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(II) Suppose the nonconducting sphere of Example 4 has a spherical cavity of radius $r_{1}$ centered at the sphere's center (Fig. 32). Assuming the charge $Q$ is distributed uniformly in the "shell" (between $r=r_{1}$ and $r=r_{0},$ determine the electric field as a function of $r$ for $(a)$ $0<r<r_{1},(b) \quad r_{1}<r<r_{0},$ and $(c)$ $r>r_{0}$

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a) $E=0$b) $E=\frac{Q}{4 \pi \epsilon_{0} r^{2}} \frac{\left(r^{3}-r_{1}^{3}\right)}{\left(r_{0}^{3}-r_{1}^{3}\right)}$c) $E=\frac{Q}{4 \pi \epsilon_{0} r^{2}}$

Physics 102 Electricity and Magnetism

Chapter 22

Gauss's Law

Electric Charge and Electric Field

Electric Potential

Cornell University

Rutgers, The State University of New Jersey

University of Winnipeg

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) Suppose the nonconduc…

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Let $P(r)=\frac{Q}{\pi R^{…

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(II) A nonconducting spher…

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A uniform sphere of radius…

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A total charge $Q$ is dist…

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Let $\rho(r)=\left(\frac{Q…

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Let $\mathrm{P}(\mathrm{r}…

12:50

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Let, $\rho(r)=\frac{Q r}{\…

Solid Nonconducting Sphere…

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Charge is distributed thro…

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A solid metallic sphere ha…

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A point charge $q$ is loca…

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A solid insulating sphere …

Okay, so we're in Chapter 22 problems. 29 here. So it says supposed the superconductor or the non conducting spear of example. Four has a spherical cavity of radius are one centered at this fears center, which is shown here. And so we have a charge. Q. Positive charge, distributed uniformly in the show between our equals r one r equals are not so. We want to determine the electric field a different scenarios here. So first, let's talk about gases hole here, where we know that electric field taught it into the area. Normal vector. If the electric field and the area vector here are perpendicular to each other, which they are, we know that they couldn't pull electric field out. Mrs. Given as e times four pi r squared the surface area, says, equaling the charge and closed over Absolutely not cool. So for part A, this is when we want to figure out for the electric field for our greater than zero less than our one. So for this area, we know that the charge and closed zero so automatically know the electric field is zero here. So that's easy enough now for Part B This is for when our is greater than our one, but less than enough. Okay, so we know that the charge is distributed uniformly with respect to volume here, so that means are charged density his uniform with respective volume. So this is given as Q over the difference in values from these studio areas. So this is for thers pie are not cute. Martinez, 4/3 high are one cute. So it's right that out as three. Q over four pi This is are not human minus our own cute. Okay, so if that's what our row is, weaken right The electric field as a function of the charge enclosed as we have before. But now we can write the charge and closed in terms of grow and this is given as p turns the volume and road times the volume in closed over for pie Absolutely r squared. So this is now road times. The volume in a closed is four pi ours 4/3 pi r cubed minus 4/3 pi r one cubed because the volume in the cavity doesn't have car. Jonas, we don't want to count that, Um and this is all over four pi slow, not R squared. So now we can plug in what row is. It's a lie even more, and we're left with three Q over for pie are not huge. Uranus are one cube times. This is 4/3 pi times R Cubed minus r. One cube and this is all over four. Pi Absalon not R squared, so it looks like a lot, but we can simplify it more than it comes down to being Q over four pot, right? Absolutely not. R squared times R Cubed minus r, one cubed over are not cute. Light us are one cute also. So that's our answer for the electric field for beat. Cool. Um, so let's move on now to see and c is now asking us for the range are great, are greater than our not so outside of the show. So now, for this one is easy because we just Cuban closed. It's all of the charge cute, So I just acts as a point charge from Galaxies low. So this is just Q over four pi solo, not R squared. Simple. Is that awesome?

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