How do I find di_L(0)/dt for a second order RLC circuit after the switch opens?

[Jebus_Chris]

http://imgur.com/6aAMV

So I need to find the current labeled as a function of time. THe switch opens at t=0 and I drew the circuit after the switch opens.

I found the initial current to be -4 and the voltage on the capacitor to be 8.


I'm having trouble trying to find $di_L(0)/dt$. I know you just need to do a simple loop around the whole circuit, $v_c+v_l+v_r=0$. The problem is I don't know the signs of the voltages, or rather I don't know what they should be. I get either $di_L(0)/dt = -40/10, -24/10$.
Same for finding the second order eq.
$1/c\int i_c+ L di_L/dt + 8 i_L = 0$ Simple loop around the outside. Are signs correct?
and
$i_c - i_L - V_2/2 = 0$ Node equation at the top node, V is equal to the voltage across the inductor and 8 ohm resistor so substitute

$i_c - i_L - 1/2 (Ldi_L/dt + 8i_L)=0$
Substitute the third into the first, do algebra, and find answer.
That part should be easy. What I need to know is whether the signs are correct. That is, when doing kichoff mesh and nodes how do you know what is positive and negative [or rather what side the element should be placed on]?

Thanks

 

[gneill]

The signs of the potential changes are determined by the assumed direction of the current. The components of capacitance and inductance have their equations defined assuming a current directed into the component and the "+" reference at the terminal where the current enters.

Note, for example, that in the above figure for the inductor that a small positive going change in current "dI" results in a positive change in the voltage across the inductor, thus giving the equation V = L dI/dt. Similarly for the capacitor, a positive current flowing into the capacitor increases the potential across the capacitor over time.

For this problem I would probably solve it using a state variable (Laplace Transform) approach, since it handles all the initial condition details easily. Having a good table of Laplace transforms helps

 

[Jebus_Chris]

I think I finally understand this + and -.
I picked the current going to the right around the loop. Labeled each element with a plus and minus, current entered the plus so each one is positive. Blah blah balh, it all worked out.