# Differential Equation Of Capacitor Energy In RC and RL Circuits?

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 ?

• 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? Jan 24, 2017 at 20:47
• @DaveCoffman I don't know how to move from th energy equation to a differential equation for the Energy Function . Jan 24, 2017 at 20:48
• 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 $E_c$. Jan 24, 2017 at 20:54
• -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? Jan 25, 2017 at 2:21
• Possible duplicate of Deriving ODE for voltage across capacitor-RC circuit or RC circuit energy. Jan 25, 2017 at 2:22

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.

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