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Help needed for a magnetism problem which is somewhat disturbing

  1. Aug 17, 2014 #1
    1. The problem statement, all variables and given/known data

    the problem is as: a very long straight solenoid has a cross section R and n turns per unit length . A direct current I flows through the solenoid . Suppose that x is the distance from the end of the solenoid , measured along its axis:
    (a) the magnetic induction B on the axis as a function of x.
    (b) the distance x' to the point on the axis at which the value of B differs by η=1% from that in the middle point of the solenoid.

    2. Relevant equations
    B=μnI[cosβ-cosω]/2
    is what I use for the (a) part , by putting the value β and ω where β is angle made by the axis and the end part of the solenoid and ω is the angle made by the axis and starting part of the solenoid.
    And n is the number of turns [per unit length.


    3. The attempt at a solution
    I am cool with the first part of the question. But the problem occurs with the second part.
    I start with the second part as:
    magnetic field at the centre of the solenoid = μnI[cos 0 - cos 180]/2=μnI
    ( What I think that for a long solenoid both the angles tend to 0 and 180 respectively for β and ω
    )
    1% deviation from this value is asked thus-
    μnI -μnI/100=μnI[cosβ-cosω]/2
    99μnI/100=μnI[b/rt.(b^2+x^2)+a/rt.(a^2+x^2)]
    now what is x'-----not clear
    is x' related to a and b (distance of that point from front and end respectively )----not clear

    so thats it .
    mooncrater
     
    Last edited: Aug 17, 2014
  2. jcsd
  3. Aug 17, 2014 #2

    BvU

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    Strange that you are happy with a) and don't need R there. (does R have the dimension of length or of area ? usually we have R for radius, D for diameter and something like S for cross section area).

    My hunch is you do need R and also the length of the solenoid in b).

    To you x' is not clear, but for me as a reader, I can't make out what a and b are, either. Could you explain? And perhaps make a drawing ?

    [Edit] Ah! distance to front and end. And rt means square root. Try the Go Advanced button; it has a √ symbol at the ready....

    By the way, the μnI can be divided out to get a clearer perspective ... then the whole thing is just a simple gonio exercise.
     
  4. Aug 17, 2014 #3

    BvU

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    Oh, and there is something wrong with the expressions for cosines. If you make a drawing, you'll see it. Hint: don't use a and b but use x' - L/2 and x' + L/2 . And be sure to maintain the - sign .
     
  5. Aug 17, 2014 #4

    BvU

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    Oh 2: just to get an impression, check how much B has dropped already at the end of the coil: ##\beta = \pi/2##, ##\omega = \pi##, gives half the B at the center ! So our x' should be well within the coil!
     
  6. Aug 18, 2014 #5
    oh 1: correction

    you are right i am mistaken in writing the distance version of the formula , I think the correct one is ;
    B=μni[b/√(b2+R2)-a/√(a2+R2)]
    where R is the radius of cross section of solenoid.
     
    Last edited: Aug 18, 2014
  7. Aug 18, 2014 #6

    BvU

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    Apart from the factor of 1/2, yes.
    The choice of coordinates suggested in the a) part of the exercise is somewhat awkward ("x is the distance to the end of the solenoid -- measured along its axis").
    It is good to read you are comfortable with that part of the exercise.
    1. What exactly is the answer you came up with ?
    2. Does it match with my wild hunch in post #4, B = μnI/2 when x = 0 ?
    3. And at what x is B = μnI ?

    For part b) you now have two variables, a and b, where there is only one degree of freedom. Some way or other, you want to write one equation with one unknown.

    Did you make a drawing ? If so, can you share it ?
     
  8. Aug 18, 2014 #7
    the drawing is attached
     

    Attached Files:

  9. Aug 18, 2014 #8
    see , the problem I am facing is that what is x' actually-
    1) distance from the edge of the coil ?
    2) distance from the centre of the coil?
    i came up with the answer-
    b/√(b2+R2)-a/√(a2+R2)=198/100
    and now I am stucked with converting a and b into x' , and while thinking about it's solution one can also think about the length of the solenoid , but the only thing about the solenoid's length given is that it's LONG. so, it's clear we can't use it's length for deriving the result we want.
     
  10. Aug 19, 2014 #9

    BvU

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    We will need the length of the solenoid. Call it L, remove b and use a + L instead, and continue.
    Usually people take the center of the coil as coordinate system origin, but in the a) part of the exercise a clear choice was already made to make x = 0 when a = 0 (or b = 0). As you have already calculated in the a) part, B is μnI/2 there (please confirm) for an infinitely long coil. So not the 99% of μnI.

    To answer b) properly, drop the infinite length and use L to get an expression for B(x'). Solve for B(x')= 0.99 B(L/2) (the latter is slightly less than μnI, which may or may not make a difference...).

    Note that μnI can be divided out, leaving a simple goniometric equation.
     
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