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Integration involving a physics problem.

  1. May 24, 2013 #1
    1. The problem statement, all variables and given/known data
    Hello all! I'm currently trying to work a problem for my Physics 2 class (for engineering and science majors). The example problem deals with "Field of a charged line segment." I conceptually understand the problem, but I am having trouble with the details involving the integration.

    The problem: "Positive charge Q is distributed uniformly along the y-axis between y= -a and y= +a. Find the electric field at point P on the x-axis at a distance x from the origin."

    I know that [itex]\stackrel{\rightarrow}{E}[/itex]= [itex]\frac{kQ}{r^{2}}[/itex]

    λ=[itex]\frac{Q}{2a}[/itex]

    dQ=λdy=[itex]\frac{Q}{2a}[/itex]dy

    r=[itex]\sqrt{x^{2}+y^{2}}[/itex]

    therfore; dE=(k)([itex]\frac{dQ}{r^{2}}[/itex])= (k)([itex]\frac{Q}{2a}[/itex])([itex]\frac{dy}{x^{2}+y^{2}}[/itex])

    E[itex]_{y}=[/itex][itex]_{-a}[/itex]∫[itex]^{+a}[/itex]dE[itex]_{y}[/itex]=0

    E[itex]_{x}=[/itex][itex]_{-a}[/itex]∫[itex]^{+a}[/itex]dE[itex]_{x}[/itex]=[itex]_{-a}[/itex]∫[itex]^{+a}[/itex](k)([itex]\frac{Q}{2a}[/itex])([itex]\frac{dy}{x^{2}+y^{2}}[/itex])(cosθ)

    cosθ=[itex]\frac{x}{\sqrt{x^{2}+y^{2}}}[/itex]

    E[itex]_{x}=[/itex][itex]_{-a}[/itex]∫[itex]^{+a}[/itex]dE[itex]_{x}[/itex]=[itex]_{-a}[/itex]∫[itex]^{+a}[/itex](k)([itex]\frac{Q}{2a}[/itex])([itex]\frac{dy}{x^{2}+y^{2}}[/itex])([itex]\frac{x}{\sqrt{x^{2}+y^{2}}}[/itex])

    simplifying and factoring out constants gives:

    ([itex]\frac{kQ}{2a}[/itex])[itex]_{-a}[/itex]∫[itex]^{+a}[/itex][itex]\frac{xdy}{(x^{2}+y^{2})^{3/2}}[/itex]

    Here is where my problem comes in.... I don't know how to integrate this. The book says "a table of integrals will help."

    The solution is given to be:

    E[itex]_{x}[/itex]=[itex]\frac{kQ}{x\sqrt{x^{2}+y^{2}}}[/itex]



    2. Relevant equations



    3. The attempt at a solution

    I do have the latest CRC book which has integral tables in it. I looked at the general forms containing: c2+x2. The one it looked the closest to was: [itex]\frac{dx}{(c^{2}+x^{2})^{n}}[/itex]. But I'm not sure... I believe the x's may be treated as constants since I'm integrating with respect to y, but I'm not exactly sure how to go about working it. I looked a U-substitution but I get bogged down and confused by the fact that I am integrating a function that includes two variables. Any help would greatly be appreciated. Again, I understand the concept, but I'm getting confused on the calculus part of it (the integration/last step). Thank you.
     
  2. jcsd
  3. May 24, 2013 #2

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    It looks like a trig substitution might work. Try ##y = x\tan\theta,\, dy=x\sec^2\theta\,d\theta,\,
    \sqrt{x^2+y^2}=x\sec\theta## and see if that does anything for you.
     
  4. May 24, 2013 #3

    NascentOxygen

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    Staff: Mentor

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