Question

A wire AB of length L has linear charge density λ=kx,where x is measured from the end A of the wire.This wire is enclosed by a Gaussian hollow surface.Find the expression for electric flux through this surface?

Answer 1

wire of length L and linear charge density is λ = Kx 
  we know,
      linear charge density means charge per unit length.
so, we can say, λ = dQ/dx
⇒ dQ = λ.dx 
⇒ ∫dQ = ∫Kx.dx    [∵  λ = Kx ]
⇒ $\bold{\int\limits^Q_0{dQ}=K\int\limits^L_0{x}\,dx}$
⇒ Q = KL²/2

 now, according to Gaussian theorem,
   Ф = Q/ε₀
  here, Ф is electric flux, Q is net charge inclosed in Gaussian surface and ε₀ is permitivity of medium [vaccum]
so, Ф = KL²/2ε₀

Hence, electric flux of this surface is KL²/2ε₀