- #1
jaumzaum
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I was trying to calculate the the charge distribution (surface charge density = σ in function of r) in a very long circular metallic plate.
I know σ not constant if we get closer to the rim of the plate
Let's say we want to calculate the E field in point Q that is x distant from the center, and we choose a point P, that is in a distance r from the center and in an angle θ to the line OQ (O is the center). This point has a area dA = r dr dθ.
PQ² = z² = r² + x² - 2 r x Cos(θ)
The field (in OQ axe) is
dE = k dA σ/z²
We know that E = 0 in any point and also ∫ σ r dr = Q/2π
Integrating only θ (and simplifying a lot of stuff) we get
∫ σ r dr = Q/2π
∫ σ dr/(r²-x²) = 0
Both integral from 0 ro R
How can I find σ from this?
I know σ not constant if we get closer to the rim of the plate
Let's say we want to calculate the E field in point Q that is x distant from the center, and we choose a point P, that is in a distance r from the center and in an angle θ to the line OQ (O is the center). This point has a area dA = r dr dθ.
PQ² = z² = r² + x² - 2 r x Cos(θ)
The field (in OQ axe) is
dE = k dA σ/z²
We know that E = 0 in any point and also ∫ σ r dr = Q/2π
Integrating only θ (and simplifying a lot of stuff) we get
∫ σ r dr = Q/2π
∫ σ dr/(r²-x²) = 0
Both integral from 0 ro R
How can I find σ from this?