A hollow cylinder with charge density λ is placed along the z-axis. What is the electric field at a point outside the cylinder?

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Q: A hollow cylinder with charge density λ is placed along the z-axis. What is the electric field at a point outside the cylinder?

Solution: For a hollow cylinder, the electric field at a point outside is E = λ/(2πε₀r) using Gauss's law.

Steps: 10

Step 1: Understand that we have a hollow cylinder with a uniform charge density (λ) placed along the z-axis.
Step 2: Recognize that we want to find the electric field (E) at a point that is outside the hollow cylinder.
Step 3: Recall Gauss's law, which relates the electric field to the charge enclosed by a surface.
Step 4: Choose a cylindrical Gaussian surface that is coaxial with the hollow cylinder and lies outside of it.
Step 5: Calculate the total charge (Q) enclosed by the Gaussian surface using the charge density (λ) and the length of the cylinder.
Step 6: Use Gauss's law: ∮E·dA = Q_enc/ε₀, where Q_enc is the charge enclosed by the Gaussian surface.
Step 7: Since the electric field is uniform and points radially outward, simplify the left side of Gauss's law to E(2πrL), where r is the radius of the Gaussian surface and L is its length.
Step 8: Set the left side equal to the right side: E(2πrL) = λL/ε₀.
Step 9: Solve for the electric field (E) by dividing both sides by (2πrL): E = λ/(2πε₀r).
Step 10: Conclude that the electric field at a point outside the hollow cylinder is E = λ/(2πε₀r).