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Homework Help: Solenoids and Toroids

  1. Jun 28, 2008 #1
    Hi everyone.
    I need help please.
    I am working on problems with solenoids and Toroids
    I have solution for the solenoid:
    B = μo i n

    And toroid:
    B = (μ o i n)/ (2Π r)
    Also, I know that the magnetic field is the function of r namely: B = B(r)

    r- radius of the Ampere’s path
    n – number of loops per unit length
    μo – constant

    My problem is:
    Using the solution for the toroid, show that for the large toroid the answer can be approximated as the solenoid on the very small piece of the toroid.
    I know that I have to play with limits. Something like:
    a - inner radius of the toroid,
    b – outer radius of the toroid,
    ∆a - the difference between radius a and radius b.
    I think I have to take a limit when ∆a goes to 0 and in this way radius a will approach radius b. in this way the solution for the toroid SHOULD be the solution for the solenoid (on the small length L of course)
    I don’t know how to set it up. How to get from the toroid solution to the solenoid solution using limits or (other technique)

    Thanks for help
    Chris W
  2. jcsd
  3. Jun 28, 2008 #2


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    You should be more specific. The B fields you gave are the B fields evaluated where exactly? And your two equations can't both be right since they don't even have the same dimensions. There is something missing in your toroid expression.

    (Hint: the two "n" do not have the same meaning)
  4. Jun 28, 2008 #3
    Thanks nrqed

    hmmm... I see it now... yeah...

    -ok the calculated B field is in the toroid between the radius a and radius b.
    -as for the meaning of n?? Yeah you are correct! there are two different n's

    -I checked my math and I see now that the correct values for the B fields are:

    for the Solenoid


    n - # of turn/ unit length
    μo - constant (mu subscript o)

    For the Toroid:


    also, the strength of the B field id the function of r: B=B(r)

    r - radius to the Ampere's Path a<r<b
    N - # total number of loops in the toroid

    I was thinking about it and looks like I can say that:

    a- inner radius of the toroid
    b- outer radius of the toroid
    (delta a) = b-a
    r - radius of the Ampere's Path a<r<b

    b=(delta a)+a

    limit when (delta a) goes to 0, than a=b
    and I am stuck here...

    The goal is to use the solution of B field for the toroid ( INSIDE the toroid) and by taking the correct limit (or using other method) to obtain the solution for the B field of the solenoid INSIDE the SOLENOID.
    So basically when (Delta a) is very small (limit goes to 0) over the small, call it; ds over solution looks like the regular, finite, straight solenoid with the B field inside.

    Once again thanks a lot nrqed
    Your comment opened my eyes a little bit.
    I am still stuck, but I feel I am a step closer.

    Any ideas? Need better explanation? I can derive the B fields for the solenoid and toroid if needed.

    Please help!!!!!
    I can be more specific if needed.

    Chris W
  5. Jun 28, 2008 #4


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    It's much simpler than this. In the limit r>>a and r >>b, you can basically say that the length of the toroid is 2 pi R, right? (That's the length of the path for Ampere's law but that's basically the length of the toroid as well in the limit r much larger than a and b).

    Now, the answer falls off directly (paying attention to the difference between "n" appearing in the solenoid formula and the "N" appearing in the toroid formula).
  6. Jun 28, 2008 #5
    WOW.... great

    This is easy!!

    Thank you so much !!!!

    you rock man!

    Chris W
  7. Jun 28, 2008 #6


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    :biggrin: You are very welcome!
  8. Jun 29, 2008 #7
    Hi all. Hi nrqed!
    I Think I have a better way to do it!!!!!
    hmmm... I was thinking about it and I think I see another way to do it.

    the objective is to prove that the solution for the B field of toroid (with small delta a) is the solution of the solenoid (for the B field inside)

    Also, I know that


    so putting all together we have:

    N= nAL n=N/AL

    B = (μo i N)/(2 Π r) B = μo i n
    B = (μo i n A L)/(2 Π r) B = (μo i N)/(A L)


    o i n A L)/(2 Π r)=(μo i N)/(A L)

    o i n A L)/(2 Π r)=(μo i N)/(A L)

    ( n A L)/(2 Π r)=( N)/(A L)

    N= nAL
    AL = 2 Π r
    This is as far as I can go

    But… if I say that AL=1 ( and I don't know WHY I can say that...lol)

    ( n )/(2 Π r)=N

    N - # total number of loops in the toroid
    n - (# of loops)/(unit length)

    let’s check the units

    N = (# of loops)/(unit length) * unit length= # of loops ≡ N

    I hope I am doing this right.
    Can someone please check it?
    Nrqed any input???


    Chris W
  9. Jun 29, 2008 #8
    This section is of course:

    N= nAL


    it came out together ....sorry
    but N is defined above so there shouldn't be any problems
  10. Jun 29, 2008 #9
    Same here:

    Toroid Solenoid
    B = (μo i N)/(2 Π r) B = μo i n
    B = (μo i n A L)/(2 Π r) B = (μo i N)/(A L)
  11. Jun 29, 2008 #10
    Man... editing in this form is a challenge

    B = (μo i N)/(2 Π r)
    B = μo i n

    B = (μo i n A L)/(2 Π r)
    B = (μo i N)/(A L)
  12. Jun 29, 2008 #11
    I hope you can follow my math here ....
    anyone? please... is this correct?
    Chris W
  13. Jun 29, 2008 #12


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    I don't follow. I don't even know what A is!

    The only way to do the problem that I can see is what we discussed yesterday. The only key point is that you must define n= N/(length of the toroid) in order to compare with the solenoid result. But of ocurse, the length of a toroid is not well-defined.

    Yesterday, I should NOT have written r>>a and r>>b, that was a mistake. Really, we are taking the limit r>>(b-a) (which also implies a>> (b-a) and b>>(b-a) . I apologize for this mistake.

    For a toroid we can define the "inner length" 2pi a, or the "middle length" 2pi r or the outer length 2pi b. In the above limit, they are equal to one another,

    For example, write r = a + (b-a)/2 Then clearly 2pi r approaches 2pi a in the limit above

    [tex] 2 \pi r = 2 \pi (a + \frac{a-b}{2}) \approx 2 \pi a [/tex]

  14. Jun 29, 2008 #13
    Thanks Patric!

    Yeah I noticed that something is wrong with r>>a and r>>b.
    All good man!


    yeah a good sport!

    Chris W
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