Electric Flux Calculation through a Closed Cylinder with Line Charge
Calculating the Total Electric Flux
Find the total electric flux through a closed cylinder containing a line charge along its axis with linear charge density λ = λ₀(1-z/h) if the cylinder and the line charge extend from z = 0 to z = h.
The total electric flux through the closed cylinder is (λ₀h / 4ε₀).
How to calculate the total electric flux?
Apply Gauss's law as follows;
Φ = ∫E dA
where;
E is the electric field
dA is the change in area
The electric field due to a line charge along its axis is given by:
E = (1 / (4πε₀))(λ / r)
Where:
E is the electric field.
ε₀ is the vacuum permittivity constant
λ is the linear charge density
r is the radial distance from the line charge.
The given parameter; λ = λ₀(1 - z/h)
The electric flux Φ through a differential area dA on the cylindrical surface at a height z. dA = 2πrdz
Φ = ∫E dA
Φ = ∫((1 / (4πε₀)) × (λ / r)) × (2πrdz)
Φ = (1 / (2ε₀)) × ∫(λ / r)dz
Now, integrate with respect to z from z = 0 to z = h:
Φ = (1 / (2ε₀)) × ∫(λ₀(1 - z/h) / r) × dz (0 to h)
Φ = (1 / (2ε₀))λ₀/r × [z - (z² / (2h))] (0 to h)
Φ = (1 / (2ε₀)) λ₀/r × [h - (h² / (2h)) - (0 - 0)]
Φ = (1 / (2ε₀))λ₀/r × [h - (h / 2)]
Φ = (1 / (2ε₀))(λ₀h/2) × (1 - 1/2)
Φ = (1 / (2ε₀))(λ₀h/2) × (1/2)
Φ = (λ₀h / 4ε₀)
Find the total electric flux through a closed cylinder containing a line charge along its axis with linear charge density λ = λ₀(1-z/h) if the cylinder and the line charge extend from z = 0 to z = h. The total electric flux through the closed cylinder is (λ₀h / 4ε₀).