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Example: Flux through a Cylinder on a Line of Charge

Suppose you have a line of charge with a uniform linear charge density of λ=15μC/m. What is the electric flux through a cylinder with radius R=0.5 m, and length l=3 m that is placed so that its axis is aligned with the line of charge? Feel free to use the electric field due to an infinite uniform line of charge: E=λ2πrϵ0ˆr (where ˆr points away from line, and r is the distance from the line).

Facts

Lacking

Representations

Φe=EdA

E=λ2πrϵ0ˆr

Charged Line and Cylindrical Surface

Approximations & Assumptions

There are a few approximations and assumptions we should make in order to simplify our model.

  • There are no other charges that contribute appreciably to the flux calculation.
  • The cylinder is aligned with respect to the line so that its bases are perpendicular to the line, and its wall is parallel (as described). This is just a geometric simplification for the model, and ensures the electric field through the cylinder wall is constant as the wall will be at a uniform distance away from the line of charge at any point.
  • Line of charge is very very long: This allows us to use the electric field equation provided in the problem statement, and ensures the electric field is constant through the wall of our cylinder.