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You are working for the summer at a research laboratory. Your research director has devised a scheme for holding small charged particles at fixed positions. The scheme is shown in the figure below. An insulating cylinder of radius a and length L ≫ a is positively charged and carries a uniform volume charge density ρ. A very thin tunnel is drilled through a diameter of the cylinder and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on opposite sides of the axis of the cylinder. Your research director has had great success with this scheme. Determine the specific value of r at which equilibrium exists. (Use the following as necessary: q and ρ.)

You are working for the summer at a research laboratory. Your research director has devised a scheme for holding small charged particles at fixed positions. The scheme is shown in the figure below. An insulating cylinder of radius a and length L ≫ a is positively charged and carries a uniform volume charge density ρ. A very thin tunnel is drilled through a diameter of the cylinder and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on opposite sides of the axis of the cylinder. Your research director has had great success with this scheme. Determine the specific value of r at which equilibrium exists. (Use the following as necessary: q and ρ.)

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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You are working for the summer at a research laboratory. Your research director has devised a scheme for holding small charged particles at fixed positions. The scheme is shown in the figure below. An insulating cylinder of radius a and length L ≫ a is positively charged and carries a uniform volume charge density ρ. A very thin tunnel is drilled through a diameter of the cylinder and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on opposite sides of the axis of the cylinder. Your research director has had great success with this scheme. Determine the specific value of r at which equilibrium exists. (Use the following as necessary: q and ρ.)

 

### Cylindrical Shell and Charge Distribution In this diagram, we have a cylindrical shell that demonstrates the distribution of an electric charge within it. The diagram is helpful in understanding concepts related to electric fields and charge densities. #### Description 1. **Cylindrical Shell**: A cylindrical shell is depicted in peach color, with a consistent radius along its length. The edge of the cylindrical shell is marked by dashed lines to indicate its boundaries. 2. **Radius (a)**: The radius of the cylinder is labeled “a”. This is the distance from the center axis of the cylinder to its surface. 3. **Charge Density (ρ)**: The symbol ρ represents the volume charge density within the cylindrical shell. 4. **Distance (r)**: There are multiple markings on the diagram showing a distance r from the central axis. It is likely referring to different radii (within the cylinder itself) at which the electric field or potential is to be measured. 5. **Point Charge (q)**: A point on the cylindrical surface is labeled with q, which might be referring to a charge or a specific point of interest on the surface for measurement or analysis. 6. **Arrows Along the Axis**: The arrows pointing outward along the central axis of the cylinder might signify the direction of the electric field or could represent the symmetry axis of the cylindrical shell, which is a common consideration in electrodynamics problems. #### Explanation of the Diagram - **Cylindrical Symmetry**: The cylindrical symmetry of this system suggests that the charge distribution is likely uniform around the shell. This symmetry helps simplify the calculations regarding the electric fields and potential. - **Applications in Gauss's Law**: This type of diagram is often used in the application of Gauss's Law. By constructing a Gaussian surface (an imaginary surface to apply Gauss's law) inside and outside of the cylindrical shell, students can determine the resulting electric field or the flux. - **Electric Field Considerations**: Students might need to consider the electric field both inside the cylinder (r < a) and outside the cylinder (r > a). Using the given charge density and symmetry, they can apply appropriate formulas and understand the behavior of the electric field with respect to different distances r. #### Educational Use This type of diagram is commonly found in topics covering electromagnetism within physics courses. It provides a visual representation to aid in understanding the distribution of electric fields due to symmetrical charge distributions,
Transcribed Image Text:### Cylindrical Shell and Charge Distribution In this diagram, we have a cylindrical shell that demonstrates the distribution of an electric charge within it. The diagram is helpful in understanding concepts related to electric fields and charge densities. #### Description 1. **Cylindrical Shell**: A cylindrical shell is depicted in peach color, with a consistent radius along its length. The edge of the cylindrical shell is marked by dashed lines to indicate its boundaries. 2. **Radius (a)**: The radius of the cylinder is labeled “a”. This is the distance from the center axis of the cylinder to its surface. 3. **Charge Density (ρ)**: The symbol ρ represents the volume charge density within the cylindrical shell. 4. **Distance (r)**: There are multiple markings on the diagram showing a distance r from the central axis. It is likely referring to different radii (within the cylinder itself) at which the electric field or potential is to be measured. 5. **Point Charge (q)**: A point on the cylindrical surface is labeled with q, which might be referring to a charge or a specific point of interest on the surface for measurement or analysis. 6. **Arrows Along the Axis**: The arrows pointing outward along the central axis of the cylinder might signify the direction of the electric field or could represent the symmetry axis of the cylindrical shell, which is a common consideration in electrodynamics problems. #### Explanation of the Diagram - **Cylindrical Symmetry**: The cylindrical symmetry of this system suggests that the charge distribution is likely uniform around the shell. This symmetry helps simplify the calculations regarding the electric fields and potential. - **Applications in Gauss's Law**: This type of diagram is often used in the application of Gauss's Law. By constructing a Gaussian surface (an imaginary surface to apply Gauss's law) inside and outside of the cylindrical shell, students can determine the resulting electric field or the flux. - **Electric Field Considerations**: Students might need to consider the electric field both inside the cylinder (r < a) and outside the cylinder (r > a). Using the given charge density and symmetry, they can apply appropriate formulas and understand the behavior of the electric field with respect to different distances r. #### Educational Use This type of diagram is commonly found in topics covering electromagnetism within physics courses. It provides a visual representation to aid in understanding the distribution of electric fields due to symmetrical charge distributions,
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