Question

In the given figure, calculate the total flux of the electrostatic field through the sphere S₁ and S₂. The wire AB, as shown here, has a linear charge density λ given by λ = kx, where x is the distance measured along the wire, from the end A.

Answer 1

Since linear charge density λ (λ = kx) depends on distance, so it is not constant. Thus consider a small element dx having charge dq, then

dq = λ dx = kx dx

∴ q = λdq = k \(\int_0^1 xdx=\frac{kl^2}{2}\)

Electric flux through Gaussian surface is:

\(\phi_1= ∮ \vec E. \vec{dS}= \frac{q}{ɛ_o} = 0\) (∵ q = 0 inside s₁)

Electric flux through Gaussian surface S₂ is:

\(\phi_2= ∮ \vec E. \vec{dS}= \frac{q}{ɛ_o} = \frac{kl^2}{2ɛ_o}\) (∵ q = \(\frac{kl^2}{2}\))

Total electric flux thorugh the spheres S₁ and S₂

Φ = Φ₁ + Φ₂

or Φ = 0 + \(\frac{kl^2}{2}\) = \(\frac{kl^2}{2ɛ_o}\)