Question

A wire AB of length L has a linear charge density (Lambda) is equals to Kx where X is measured from the end A of the wire . The wire is enclosed by gaussian Hollow surface. find the expression for the electric flux through this Surface

Answer 1

a 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 ]
⇒ $\bold{ \int\limits^Q_0 {dQ} \, =\int\limits^L_0{Kx} \,dx}$
⇒ 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

Answer 2

linear charge density means charge per unit length.

so, we can say, λ = dQ/dx

⇒ dQ = λ.dx 

⇒ ∫dQ = ∫Kx.dx    [∵  λ = Kx ]

⇒ 

⇒ 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