mathslover
  • mathslover
The radius of the circular conducting loop is R. Magnetic Field is decreasing at a constant rate \(\alpha\) (inwards to the plane of loop). Resistance per unit length of the loop is \(\rho\). The current in wire AB is? (AB is one of the diameters)
Physics
katieb
  • katieb
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mathslover
  • mathslover
mathslover
  • mathslover
This is what I've tried : \(i = \cfrac{1}{r} \cfrac{d\phi}{dt} = \cfrac{1}{r} \cfrac{S dB}{dt} \) where \(\phi\) is magnetic flux. \(r\) is resistance. \(S\) is the area A of the circular loop. I've been given with : \(\cfrac{r}{2 \pi l } = \rho \implies r = \rho \times 2 \pi l= \rho \times 2 \pi (2R)\) and \(S = \pi (R^2)\) So, we have : \( i = \cfrac{1}{\rho \times 4 \pi R} \times \pi (R^2) \times (- \alpha) \) \(i = \cfrac{R \alpha}{4 \rho } \) And direction of current will be from B to A |dw:1433312077316:dw|
mathslover
  • mathslover
While the correct answer is "zero" Can anyone help me with this?

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mathslover
  • mathslover
Sorry, didn't get you. We have \(\phi = \bf{B} . \bf{S} \) Right? And the angle between S and B is 0 degrees. So, why would it be zero?
mathslover
  • mathslover
Ohk! Take your time buddy.
mathslover
  • mathslover
What I'm thinking is, we have to calculate current through AB.... so, that should make a difference, will it? @dan815
mathslover
  • mathslover
Yeah... and perpendicular to area vector is parallel to B vector.. ain't it?
mathslover
  • mathslover
Ouch! :P The diameter has no surface, that's why..?
mathslover
  • mathslover
Oh, so, even if it was for the whole circular loop, the answer would have been zero..?
dan815
  • dan815
starting over lol
mathslover
  • mathslover
uhmm...? :/ confused
dan815
  • dan815
okay so, let me see if i got this question right, does it mean mag field is increasing over time, or its decreasing constnatly spatially as u get closed to the center
abb0t
  • abb0t
mathslover
  • mathslover
It simply means - \(\cfrac{dB}{dt} = - \alpha \) Now if you differentiate it again, it will give you "zero" ... So, it is decreasing constantly !
mathslover
  • mathslover
Ignore that "-" before dB/dt
dan815
  • dan815
are u sure, the way the wrote it can be interpreted both ways
mathslover
  • mathslover
"According to the Lenz's law emf's of the same magnitude in the clockwise direction are induced in the two loops into which the figure is divided. So, current is induced in the clockwise direction in the outer boundary but no current in wire AB" This is what I got when I searched (just now) on google... https://books.google.co.in/books?id=zlWRMWEcJG0C&pg=PA349&lpg=PA349&dq=The+radius+of+the+circular+conducting+loop+is+R.+Magnetic+Field+is+decreasing+at+a+constant+rate&source=bl&ots=saTfbsAdfw&sig=cETI6m4U9WxnEfIW1hiHEJlQLJU&hl=en&sa=X&ei=U6FuVeq0CtGUuATVmIKwBQ&ved=0CB4Q6AEwAA#v=onepage&q=The%20radius%20of%20the%20circular%20conducting%20loop%20is%20R.%20Magnetic%20Field%20is%20decreasing%20at%20a%20constant%20rate&f=false
dan815
  • dan815
|dw:1433313628013:dw|
dan815
  • dan815
yah there is a current induced in the outerloop
dan815
  • dan815
but in the center wire i think the current tries to flow both ways and cancels out u have no flow
mathslover
  • mathslover
Why both ways? :/
dan815
  • dan815
|dw:1433313980505:dw|
dan815
  • dan815
it always wants ot go in counter clockwise or clockwise whatever result u got for the outerloop
dan815
  • dan815
but if the top part wants to go clockwise and bottom loop too , and the total current is equal everywhere
dan815
  • dan815
there u must have no flow in the middle
mathslover
  • mathslover
Oh WOW! Excellent. Thanks a lot Dan! :)
dan815
  • dan815
sure thing :D
dan815
  • dan815
make sure u understand why this argument works only for electromagnetic induced currents
dan815
  • dan815
normally this wont happen in a circuit
dan815
  • dan815
but since its EMF induced they wanna maintain that clockwise or ccw flow
mathslover
  • mathslover
Yeah, I understand that point. Thanks again! :)
dan815
  • dan815
ok also u can show it mathematically
dan815
  • dan815
if u calculate the 2 semicircle loops separetely
dan815
  • dan815
|dw:1433314421990:dw|
dan815
  • dan815
u can solve for each semi circle given that B field
dan815
  • dan815
and ull end up with zero again
dan815
  • dan815
cuz again ull get the current flow in opposite directions, which is more rigorous
mathslover
  • mathslover
I see! Thanks for making it clear to me from every angle. :) Mr. Lifesaver :D

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