Why exactly do capacitors charge and discharge exponentially? Hi this is my first time posting so please feel free to correct me on any formatting errors I may make.
I understand that as a capacitor charges, the amount of electrons that are deposited on one plate increases, thereby the overall voltage across the capacitor increases. And I kind of understand that because of that, the rate at which 1 coulomb of charge flows in the circuit starts to fall because of this. But what I don't understand is why this decrease in current is exponential, or how any relationship between these variables are exponential. I don't know if I was doing this wrong but I couldn't find a true explanation with a google search, so I was hoping my question could be answered here, thanks in advance.
 A: As a capacitor charges, electrons are pulled from the positive plate and pushed onto the negative plate by the battery that is doing the charging.  Looking just at the negative plate, note that electrons repel each other, so they will spread out evenly on the negative plate as they accumulate.  Since electrons repel each other, as more electrons accumulate on the negative plate, it becomes increasingly more difficult to add the next electron to that plate because the repulsive force on the next electron that you are trying to add to the negative plate is directly proportional to the number of electrons that are already on that plate.  The differential equation that describes this effect is formulated such that the rate of charging is negatively affected by the charge that is already on the negative plate, and is proportional to the charge that is already on that plate.  The solution of the differential equation results in an exponential mathematical form with a negative exponent.
A similar argument can be made for discharging the negative plate, as the repulsive force on an electron leaving the negative plate is proportional to the charge on that plate.  Likewise, a similar argument can be made for the positive plate regarding how easy it is to either remove or add electrons to that plate as the capacitor is charging or discharging.
Note that there are many instances in nature of a rate depending on how much of some substance or energy already exists (e.g., Newton's Law of cooling), and for that reason, the exponential mathematical form noted above occurs frequently.
A: When a capacitor discharges through a simple resistor, the current is proportional to the voltage (Ohm's law).
That current means a decreasing charge in the capacitor, so a decreasing voltage.
Which makes that the current is smaller.
One could write this up as a differential equation, but that is calculus.
One can also reason that when half of the charge is gone after a certain time $\tau_{1/2}$, the current is half as large, so it again it will take the same $\tau_{1/2}$ to halve again.
This is exponential decay, in the same way as radioactive decay.
(But it is not true when you connect an LED to the capacitor.)
