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**Problem 4: Electric Field in a Cylindrical Shell** A hollow, very long, non-conducting cylindrical shell has an inner radius \( R_1 \) and an outer radius \( R_2 \). A very thin wire with a linear charge density \( \lambda_i \) is located at the center of the shell. The shell itself carries a cylindrically symmetric charge density \( \rho = br \) for \( R_1 < r < R_2 \), which increases linearly with the radius (but does not change along the length). Here, \( b \) is a constant of proportionality. **Task:** - **Draw and label** a Gaussian surface. - Use Gauss's Law to find the radial electric field in the region \( r < R_1 \). **Graph/Diagram Explanation:** The diagram displays a cross-section of the cylindrical shell. The inner circle, marked as \( R_1 \), represents the inner radius. The outer dashed circle represents the outer radius, \( R_2 \). A Gaussian surface is depicted as a dashed cylinder within the shell, which is designed to enclose the central wire. The central wire is drawn along the axis of the cylindrical shell, indicating where the linear charge density \( \lambda_i \) is located. You may take the positive direction as outward when calculating using Gauss's Law. The challenge is to understand how the charge distribution affects the electric field in specified regions using these visual and procedural aids.
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Transcribed Image Text:**Problem 4: Electric Field in a Cylindrical Shell** A hollow, very long, non-conducting cylindrical shell has an inner radius \( R_1 \) and an outer radius \( R_2 \). A very thin wire with a linear charge density \( \lambda_i \) is located at the center of the shell. The shell itself carries a cylindrically symmetric charge density \( \rho = br \) for \( R_1 < r < R_2 \), which increases linearly with the radius (but does not change along the length). Here, \( b \) is a constant of proportionality. **Task:** - **Draw and label** a Gaussian surface. - Use Gauss's Law to find the radial electric field in the region \( r < R_1 \). **Graph/Diagram Explanation:** The diagram displays a cross-section of the cylindrical shell. The inner circle, marked as \( R_1 \), represents the inner radius. The outer dashed circle represents the outer radius, \( R_2 \). A Gaussian surface is depicted as a dashed cylinder within the shell, which is designed to enclose the central wire. The central wire is drawn along the axis of the cylindrical shell, indicating where the linear charge density \( \lambda_i \) is located. You may take the positive direction as outward when calculating using Gauss's Law. The challenge is to understand how the charge distribution affects the electric field in specified regions using these visual and procedural aids.
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