Deriving Capacitor Formulas?

I understand that for charging:

Vcap = E(1 - exp(-t/RC)

and for discharging:

Vcap = E*exp(-t/RC)

but... How is this actually derived? I've looked through all of my textbooks, and most of them say "With further mathematical analysis we can derive..." and stuff like that. That doesn't help me though.

Any suggestions? (Note: I do have 3 calculus classes behind me, so don't be afraid to bring out the big guns for the math. I'm sure it'll be blatanly obvious when it's pointed out how to do this, but I'm having a serious mental block today).

Hmm... I seem to have found the answer, and it's more complicated than I though

https://scienceworld.wolfram.com/physics/CRCircuit.html

This states that it's a differential equation

https://mathworld.wolfram.com/OrdinaryDifferentialEquation.html

describes the process a bit... A little over my head (Differential Equations is the next calculus class on my agenda )

The equations that you mentioned are simply solutions to the following differential equation, which relates current in a cap to change in voltage:

I=C(dV/dt)

now since I=-V/R (the charging cap has a resistor connected to it in a loop)

you get:

-V/R=C(dV/dt)

, when you colve this equation you get the results you mentioned (and I don't think that gets covered in Calc III) - you probably have to wait until you take Differential Equations (that's right after Calc 3)

eemage21 said:

..snip..

you probably have to wait until you take Differential Equations (that's right after Calc 3)

Click to expand...

Indeed. Thanks for your help though. I'm taking Differential Equations next semester, so maybe it will be a little more clear after that.