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Find the magnetic field at the point

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  1. Mar 27, 2017 #1
    1. The problem statement, all variables and given/known data
    A long, straight wire lies along the z−axis and carries a 4.20 −A current in the +z−direction. Find the magnetic field (magnitude and direction) produced at the following points by a 0.400 −mm segment of the wire centered at the origin.

    2. Relevant equations

    ## \vec B ## = 2 ## \frac {μ_0} {4π} ## ## \int_0^.0004 ## ## \frac {I*d\vec s \times \hat r} {r^2}##

    x = 2 meters, y = 2meters, z=0

    ## d\vec s \times \hat r ## = ds*sin(Θ)

    3. The attempt at a solution
    I took the original equation, which is from the Biot-Savart Law, and simplified the cross product using the algebraic definition above, such that the problem simplifies into:

    ## \vec B ## = 2 ## \frac {μ_0 I} {4π} ## ## \int_0^.0004 ## ## \frac {ds*sin(Θ)} {r^2}##

    Where the angle is that between the z-axis and the hypotenuse, such that sin(Θ) can be described as:
    sin(Θ) = ##\frac {\sqrt (x^2+y^2)} {r}##

    which gives

    ## \vec B ## = 2 ## \frac {μ_0 I \sqrt(8)} {4π} ## ## \int_0^.0004 ## ## \frac {ds} {\sqrt(s^2 +2^2 +2^2)^3}##

    Which, when I solve gives me the answer for the magnitude and direction of the magnetic field of
    ##8.4*10^{-15} T## which doesn't seem right at all, since previous answers to some other parts of this problem were in around ^-11. Any help is greatly appreciated!
     
  2. jcsd
  3. Mar 28, 2017 #2

    haruspex

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    It says centered at the origin, so I think your integration range is wrong.

    The wire length is so small compared to √8 m that I don't think you need to integrate at all. Just treat the whole segment as being at the origin.
     
  4. Mar 28, 2017 #3
    Since it it centered at the origin, the integration would be ##\int_-.0002^.0002 ## which I believed could be changed to 2 ##\int_0^.0004## Though that is wrong because the function is odd, not even.


    Also, since the wire length is so small compared to ##\sqrt 8##, would it be possible to solve this using Ampere's Law to find the magnitude?
     
  5. Mar 28, 2017 #4

    haruspex

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    No, it is even.
    Just drop the s2 in the denominator, making it constant over the range. Assuming everything else is right.
     
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