A cylindrical Gaussian surface of length L and radius R encloses a charge Q uniformly distributed along its length. What is the electric field 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 uniformly distributed along its length. What is the electric field at a distance R from the axis of the cylinder?
Q/(2πε₀R)
Q/(4πε₀R²)
0
Q/(ε₀L)
Using Gauss's law, the electric field outside the cylinder is E = Q/(2πε₀R).
Questions & Step-by-step Solutions
1 item
Q
Q: A cylindrical Gaussian surface of length L and radius R encloses a charge Q uniformly distributed along its length. What is the electric field at a distance R from the axis of the cylinder?
Solution: Using Gauss's law, the electric field outside the cylinder is E = Q/(2πε₀R).
Steps: 8
Step 1: Understand that we have a cylindrical Gaussian surface with a charge Q distributed along its length.
Step 2: Identify that we want to find the electric field 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 cylindrical surface, the electric field (E) is uniform and points radially outward, so we can express the electric flux as Φ = E × A, where A is the surface area of the cylindrical Gaussian surface.
Step 5: The surface area A of the cylindrical Gaussian surface is 2πRL, where R is the radius and L is the length of the cylinder.
Step 6: Set up Gauss's law: Φ = Q_enc / ε₀, where Q_enc is the charge enclosed by the Gaussian surface. Since the charge is uniformly distributed, Q_enc = Q.
Step 7: Substitute the expression for electric flux into Gauss's law: E × (2πRL) = Q / ε₀.
Step 8: Solve for the electric field E: E = Q / (2πε₀R).
