A charge of 170μC is at the center of a cube of edge 80.0...

A charge of 170$\mu \mathrm{C}$ is at the center of a cube of edge 80.0 cm. No other charges are nearby. (a) Find the flux through each face of the cube. (b) Find the flux through the whole surface of the cube. (c) What If? Would your answers to either part (a) or part (b) change if the charge were not at the center? Explain.

Key Concepts

Gauss's Law

Gauss's Law is one of Maxwell's equations which relates the net electric flux through a closed surface to the charge enclosed by that surface. It is expressed mathematically as ? = Q_enc/??, where Q_enc is the enclosed charge and ?? is the permittivity of free space. This principle is especially useful for calculating electric fields and flux when there is high symmetry in the configuration of charges or the geometry of the enclosing surface.

Electric Flux

Electric flux is a measure of the number of electric field lines penetrating a surface. It is determined by integrating the electric field over the area of the surface. In the context of Gaussian surfaces, the net flux gives a direct measure of the enclosed charge when applying Gauss’s Law, simplifying calculations under conditions of symmetry.

Symmetry

Symmetry in the context of electric fields is a powerful concept that simplifies the analysis of field configurations. When a charge is placed at the center of a symmetric object, such as a cube, the field distribution and, hence, the flux through each symmetric face will be identical. Loss of symmetry, such as when a charge is not centrally located, means the flux distribution among the faces may no longer be uniform, although the total flux through the closed surface remains determined solely by the enclosed charge.

Charge Distribution and Enclosure

The concept of charge distribution and enclosure pertains to understanding how the positioning of a charge within an enclosing surface affects the local field outcomes without changing the total flux determined by Gauss’s law. When a charge is fully inside a Gaussian surface, the total flux remains constant regardless of the charge's exact location within the surface; however, the distribution of flux on different parts of the surface can vary if the symmetrical condition is disrupted.

Transcript

00:01 Okay, so in this problem we have a cube.

00:05 So let's draw here our cube.

00:23 So this is our cube and you know that we have a charge inside this cube, precisely at the center of the cube.

00:34 And in the first item of this problem, we have to calculate what is the electrical flux from each one of the faces of the cube.

00:46 We must first remember about the physics involving this problem.

00:51 Since we know that the charge is precisely at the center of the cube, we can say that each face of the cube is going to see passing through her the same amount of flux.

01:07 So we can say that the electrical flux from one face of the cube is going to be one -sixth of the total electrical flux that passes from this from this from this from this from this and what is the definition of electrical flux when we have a gaussian surface well is just the charge inside oops charge inside divided by epsilon zero okay so we can say that this is going to be 160 of the charge divided by epsilon 0.

02:04 So that's the flux from one face of the cube.

02:08 It's going to be equal for each one of the faces.

02:13 So let's calculate this.

02:15 We can say that the flux, the electrical flux from one face is going to be 16.

02:28 Of the charge inside which is 1 .7 times 10 to the minus 4 divided by epsilon 0 which is 8 .8542 times 10 to the minus 12.

02:51 So if we calculate this we will find that the flux from one surface is going to be equal 3 .2 times 10 to the 6 newtons per meter square, newton's meter square per kulem.

03:13 So that's the answer to the first item.

03:15 The second item, the second item is going to be, let's see...

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