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)