### Magnetic Flux Through a Cube#### Problem DescriptionA cube with edge length \( \ell = 6.0 \, \text{cm} \) is depicted, oriented with its edges along the x, y, and z axes. A uniform magnetic field exists in the region, characterized by the components:- \( B_x = +3.0 \, \text{T} \)- \( B_y = +8.0 \, \text{T} \)- \( B_z = +5.0 \, \text{T} \)#### DiagramThe diagram shows a three-dimensional cube where:- The cube is situated in space with one vertex at the origin.- The faces of the cube are parallel to the coordinate planes (xy, yz, zx planes).- A magnetic field vector \( \mathbf{B} \) is indicated with an arrow, showing a general orientation.#### Tasks(a) **Calculate the Magnetic Flux Through the Shaded Face**Given:- A specific face of the cube is shaded.- Calculate the flux \( \Phi \) through this face using the formula: \[ \Phi = B \cdot A \cdot \cos(\theta) \] where \( B \) is the component of the magnetic field perpendicular to the face, \( A \) is the area of the face, and \( \theta \) is the angle between the field lines and the normal to the face.- **Area calculation:** \[ A = \ell \cdot \ell = (0.06 \, \text{m})^2 \]- **Identify the correct \( B \) for the face:** Assume the shaded face is perpendicular to the x-axis: \[ \Phi = B_x \cdot A = 3.0 \, \text{T} \times (0.06 \, \text{m})^2 \](b) **Net Flux Through the Entire Cube**- The net magnetic flux through a closed surface (such as the cube) is given by Gauss's law for magnetism: \[ \text{Net Flux} = 0 \] Since it's a closed surface and magnetic field lines do not begin or end inside the material, they continue through, making the net flux zero.#### Calculation Results- **Flux

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A cube of edge length { = 6.0 cm is positioned as shown in the figure below. There is a uniform magnetic field throughout the region with components B, = +3.0 T, B, = +8.0 T, and B, = +5.0 T. (a) Calculate the flux through the shaded face of the cube. T.m2 (b) What is the net flux emerging from the volume enclosed by the cube (i.e., the net flux through all six faces)? T.m2

A cube of edge length { = 6.0 cm is positioned as shown in the figure below. There is a uniform magnetic field throughout the region with components B, = +3.0 T, B, = +8.0 T, and B, = +5.0 T. (a) Calculate the flux through the shaded face of the cube. T.m2 (b) What is the net flux emerging from the volume enclosed by the cube (i.e., the net flux through all six faces)? T.m2

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

### Magnetic Flux Through a Cube

#### Problem Description

A cube with edge length \( \ell = 6.0 \, \text{cm} \) is depicted, oriented with its edges along the x, y, and z axes. A uniform magnetic field exists in the region, characterized by the components:
- \( B_x = +3.0 \, \text{T} \)
- \( B_y = +8.0 \, \text{T} \)
- \( B_z = +5.0 \, \text{T} \)

#### Diagram

The diagram shows a three-dimensional cube where:
- The cube is situated in space with one vertex at the origin.
- The faces of the cube are parallel to the coordinate planes (xy, yz, zx planes).
- A magnetic field vector \( \mathbf{B} \) is indicated with an arrow, showing a general orientation.

#### Tasks

(a) **Calculate the Magnetic Flux Through the Shaded Face**

Given:
- A specific face of the cube is shaded.
- Calculate the flux \( \Phi \) through this face using the formula:
\[
\Phi = B \cdot A \cdot \cos(\theta)
\]
where \( B \) is the component of the magnetic field perpendicular to the face, \( A \) is the area of the face, and \( \theta \) is the angle between the field lines and the normal to the face.

- **Area calculation:**
\[
A = \ell \cdot \ell = (0.06 \, \text{m})^2
\]

- **Identify the correct \( B \) for the face:**
Assume the shaded face is perpendicular to the x-axis:
\[
\Phi = B_x \cdot A = 3.0 \, \text{T} \times (0.06 \, \text{m})^2
\]

(b) **Net Flux Through the Entire Cube**

- The net magnetic flux through a closed surface (such as the cube) is given by Gauss's law for magnetism:
\[
\text{Net Flux} = 0
\]
Since it's a closed surface and magnetic field lines do not begin or end inside the material, they continue through, making the net flux zero.

#### Calculation Results

- **Flux

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