Deriving the RC Exponential

A plain resistor circuit gives you one equation with no calculus: V = IR, solve and done. Add a capacitor and the current isn't constant anymore, because the charge on the capacitor keeps changing, and the voltage across it depends on how much charge is already sitting there. The loop rule now has a term that depends on q and a term that depends on dq/dt in the same equation. That mix is exactly what a differential equation is: a statement about how a quantity and its own rate of change relate to each other.

Start with the loop: EMF - IR - q/C = 0. Current is the rate charge flows onto the plates, so I = dq/dt. Substitute and rearrange: dq/dt + q/(RC) = EMF/R. To solve it, separate the variables, putting every q on one side and every t on the other: dq/(EMF·C - q) = dt/(RC). Integrate both sides, using the starting condition q = 0 at t = 0 to pin down the constant. The result is q(t) = CV(1 - e^−t/RC), charge climbing from zero toward the full value CV, with RC setting how fast.

Look at the shape of that solution: the rate dq/dt is largest when q is farthest from its final value CV, and shrinks to zero as q approaches it. That same pattern — rate of change proportional to distance from equilibrium — is the entire content of radioactive decay (dN/dt = -lambda·N) and Newton's law of cooling (dT/dt proportional to T - T_room). Once you recognize the pattern, you don't need to rederive the exponential every time; you just match the physical setup to the same differential equation and reuse the solution.