What is the reason behind the lag of current in inductor, lead in capacitor?

Inductor: Inductors are storage elements which store energy in the form of magnetic fields. The current flowing through the inductor is the reason for the establishment of the magnetic field.

Lenz’s law states that the effect tends to oppose the cause. Try to follow my reasoning here. A voltage was applied across the inductor. This causes current to flow through it. The current passing through the inductor establishes a magnetic field(cork screw rule). According to Faraday’s law, when a current carrying conductor interacts with a magnetic field, there is an EMF induced in the conductor. So the current flowing through the inductor and the magnetic field established due to the current interact with each other to induce an EMF in the inductor, which opposes the applied voltage(effect opposes cause) according to Lenz’s law. This EMF causes a current to flow, which opposes the main current. This causes a reduction in main current.
Let us analyse what happened above. If the voltage was applied to the inductor for one half cycle and removed, it would undergo the above process and settle in a final state of zero current. As long as there is a change in voltage, there would be a change in current, which would in turn induce an EMF to oppose the change, thereby inducing an opposing current. This causes the current to lag the applied voltage. Any change in voltage would finally affect the current passing through the coil. This can be explained using Faraday’s equations for electromagnetic induction.
There is a differentiation involved, which changes the phase of the current with respect to the applied voltage.

Capacitors behave the same way, but they store energy in the form of electric fields. The EMF induced due to the current flowing involves an integral, which changes the phase of the voltage with respect to the current.
P.S: I know that I wasn’t too clear in explaining the above, but I really require a blackboard to teach this. The concept of Lenz’s law is a tricky one.