The Current-Voltage Laws of Capacitor and Inductor
Current is proportional to how fast the voltage changes
Apply a triangular voltage to a capacitor and slide its slope — the rate of change of voltage, dv/dt. The capacitor’s current is proportional not to the voltage at the moment but to how fast the voltage changes: i = C·dv/dt. While the voltage rises at a steady slope the current stays flat and constant, and the steeper the slope the larger the current. Set the slope so the current reaches the target of 4.
A different arithmetic from the resistor
A resistor ties this instant’s voltage and current directly through Ohm’s law, v = iR. A capacitor does not. The charge q piled on its two plates is proportional to the voltage (q = Cv), and the current is how fast that charge flows in and out (i = dq/dt). Put them together and i = C·dv/dt. The current depends not on what the voltage is now but on how fast it changes. A high but steady voltage carries no current; a low but fast-changing voltage carries a large one.
A steady voltage blocks the current
When dv/dt is zero — when the voltage does not change — the current is zero too. Hold a DC voltage and once the capacitor is fully charged it carries no current; this is why a capacitor blocks DC. Conversely a fast-swinging voltage carries a large current, so fast signals pass easily. A triangular wave shows this clearly: while the voltage rises at a straight slope the current is flat and constant, and when the slope bends negative the current at once flips sign, drawing a square wave. The current waveform is simply the slope of the voltage waveform carried over.
The inductor is the mirror
The inductor is the mirror of the capacitor, with voltage and current swapped. Where the capacitor had i = C·dv/dt, the inductor has v = L·di/dt: the rate of change of current makes the voltage. So an inductor’s current cannot change suddenly, because that would require an infinite voltage. Likewise a capacitor’s voltage cannot change suddenly, because that would require an infinite current. These two differential laws are the starting point of every transient, and the time constant τ, and the impedances jωL and 1/jωC seen for sinusoids, are all just what you get by putting a sinusoid into this one pair of laws.
Back to the first screen
As you slid the slope dv/dt, the capacitor current grew in proportion, reaching the target of 4 when the slope was 2 (C = 2). While the voltage rose straight the current was flat, the steeper the slope the larger the current, and when the slope bent negative the current changed sign. The current watches not the voltage of the moment but how fast it changes — i = C·dv/dt was the whole of it. The inductor is the mirror with the two swapped, v = L·di/dt. Where the resistor tied values, the capacitor and inductor tie change, and this one pair of differential laws is the root of transients and reactance.