LC Problem 37

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

VC=L1I˙L1VC=L2I˙L2IL1IL2=CV˙CV_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:

I˙L1I˙L2=CV¨CVCL1VCL2=CV¨CV¨C=(1L1C+1L2C)VC=ω2VC-\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 ω \omega . The general form is:

VC=Asin(ωt)+Bcos(ωt)V˙C=Aωcos(ωt)Bωsin(ωt) 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:

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

Applying initial conditions:

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

Finally,

VC=EωrCsin(ωt)ω=1L1C+1L2C 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 VC V_C is:

VC(max)=E/(rC)1L1C+1L2C V_C (max) = \frac{E / (r C)}{\sqrt{\frac{1}{L_1 C} + \frac{1}{L_2 C}} }

Knowing the expression for VCV_C , the expression for IL2 I_{L2} can be calculated fairly easily by integrating:

VC=L2I˙L2I˙L2=VCL2V_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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@Lil Doug Here it is

Steven Chase - 1 week, 1 day ago

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@Steven Chase it was easy, Thanks

Lil Doug - 1 week, 1 day ago

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

Lil Doug - 1 week, 1 day ago

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

Steven Chase - 1 week, 1 day ago

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

Lil Doug - 1 week, 1 day ago

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

Lil Doug - 1 week, 1 day ago

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

Steven Chase - 1 week, 1 day ago

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@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?

Lil Doug - 1 week, 1 day ago

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

Lil Doug - 1 week, 1 day ago

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

Lil Doug - 1 week, 1 day ago

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@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.

Lil Doug - 1 week ago

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

Steven Chase - 1 week ago

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

Lil Doug - 1 week ago

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@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 - 6 days, 7 hours ago

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@Aaghaz Mahajan Physics pathfinder, Irodov, Krotov .

Lil Doug - 6 days, 3 hours ago

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@Lil Doug Oooh thanks!

Aaghaz Mahajan - 5 days, 21 hours ago

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@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.?

Lil Doug - 1 week ago

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I think it is just a coincidence

Steven Chase - 1 week ago

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@Steven Chase you are a very lucky guy :)

Lil Doug - 1 week ago

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@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.

Lil Doug - 1 week ago

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

Steven Chase - 1 week ago

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@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.

Lil Doug - 1 week ago

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

Steven Chase - 1 week ago

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@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.

Lil Doug - 6 days, 21 hours ago

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

Steven Chase - 6 days, 19 hours ago

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@Steven Chase Yes it will work :)

Lil Doug - 6 days, 19 hours ago

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

Lil Doug - 6 days, 19 hours ago

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