A capacitor made of two parallel plates each of plate area A and separations d, being charged by an external ac source. How is the displacement current inside the capacitor the same as the current charging the capacitor?

Let the altering emf charging the plates of the capacitor be E = E₀sinωt
Charge on the capacitor q = EC = CE₀sinωt
I = dq/dt = d/dt(CE₀sinωt) = ωCE₀cosωt = I₀cosωt (where I₀ = ωCE₀)
Displacement current ID = ε₀dΦE/dt = ε₀A dE/dt = ε₀A d(q/ε₀A)/dt = d/dt(CE₀sinωt) = ωCE₀cosωt = I₀cosωt
Thus the displacement current inside the capacitor is the same as the current charging the capacitor.