Why does the current gradually increase in the inductor?

When a current starts to flow in the inductor, a magnetic field is generated around the inductor at the same time.

This change in magnetic field induces electromagnetic induction, and the inductor produces electromotive force. Due to the interruption of this electromotive force, the current gradually increases.

When direct current is supplied to the inductor, the magnetic field is no longer changed. Therefore, the inductor becomes a normal conductor, allowing the current to pass through well.

However, when an alternating current is supplied to an inductor, the inductor continues to generate electromotive force, which interrupts the flow of current.

Therefore, an inductor can pass a direct current well but not an alternating current.

Why does not the current keep flowing when I switch on the capacitor on direct current?

Essentially, the capacitor is wire that is disconnected.

When I switch on the capacitor circuit, current can flow while the capacitor is charging. However, when the charging of the capacitor is completed, the current can no longer flow.

If AC power is supplied, the capacitor will continue charging and discharging repeatedly. Therefore, the capacitor can pass the alternating current, but not pass the direct current well.

Where does this feature apply?

Inductors pass low frequency currents and capacitors pass high frequency currents.

Because of these frequency characteristics, the inductor and capacitor can filter out the desired frequency of the electrical signal or prevent unwanted frequency currents. This is very useful in our surroundings, such as in wireless communications (for example, finding the frequency of a TV or radio)