A long straight wire carries a uniform linear charge density λ. What is the electric field at a distance r from the wire?

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Q: A long straight wire carries a uniform linear charge density λ. What is the electric field at a distance r from the wire?

Solution: Using Gauss's law for a cylindrical surface around the wire, the electric field E at a distance r is E = λ/(2πε₀r).

Steps: 8

Step 1: Understand that we have a long straight wire with a uniform charge distributed along its length. This means that the charge density is constant.
Step 2: Identify the distance 'r' from the wire where we want to find the electric field.
Step 3: Recall Gauss's law, which relates the electric field to the charge enclosed by a surface. For a long wire, we can use a cylindrical surface (Gaussian surface) around the wire.
Step 4: Calculate the total charge enclosed by the Gaussian surface. The charge enclosed is equal to the linear charge density (λ) multiplied by the length of the wire segment inside the cylinder.
Step 5: Use the symmetry of the problem. The electric field will be the same at all points on the cylindrical surface and will point radially outward from the wire.
Step 6: Apply Gauss's law: The electric flux through the cylindrical surface is equal to the electric field (E) times the surface area of the cylinder (2πrL), where L is the length of the cylinder.
Step 7: Set the electric flux equal to the charge enclosed divided by the permittivity of free space (ε₀): E * (2πrL) = λL/ε₀.
Step 8: Simplify the equation to solve for the electric field E: E = λ/(2πε₀r).