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mathslover

  • one year ago

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)

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  1. mathslover
    • one year ago
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    @ganeshie8 @Miracrown @iambatman @dan815

  2. mathslover
    • one year ago
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    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|

  3. mathslover
    • one year ago
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    While the correct answer is "zero" Can anyone help me with this?

  4. mathslover
    • one year ago
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    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?

  5. mathslover
    • one year ago
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    Ohk! Take your time buddy.

  6. mathslover
    • one year ago
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    What I'm thinking is, we have to calculate current through AB.... so, that should make a difference, will it? @dan815

  7. mathslover
    • one year ago
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    Yeah... and perpendicular to area vector is parallel to B vector.. ain't it?

  8. mathslover
    • one year ago
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    Ouch! :P The diameter has no surface, that's why..?

  9. mathslover
    • one year ago
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    Oh, so, even if it was for the whole circular loop, the answer would have been zero..?

  10. dan815
    • one year ago
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    starting over lol

  11. mathslover
    • one year ago
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    uhmm...? :/ confused

  12. dan815
    • one year ago
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    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

  13. abb0t
    • one year ago
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    @Jhannybean

  14. mathslover
    • one year ago
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    It simply means - \(\cfrac{dB}{dt} = - \alpha \) Now if you differentiate it again, it will give you "zero" ... So, it is decreasing constantly !

  15. mathslover
    • one year ago
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    Ignore that "-" before dB/dt

  16. dan815
    • one year ago
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    are u sure, the way the wrote it can be interpreted both ways

  17. mathslover
    • one year ago
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    "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

  18. dan815
    • one year ago
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    |dw:1433313628013:dw|

  19. dan815
    • one year ago
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    yah there is a current induced in the outerloop

  20. dan815
    • one year ago
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    but in the center wire i think the current tries to flow both ways and cancels out u have no flow

  21. mathslover
    • one year ago
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    Why both ways? :/

  22. dan815
    • one year ago
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    |dw:1433313980505:dw|

  23. dan815
    • one year ago
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    it always wants ot go in counter clockwise or clockwise whatever result u got for the outerloop

  24. dan815
    • one year ago
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    but if the top part wants to go clockwise and bottom loop too , and the total current is equal everywhere

  25. dan815
    • one year ago
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    there u must have no flow in the middle

  26. mathslover
    • one year ago
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    Oh WOW! Excellent. Thanks a lot Dan! :)

  27. dan815
    • one year ago
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    sure thing :D

  28. dan815
    • one year ago
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    make sure u understand why this argument works only for electromagnetic induced currents

  29. dan815
    • one year ago
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    normally this wont happen in a circuit

  30. dan815
    • one year ago
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    but since its EMF induced they wanna maintain that clockwise or ccw flow

  31. mathslover
    • one year ago
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    Yeah, I understand that point. Thanks again! :)

  32. dan815
    • one year ago
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    ok also u can show it mathematically

  33. dan815
    • one year ago
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    if u calculate the 2 semicircle loops separetely

  34. dan815
    • one year ago
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    |dw:1433314421990:dw|

  35. dan815
    • one year ago
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    u can solve for each semi circle given that B field

  36. dan815
    • one year ago
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    and ull end up with zero again

  37. dan815
    • one year ago
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    cuz again ull get the current flow in opposite directions, which is more rigorous

  38. mathslover
    • one year ago
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    I see! Thanks for making it clear to me from every angle. :) Mr. Lifesaver :D

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