicular to the surface at all points. So that's radio and positiveness tells us it's radio outward. Awesome. So there's your answer for part A. So now, for part B, it asks What is the electric field for inside of the cylinder? And at this point, we can see the enclosed area zero, because if we draw our calc and cylinder inside, there's no there's no area there. There's nothing no area with charge accumulated. So if area enclosed zero, you know that the electric field is also zero moving on the part. See, it says, compared to the result for a long line of charge, which is example six of gases law. So the field for our greater than are not due to a shell is the same as the field due to a long line of charge. If you substitute land for two, pi are not signal. So if we replace that, then we should see that e well, First of all, right out was Sigma are not over. Absalon are so if we replace signal there, look, if we replace Lambda with that, then Sigma are not as Landover to pie and we plug it in and get him over to pie. Absolutely R, which is exactly what it is for a line charge. So it's pretty similar if we substitute that. And when we're talking about distances far away, it really does act like that, and that's usually what we can do. We could approximate the surface charge of this cylinder as a linear charged up city, but when we're expecting a closer recon, see that the answer's really do compare Similarly so

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