I've looked on the internet but haven't really read anything that intuitively explains why the voltage across a capacitor and the current through an inductor can't change instantaneously. Can anybody explain it so that I can easily understand what is going on?
Hey Riley, the math explanation is actually pretty straightforward even if you don't know basic calculus.
The current through a capacitor is given as the capacitance times the change of the voltage versus time. So, the faster the voltage changes, the greater the current. Also, the greater the current, the greater the voltage change. If the voltage were to change instantly, that would mean that the current would be extremely large - infinitely large.
The opposite for the inductor. The voltage across an inductor is given by the inductance times the change in the current versus time. So, the faster the current changes, the greater the voltage. If the current were to change instantly, the voltage would be infinitely large.
Hopefully this helps you understand the limitations more intuitively, at least from a mathematical perspective.
For those old enough to have worked on cars with points, that's how the coil is able to generate 1000's volts for the spark plug. The points open up (very fast) and the coil 'must keep the circuit connected' thus raising the voltage until the spark plug literally ionizes the gas in the cylinder thus completing the circuit.