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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, = +7.0 T, B, = +6.0 T, and B, = +7.0 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, = +7.0 T, B, = +6.0 T, and B, = +7.0 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
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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### Magnetic Flux through a Cube **Introduction:** Consider a cube with an edge length of \( \ell = 6.0 \, \text{cm} \) positioned as shown in the diagram. The system is subject to a uniform magnetic field with the following components: - \( B_x = +7.0 \, \text{T} \) - \( B_y = +6.0 \, \text{T} \) - \( B_z = +7.0 \, \text{T} \) The objective is to calculate the magnetic flux for specific scenarios related to this cube. **Diagram Explanation:** The diagram features a cube situated in a three-dimensional coordinate system with the axes labeled as \( x \), \( y \), and \( z \). The magnetic field \( \vec{B} \) is indicated with a green arrow penetrating one of the cube's faces, which is shaded. The dimensions along each axis are labeled as \( \ell \). **Problems:** (a) **Calculate the flux through the shaded face of the cube.** - Enter the flux value in \( \text{T} \cdot \text{m}^2 \). (b) **Determine the net flux emerging from the volume enclosed by the cube (i.e., the net flux through all six faces).** - Enter the net flux value in \( \text{T} \cdot \text{m}^2 \). ### Additional Information To solve these problems: - Use the formula for magnetic flux through a surface: \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field strength, \( A \) is the area of the face, and \( \theta \) is the angle between \( \vec{B} \) and the normal to the surface. - For part (b), apply Gauss's Law for Magnetism, noting that the net magnetic flux through a closed surface is always zero.
Transcribed Image Text:### Magnetic Flux through a Cube **Introduction:** Consider a cube with an edge length of \( \ell = 6.0 \, \text{cm} \) positioned as shown in the diagram. The system is subject to a uniform magnetic field with the following components: - \( B_x = +7.0 \, \text{T} \) - \( B_y = +6.0 \, \text{T} \) - \( B_z = +7.0 \, \text{T} \) The objective is to calculate the magnetic flux for specific scenarios related to this cube. **Diagram Explanation:** The diagram features a cube situated in a three-dimensional coordinate system with the axes labeled as \( x \), \( y \), and \( z \). The magnetic field \( \vec{B} \) is indicated with a green arrow penetrating one of the cube's faces, which is shaded. The dimensions along each axis are labeled as \( \ell \). **Problems:** (a) **Calculate the flux through the shaded face of the cube.** - Enter the flux value in \( \text{T} \cdot \text{m}^2 \). (b) **Determine the net flux emerging from the volume enclosed by the cube (i.e., the net flux through all six faces).** - Enter the net flux value in \( \text{T} \cdot \text{m}^2 \). ### Additional Information To solve these problems: - Use the formula for magnetic flux through a surface: \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field strength, \( A \) is the area of the face, and \( \theta \) is the angle between \( \vec{B} \) and the normal to the surface. - For part (b), apply Gauss's Law for Magnetism, noting that the net magnetic flux through a closed surface is always zero.
**Transcription for Educational Website:** Transcranial magnetic stimulation (TMS) is a noninvasive technique used to stimulate regions of the human brain. A small coil is placed on the scalp, and a brief burst of current in the coil produces a rapidly changing magnetic field inside the brain. The induced emf can be sufficient to stimulate neuronal activity. One such device generates a magnetic field within the brain that rises from zero to 1.2 T in 100 ms. Determine the magnitude of the induced emf within a circle of tissue of radius 1.6 mm and that is perpendicular to the direction of the field. \[ \mathcal{E} = \_\_\_\_\_\_ \text{ mV} \] **Explanation:** This text discusses the principles of TMS, focusing on the stimulation of brain regions through the use of a rapidly changing magnetic field. The problem involves calculating the magnitude of the induced electromotive force (emf) within a specific area of brain tissue, emphasizing the magnetic field's parameters and the geometric constraints of the situation.
Transcribed Image Text:**Transcription for Educational Website:** Transcranial magnetic stimulation (TMS) is a noninvasive technique used to stimulate regions of the human brain. A small coil is placed on the scalp, and a brief burst of current in the coil produces a rapidly changing magnetic field inside the brain. The induced emf can be sufficient to stimulate neuronal activity. One such device generates a magnetic field within the brain that rises from zero to 1.2 T in 100 ms. Determine the magnitude of the induced emf within a circle of tissue of radius 1.6 mm and that is perpendicular to the direction of the field. \[ \mathcal{E} = \_\_\_\_\_\_ \text{ mV} \] **Explanation:** This text discusses the principles of TMS, focusing on the stimulation of brain regions through the use of a rapidly changing magnetic field. The problem involves calculating the magnitude of the induced electromotive force (emf) within a specific area of brain tissue, emphasizing the magnetic field's parameters and the geometric constraints of the situation.
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