A cylindrical Gaussian surface of length L and radius R encloses a charge Q. What is the electric field E at a distance R from the axis of the cylinder?
Practice Questions
1 question
Q1
A cylindrical Gaussian surface of length L and radius R encloses a charge Q. What is the electric field E at a distance R from the axis of the cylinder?
Q/(2πε₀R)
Q/(4πε₀R²)
Q/(ε₀L)
0
Using Gauss's law, the electric field E at a distance R from the axis of a long charged cylinder is E = Q/(2πε₀L) for points outside the cylinder.
Questions & Step-by-step Solutions
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Q
Q: A cylindrical Gaussian surface of length L and radius R encloses a charge Q. What is the electric field E at a distance R from the axis of the cylinder?
Solution: Using Gauss's law, the electric field E at a distance R from the axis of a long charged cylinder is E = Q/(2πε₀L) for points outside the cylinder.
Steps: 9
Step 1: Understand that we have a cylindrical Gaussian surface with a charge Q inside it.
Step 2: Recognize that the electric field E we want to find is at a distance R from the axis of the cylinder.
Step 3: Recall Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (ε₀).
Step 4: For a long charged cylinder, the electric field is uniform and points radially outward from the axis.
Step 5: Set up the Gaussian surface: a cylinder of length L and radius R that surrounds the charged cylinder.
Step 6: Calculate the electric flux through the curved surface of the Gaussian cylinder, which is E times the surface area (2πRL).
Step 7: According to Gauss's law, set the electric flux equal to the enclosed charge divided by ε₀: E(2πRL) = Q/ε₀.
Step 8: Solve for the electric field E: E = Q/(2πε₀L).
Step 9: Conclude that this formula gives the electric field E at a distance R from the axis of the cylinder for points outside the cylinder.
