Prove that the total current which is the sum of conduction current and displacement current is always continuous and any loss in conduction current `(I_C)` appears as displacement current `(I_D)`.

Explanation

Consider a volume V in a medium through which
currents are flowing. Let `I_c` be the conduction
current entering the volume V and `I_c'` be the
conduction current leaving the volume V. Then
total charges entering and leaving the volume in
time dt will be `I_c` and `I_c'`dt. Therefore, the
charge accumulated inside the volume V during
time dt is given by dq(inside V)=`I_cdt-I_c'dt`
or `(dq)/(dt)=I_c-I_c'....(i)`
From Gauss's Theoram in electrostatics, we have
`phi_E=ointvecE.vec(ds)=q/(in_0) ("inside") or in_0 phi_E=q`
or `I_d=in_0 (dphi_E)/(dt)=(d (in_0phi_E))/(dt)=(dq)/(dt) ......(ii)`
From (i) and (ii),
`I_c-I_c'=I_d or I_c=I_d+I_c'`
Thus we conclude that the loss of conduction
current `(=I_c-I_c')` appears as the displacement
current `(I_d)` and conduction current plus
displacement current remains constant i.e.,
`I_c'+I_d`= a constant.