Calculate the capacitance of a parallel plate capacitor with plate area A and distance between plates d, when filled with a dielectric whose dielectric constant varies as
K[x]=e0 + beta* x , 0<x<d/2
K[x]= e0 + beta *[d-x] , d/2<x<d
For what value of beta would the capacity of the capacitor be twice that when it is without any dielectric.
My problem is that how should I introduce the small change in x i.e. dx .
In my disussion i mean k=K[x]
and e0 means that stupid epsilon
Well chandni its really easy!!!!!!!!!
divide the CAPACITOR in numerous capacitors which r in series. Hope u got it da.Basically the reason for dividing my capicitor like this is that i would have a series arrangement
Now the effective capacitance is
1/C=integral[1/dc]
(i hate it when we cant type integral symbols)
Now dc is ke0A/dx after this substitution u see that ur dx comes to the numerator and that u can integrate ur expression.
Well this the dumbest method method of solving things,
which does lead u to the answer.The answer is C=beta*e0A/(2ln[(e0+beta*d/2)/e0])
Another method is by ur way i.e introducing a small dx but not considering dc
Well its like this girl......
Assume a conjugate pair charge on ur plates and find the work done in moving a unit chage from the -ve plate 2 the positive plate.
U should be knowing that whatever be the medium the electric feild lines are decreased by a factor of k to those in air.This wala has an experimental proof
hence the W.D is integral(F.ds) here force on the unit charge is the feild itself hence integral(E.dx)
Now E is Q/kAe0 . now as theres a discotinuity in the variation of k , split up ur integral at those pts and integrate.(dont forget 2 write
K[x]=e0 + beta* x , 0<x<d/2
K[x]= e0 + beta *[d-x] , d/2<x<d
Now capaciance is simply Q/P.d
the P.D itself being the W.D in moving a Unit Charge from -ve plate 2 the +ve plate.
AND EUREKA!!!!!!!!!!!!!!!!!!
this time instead of saying i am gr8, which i already am as a stupid
,its enough if u just thank me for wastin my time typin my solution QUALITATIVELY.
Well u could thank me by becoming a member of orkut.
AND about that integration pbm i was not aware parameter differetiation thats reason i was not bl 2 crack it any ways thanks for takin pains 2 prsnt ur solution twice
Well the 1st method is dumb but it really helps to arrive at the solution
Manoj,thanks a lot for replying and giving the solution.I am really very sorry that I am giving my reply so late.Congratulations for getting average as 60% in the tests taken over whole year. I am impressed by how you approach to solve physics problems.Concept knowing is another thing but applying them is what is really important .
Avdesh, thanks to you also for taking interest in my problem.
ya thanks i needed that!!! My morale was too low as i couldnt beat my sis in the sunday test yesterday at my place i bitterly needed a morale boost!! Oh and about my tests those r useless!!! i hardly get 40 in chem cos u see i hate chem as equal as i hate my sis gettin ahead of me!! all my marks r due 2 93-80 in phy and 70 -90 in maths and dont worry my name will soon be out of the list as i have just written 4 chem test and got around 45-50 in all 4 of em !! i JUST HATE chem and my sis , she even got selected for the second phase of chem olympiad ! shes a walkin chem and physics genius!! My physics talents r due to my sis. We both study in the same institute and we have a hell of a competition with each other!! Shes goin 2 write her iit along with me, I hope i really get ahead of her !!! I was just tellin u all this cos i just wanna say my sis is the secret of my sucesssssss!!!!!!!!!!
Oh and there is another method for solving this It was invented by the person you know ............ its me
But you wont be needing it though ,i think its out our scope anyways i'll tell you later!!!!!!! i know you'll definitely make it into the IIT and me!!!! may be , so if we meet there i'll tell you this method !!!! its really an intresting one but lengthy though!!! So if you wanna be friend all you got to do is mail me!!! Well i am dying to make friendship with someone who really appreciates me!!!! This is because Though i am an achiever my sister is an over achiever so she gets all the praise !!!!!!!I really need some morale boost to clear the IIT
I think grades only give the correct measurement of your potentials and my performance is always between 45-61%,since 61% is the maximum I have got till now and minimum, just don't ask about that. Anyways,I will try my best and you also try for your best.
let a small element dx verticlally here dC will be addition of two elementary capacitors in series in two different dielectric slabs,
then find 1/dC and integrate with limits (0 to d), to add all parrallel elementary elements to get the answer.
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