What is the electric field due to a uniformly charged line of charge with linear charge density λ at a distance r from the line?

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Q: What is the electric field due to a uniformly charged line of charge with linear charge density λ at a distance r from the line?

Solution: Using Gauss's law, the electric field due to a uniformly charged line of charge is E = λ/(2πε₀r).

Steps: 11

Step 1: Understand that a uniformly charged line of charge has a constant charge per unit length, called linear charge density (λ).
Step 2: Recognize that we want to find the electric field (E) at a distance (r) from this line of charge.
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 surrounds the line of charge. The radius of this cylinder is r.
Step 5: Calculate the total charge (Q) enclosed by the Gaussian surface. If the length of the line of charge is L, then Q = λ * L.
Step 6: Apply Gauss's law: The electric flux through the surface equals the charge enclosed divided by the permittivity of free space (ε₀).
Step 7: The electric field (E) is uniform along the curved surface of the cylinder, so the electric flux is E times the surface area of the curved part of the cylinder (2πrL).
Step 8: Set up the equation: E * (2πrL) = Q/ε₀.
Step 9: Substitute Q with λ * L in the equation: E * (2πrL) = (λ * L)/ε₀.
Step 10: Simplify the equation by canceling L from both sides: E * (2πr) = λ/ε₀.
Step 11: Solve for E: E = λ/(2πε₀r).