A simple, series R-C circuit is yet another one of those basic, ideal
examples incorporated into almost every calculus-based introductory physics
course known to man.

The idea is that one can use the potential difference across a battery (or any
other DC power source for that matter) to charge a capacitor by allowing current to
flow through a single-loop circuit of resistance **R**. (The resistance can come
either in the form of the intrinsic resistance of the wire comprising the
circuit, or of some component resistor added as a part of the system.) The potential
difference across the battery
(**V**_{tot}) will be split between the
resistor and the capacitor, so that the voltage over the resistor
(**V**_{R}) will equal

V_{R} = I * R

where **I** is the current, and the voltage across the capacitor
(**V**_{C}) will equal

V_{C} = Q / C

where **Q** is the charge stored on the capacitor, and **C** is the capacitor's
capacitance. (V_{tot} =
V_{R} +
V_{C}).

The circuit will behave according to two distinct modes of operation: first,
there is the case where one charges the capacitor from 0 to some maximum charge
(**Q**_{max} =
V_{tot} * C). In this scenario, the
standard convention is that at time t=0, a switch is thrown and the circuit is completed.
Because Q=0 at t=0, there is no potential difference across the capacitor
(V_{C} = 0), and the entirety of the voltage
of the battery is spent over the resistor
(V_{R} =
V_{tot}). As such, the initial current
**I**_{0} can be found

I_{0} = V_{tot} / R

Charge will slowly accumulate on the plates of the capacitor, increasing
V_{C} and decreasing
V_{R}. As more and more of the potential
difference is spent over the plates of the capacitor, the current in the circuit will decrease
until the capacitor reaches its maximum charge, at which point the current ceases
completely.

One can obtain a quantitative description of both the current in the circuit
and the charge being stored on the plates of the capacitor at any point as a function of
time by applying Kirchoff's Law and recalling that current is equivalent to the rate of
charge transferred over a given amount of time (I = dQ/dt)

V_{tot} = IR + Q/C = R * dQ/dt + Q/C

Solving the differential equation, one finds that the charge

Q(t) = Q_{max} (1-
e^{-t/T})

Where **T** is the capacitor time constant

T = RC

The current flowing through the circuit is

I(t) =
I_{0}
e^{-t/T}

The other mode of operation for the circuit is that in which the charged
capacitor is decoupled from the battery and allowed to discharge over the resistor. In
this case, the capacitor will behave just like a battery, and its voltage
V_{C} will take on the role of the battery in
the circuit (V_{R} =
V_{C}). The charge on the capacitor will
bleed off into the circuit

Q(t) =
Q_{0}
e^{-t/T}

where Q_{0} is the initial charge. The
current will begin at a maximum value I_{0}
= V_{C} /R, and fall off according to the
relationship

I(t) =
I_{0}
e^{-t/T}

until all of the charge is gone, and the current falls to zero.

... All in all, this is basically one of those horrible nightmares that keeps
introductory physics students up with the cold sweats nights before the exam, but that
looks really really really really easy when you try it again four years later.

*(Really, honestly, I promise.)*