A solid insulating sphere of radius a=5.00 cm carries a net positive charge of Q=3.00 μC uniform

A solid insulating sphere of radius a=5.00 cm carries a net...

Key Concepts

Superposition Principle

The superposition principle states that when multiple charge distributions are present, the total electric field is the vector sum of the fields produced by each individual distribution. This allows for the separate analysis of different regions of the system and then combining the results to produce the overall electric field profile.

Graphical Analysis of Electric Fields

Graphing the electric field versus distance requires identifying distinct regions where the behavior of the field changes due to different charge distributions or material properties. By analyzing each region separately—inside the insulating sphere, between the sphere and conducting shell, within the conductor, and outside the conductor—the complete electric field profile can be accurately represented.

Conductors and Electrostatic Equilibrium

In conductors at electrostatic equilibrium, the electric field within the material is zero, and excess charge resides on its surfaces. This concept is crucial for understanding how the electric field behaves in and around conducting shells, particularly in regions where the material itself can cancel internal fields.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through any closed surface to the charge enclosed by that surface. In problems with spherical symmetry, it allows for simple calculations of the electric field by using a Gaussian surface whose symmetry matches that of the charge distribution.

Uniform Charge Distribution

A uniform charge distribution means that the charge is spread evenly throughout a given volume, resulting in a constant charge density. This uniformity simplifies the analysis of the electric field both inside and outside the material since the symmetry allows the use of integral forms of Gauss's Law with ease.

Electric Field of a Spherical Charge Distribution

For a sphere with a uniform charge distribution, the electric field inside the sphere increases linearly with distance from the center, while outside the sphere it behaves as if all the charge were concentrated at the center, following an inverse-square law. This distinction is critical when constructing a graph of the electric field that highlights the different regimes based on the radius.

Transcript

00:01 Okay, so this problem, first of all, let's try to draw this system, because when we have the system well established, everything is going to be easier to see.

00:15 So first of all, we have an insulated, an insulating sphere.

00:22 So let's put here an insulating sphere with radius a.

00:29 So you have an insulated sphere with radius a and charge q.

00:36 That means that the charge is distributed uniform through the entire volume of the sphere.

00:47 Then we have a spherical shell.

00:51 So here we have a spherical shell, concentric spherical shell.

00:57 This is not concentric, but i will try my best to be concentric.

01:01 So this is the concentric spherical shell with a second radius.

01:09 So we have here a radius b.

01:12 And we have the outer spherical concentric shell.

01:18 All this is concentric, okay.

01:20 It's not a good draw, but...

01:24 And this second radius has length c.

01:30 So this is the radius c of the outer shell.

01:33 The inner shell has radiod b and the insulated sphere has charge q and radius a.

01:45 And we finally know and we also know that this shell, this is vertical shell, has a net charge, which means in both sides we have a net charge with negative, negative q.

02:03 Actually, let's put this way.

02:05 We have a q, which is negative 1 microculum.

02:15 Okay, so this is our system.

02:18 We have the insulating sphere inside with charge capital q, and we have this concentric sphertriced spherical shell with our inner radius and an outer radius, which means that this is a capacitor, because we have the inside radius and the outer radius.

02:36 So what we have to do in this problem is to describe a graph of the magnitude of the electric field in this configuration when we go from the radius to 0 to 55 centimeters.

02:53 So we just want to describe the graph, let's put here, of the electric field by the radius.

03:11 Okay? so how are we going to do this? to describe the graph we must know what is the behavior of the electric field and to calculate the behavior of the electric field we must do a gaussian surface in in each one of the regions.

03:33 So we have to make a gaussian surface inside the sphere.

03:38 We have to make a gaussian surface between the insulating surface.

03:43 Sphere and the spherical shell, we have to make a gaussian surface between both of the shells.

03:55 Let's put here.

03:57 And we have to make a gaussian surface outside all of this.

04:02 Okay? so it's kind of polluted the system, but essentially we just need to make a calculation for each one of the regions.

04:14 So, we already make these calculations in previous problems, so let's simplify a little.

04:20 And let's think about the first region.

04:23 In the first region, the first region is when the radius go from zero to a...

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