Deriving the damped circuit equation and energy dissipated through a resistor.

[LogicX]

Homework Statement

I figure I will just combine these two questions into one topic.

1) The energy stored in a capacitor is .5CE^2, where E is the voltage. Wat is the instantaneous power dissipated in a resistor R through which this capacitor discharges? Show that the total energy dissipated through the resistor is exactly .5CE^2

2) E=R dq/dt + q/C Show that the solution to this equation is q(t)=EC (1-e^(-t/RC))

Homework Equations

I=dq/dt=-Q/RC e^(-t/RC)

Voltage across a resistor in a series RC circuit: V=Ee^(-t/RC)

Both problems require calculus.

The Attempt at a Solution

For number 1... P=i^2 R ? I'm not sure what to do with this.

The second question I am just totally clueless about. I've never seen a derivative taken where you end up with e when you didn't start with e. I think this may be another example of "hey, we are going to give you this question to struggle with even though you have never learned the math. Good luck on the exam."

This is my last ditch attempt before my exam tomorrow.

 

[Quinzio]

I'll write the solution for 2) as you say it's out of your reach.

$$E=R \ dq/dt + q/C$$

$$E - {q \over C} = R \ {dq \over dt}$$

$$dt = R \ {dq \over (E - {q \over C})}$$

$$\int_0^{\infty} {dt} = \int_0^{\infty} R\ { dq \over {(E - {q \over C}) } }$$

$$t = -{RC} \ ln {(E - {q \over C})} +K$$

Then with some passages you arrive to the final solution.