Solving the Equation for a Simple LC Circuit

[serverxeon]

In a simple LC circuit,

How do i set up the differential equation?

Following Walter Lewin's teaching that
0) Use Maxwell-Faraday Equation instead of Kirchhoff Rule
1) if E field and dl are in the same direction, then I write a positive term.
2) There is no electric field in an inductor
3) Write the induced EMF on the right as -L dI/dt

In the image above, current will run clockwise, so I move my dl clockwise.
I'll arrive at the following equation.

-Vc = - L dI/dt

which is incorrect. The correct equation only has negative on either side.
What is wrong with my steps?

The steps has served me well for all other types of circuits.. Just LC which is giving me a problem

 

[BruceW]

I'm not familiar with these steps. But I'll try to help. Um. So dl is clockwise, and E is clockwise too. So you write a positive term.. which term should that be?

 

[serverxeon]

when coming to the capacitor,
the E field inside the cap is pointing down.
But my dl is going up.
that gives a negative term

 

[BruceW]

ok, I haven't used that method myself. It seems to have worked though. Vc = L dI/dt is correct, I think. For example, the current will start to flow clockwise, so taking clockwise as positive, dI/dt is positive, (and of course L is positive), and Vc is positive in the clockwise direction, so you have an equation containing all positive terms. It looks good to me.

 

[serverxeon]

Vc = L dI/dt is incorrect!

One of the terms need to be negative, so that when I shift all to one side I get a SHM D.E.!

I can't make out where the negative went!

 

[BruceW]

ah, yes, you're right. one of the terms must be negative. I can only think that one of the steps in the method went wrong. I have not used this method before, so I am not sure what it is supposed to be. But I think maybe it is because step 3) is incorrect. I found in a pdf online about Walter Lewin's lectures that: "If an inductor is traversed in the direction moving with the current, the change in potential is -l dI/dt; if it is traversed in the direction opposite the current, the change in potential is +L dI/dt" So maybe this is where you went wrong?

 

[serverxeon]

ah, thanks for the pointer to the pdf.

in his document, I do see the 'incorrect' equation shown (11.5.4)
(Well that means it isn't incorrect!)

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From my understanding, the negative indicate that the charge on the cap is decreasing with time
I should be subbing I= - dQ/dt, which will hence make the term positive.

does that explanation sounds right to you?

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Den it comes the problem why would I not need to flip signs in equations for RC circuits... hmm

 

[BruceW]

yeah, It seems that here, he is defining Q and I to be strictly positive quantities. And in this case, the capacitor is going to be losing its charge, therefore you have to adopt the convention I = - dQ/dt for this case, so that both quantities can be positive. And then 11.5.4 follows from this definition.

In the RC case, again the cap is losing its charge, so I = - dQ/dt and the equation for an RC circuit is Q/c = IR from here, you can work out the answer for an RC circuit. So there is still 'sign flipping' going on here. Or maybe he gives a different method for RC circuits?