I'm having a hard time finding the differential equation for enregy in a capacitor for an RC basic circuit which contains a resistor and a capacitor + the source .

I have to start using the Kirchhoff’s voltage law.

I know that :

$$E_c = \frac{1}{2}CU_c^2$$

Where $E_c$ is the energy stored in a capcitor and $U_c$ is the voltage across the capacitroe .

I tried to find the same equation for an RL circuits but I couldn't so how should I approach it ?

  • $\begingroup$ Is the problem that you don't know the equation for energy stored in an inductor, or that you don't know how to move from the energy equation to the differential form? $\endgroup$ Jan 24, 2017 at 20:47
  • $\begingroup$ @DaveCoffman I don't know how to move from th energy equation to a differential equation for the Energy Function . $\endgroup$ Jan 24, 2017 at 20:48
  • $\begingroup$ I'm afraid that it's been a few years since I've done this sort of problem, so I probably can't provide much help. One thing to try, though, is finding the equation for charge on the capacitor as a function of current, and then using the charge equation to find $U_c$, and thus $E_c$. $\endgroup$ Jan 24, 2017 at 20:54
  • $\begingroup$ -1. There is a lot of material available on the internet on AC theory and LRC circuits. Why aren't you making use of it? Or are you working from a textbook? $\endgroup$ Jan 25, 2017 at 2:21
  • $\begingroup$ Possible duplicate of Deriving ODE for voltage across capacitor-RC circuit or RC circuit energy. $\endgroup$ Jan 25, 2017 at 2:22

1 Answer 1


Since you're asking for a ode of the energy across a capacitor, lets proceed as follows:

We know that : $V = R.q' + \frac{q}{C}$ ($q' = \frac{dq}{dt}$)

Solving for q, we have:

$q = CV - R.C.q'$

Or $\frac{q^2}{2C} = \frac{CV^2}{2} + \frac{CR^2.q'^2}{2} - RCV.q'$

This is the required differential eqn. Since the lhs gives energy across a cpacitor with charge q.

  • $\begingroup$ I need to solve it for E (energy) not Q or U . $\endgroup$ Jan 25, 2017 at 11:01
  • 2
    $\begingroup$ Q^2/2C gives the capacitors energy at any instant $\endgroup$
    – Lelouch
    Jan 25, 2017 at 12:02

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