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:

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

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Lil Doug Here it is

Steven Chase - 9 months, 1 week ago

@Steven Chase it was easy, Thanks

Talulah Riley - 9 months, 1 week ago

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

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

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

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

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

@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?

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

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

@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.

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

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

Talulah Riley - 9 months 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?

Aaghaz Mahajan - 9 months ago

@Aaghaz Mahajan Physics pathfinder, Irodov, Krotov .

@Talulah Riley – Oooh thanks!

@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.?

I think it is just a coincidence

Steven Chase - 9 months ago

@Steven Chase you are a very lucky guy :)

Actually, that was a part 2 of the original note. So it is most likely a coincidence, that the part 1 note was filled with so many comments that a part 2 note was created exactly when Mr. Chase lost his latex.

A Former Brilliant Member - 8 months, 2 weeks 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.

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

@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.

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

@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.
Thanks in advance.
Hope I am not disturbing you.

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

@Steven Chase Yes it will work :)

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

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$