Question

A charged wire AB of length l is placed inside a sphere.linear charge density of the wire is equal to kx, where x is the distance measured along the wire from end A and k Is constant .calculate the flux through the sphere?

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