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s3a
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Homework Statement
Problem statement:
An infinitely long, straight line has a uniform charge distribution of ρ C/m. Use Gauss' law to find the electric field at a point r m away from it.
Solution:
Consider a cylindrical volume of height ℓ with circular cross sectional area of radius r, which has the line as its axis. The volume contains a total charge of Q = ρℓ. By symmetry, the E field is radial in direction and has the same magnitude on the surface of the cylinder. The total flux through the surface is Ψ = ϵE × ℓ × πr^2. By Gauss' law, Ψ = Q from which E = ρ/(πϵr^2).
Homework Equations
Gauss' Law.
Cylinder.
The Attempt at a Solution
I'm trying to understand the given solution.
Is Q = ρℓ instead of Q = ρV true, because the charge is distributed along the real-world line instead of the imaginary cylinder?
And, what about Gauss' Law indicates that Ψ = Q?
Also, I keep seeing the capital letter phi, Φ, in the context of Gauss' Law. Is Φ from what I see on-line the same thing as the Ψ I see in my book?
Lastly, how was the total flux found to be Ψ = ϵE × ℓ × πr^2?
For what it's worth, given Ψ = Q, Q = ρℓ and Ψ = ϵE × ℓ × πr^2, I do know how to get E = ρ/(πϵr^2).
Any input would be GREATLY appreciated!