# LC Problem 37

Before the switch flips, the current in inductor $1$ is $\frac{E}{r}$, and the current in inductor $2$ is zero. After the switch flips, the basic equations are:

$V_C = L_1 \dot{I}_{L1} \\ V_C = L_2 \dot{I}_{L2} \\ -I_{L1} - I_{L2} = C \dot{V}_C$

Differentiating the last equation and substituting:

$-\dot{I}_{L1} - \dot{I}_{L2} = C \ddot{V}_C \\ - \frac{V_C}{L_1} - \frac{V_C}{L_2} = C \ddot{V}_C \\ \ddot{V}_C = - \Big( \frac{1}{L_1 C} + \frac{1}{L_2 C} \Big) V_C = -\omega^2 V_C$

This corresponds to SHM with angular frequency $\omega$. The general form is:

$V_C = A \sin( \omega t ) + B \cos( \omega t ) \\ \dot{V}_C = A \omega \cos( \omega t ) - B \omega \sin ( \omega t )$

Initial conditions:

$V_C (0) = 0 \\ \dot{V}_C (0) = - \frac{E}{r C}$

Applying initial conditions:

$0 = B \\ - \frac{E}{r C} = A \omega$

Finally,

$V_C = - \frac{E}{\omega r C} \sin( \omega t ) \\ \omega = \sqrt{\frac{1}{L_1 C} + \frac{1}{L_2 C}}$

So the max value of $V_C$ is:

$V_C (max) = \frac{E / (r C)}{\sqrt{\frac{1}{L_1 C} + \frac{1}{L_2 C}} }$

Knowing the expression for $V_C$, the expression for $I_{L2}$ can be calculated fairly easily by integrating:

$V_C = L_2 \dot{I}_{L2} \\ \dot{I}_{L2} = \frac{V_C}{L_2}$

Note by Steven Chase
1 week, 1 day ago

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Sort by:

@Lil Doug Here it is

- 1 week, 1 day ago

@Steven Chase it was easy, Thanks

- 1 week, 1 day ago

@Steven Chase how do you mention my name???

- 1 week, 1 day ago

I just typed the "@" symbol, started typing your name, and let the auto complete finish it

- 1 week, 1 day ago

@Steven Chase when I do this in a new note made by me, that thing doesn't work

- 1 week, 1 day ago

@Steven Chase have a look in this problem, can we try analytically.

- 1 week, 1 day ago

I've got expressions for the velocities. It will take some time to write up, so I'll do it tomorrow.

- 1 week, 1 day ago

@Steven Chase sir, two rod (one is conducting and another one not conducting) are rotating about their end points , what is the emf induced in them?

- 1 week, 1 day ago

@Steven Chase whenever you have free time, take a look in this problem
As always, Thanks in advance :)

- 1 week, 1 day ago

@Steven Chase I am not able to solve the 36th problem.

- 1 week, 1 day ago

@Steven Chase sir I am not able to solve problem 31. :(
Sir last night you said you will write the solution of above problem (particle of charge q and mass m.............)
Still it is upto you only, because “You didn't owe me anything”
But there is still request :) Bye.

- 1 week ago

Yes, I still plan to post a solution to the damped charge problem. Probably tomorrow. It is a very tedious analytical solution

- 1 week ago

@Steven Chase are you the solver of mechanics problem (spaceship mechanics)?

- 1 week ago

@Lil Doug Hey if you dont mind me asking, which book do you get these problems from? Your latest electrostatic one, or that spaceship one?

- 6 days, 7 hours ago

@Aaghaz Mahajan Physics pathfinder, Irodov, Krotov .

- 6 days, 3 hours ago

Oooh thanks!

- 5 days, 21 hours ago

@Steven Chase sir is it a coincidence that when you lost a huge latex work, at that time only staff posted a new discussion for grabbing attention.?

- 1 week ago

I think it is just a coincidence

- 1 week ago

@Steven Chase you are a very lucky guy :)

- 1 week ago

@Steven Chase if you are free you can help me in 31 th problem.
In my opinion you can use pen and page also.
If you use that, staff will feel bad,you are ignoring latex due to that lost work.
So he will try to fix the problem as fast as possible. Nice trick.

- 1 week ago

I doubt it. The community section is not a high priority anymore. Almost all of the focus is on course content

- 1 week ago

@Steven Chase what does course content mean?
Does it mean the content which Brilliant provide to premium member.
I think community section should have the most priority, because a guy named mention[1247521:Steven Chase] who has exceptional skills to solve a problem is on there.

- 1 week ago

Yeah, its the subject based course content. I don't think you have to have Premium to see it though

- 1 week ago

@Steven Chase Sir I need your help in 36th problem. Please the way you use to write solutions 3 to 4 years ago like using pen and paper , then scanning the page.
Please use that technique only, it is very easy and convenient and fast to provide a solution.
Hope I am not disturbing you.

- 6 days, 21 hours ago

I can upload the differential equations that describe the system, and you can take it from there. Will that work?

- 6 days, 19 hours ago

@Steven Chase Yes it will work :)

- 6 days, 19 hours ago

@Steven Chase I am. Just confused in rotating the velocity vector by $2π$ ?
How to deal with that?

- 6 days, 19 hours ago