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Homework Help: Electric Field at the Center of a Triangle

  1. Sep 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Given an equilateral triangle with sides 20-cm, with point charges at each point:
    q1 at the top with a charge of +4.0uM,
    q2 at the bottom right with a charge of -4.0uM, and
    q3 at the bottom left with a charge of +4.0uM.

    k is assumed to be exactly 9.00x109 N-m2/c2

    Question: What is the electric field at the center of the triangle?


    2. Relevant equations

    E = (kq)/r2

    3. The attempt at a solution

    First and foremost, I employed a little geometry find the center of the triangle:

    r = sqrt( (0.20-m)2 - (0.10-m)2 ) / 2
    r = 0.866-m, rounded to
    r = 0.90-m

    E1 = (9.00x106 N-m2/c2)(4.0x10-6C) / (0.09-m)2
    E1 = 4.4x106 N/C

    The magnitudes of all three charges are equivalent, so:

    E1 = E2 = E3

    So the x and y components would be:

    Ex = (4.4x106 N/C)cos30° + (4.4x106 N/C)cos330°
    Ex = 7.6 x 106 N/C

    Ey = (4.4x106 N/C)sin270° + (4.4x106 N/C)sin30° + (4.4x106 N/C)sin330°
    Ey = -4.4x106 N/C

    E = sqrt[ (7.6 x 106 N/C)2 - (4.4x106 N/C)2 ]
    E = 6.2 x 106 N/C

    So, my answer is 6.2 x 106 N, but the book answer key shows 5.4 x 106 N/C. That is quite a significant difference, but I can't seem to find where I'm making a mistake. It's driving me mad!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 4, 2010 #2

    ehild

    User Avatar
    Homework Helper



    Check your geometry. How far are the corners of an equilateral triangle from the centre?

    You rounded r too much, just at the beginning of your calculations.
    The value of r has got 4% error because of rounding. It is squared, that doubles the error to 8% which increases further during the rest of the calculations.

    You must not round so much during the calculations. Keep 3-4 significant digits.

    You have r2 in the formula for the electric field intensity. So you do not need the square root at all and.


    ehild
     
  4. Sep 4, 2010 #3
    I must be daft!

    And I made a typo, it's 0.09 and not 0.90. But that's beyond the point!

    The distance from one corner of an equilateral triangle is equal to a hypotenuse made from a right triangle with one leg half the length of teh sides of the triangle and the other leg equal to 1/3 the value of the distance from one corner to the midpoint of the opposing side!

    SO

    r2 = [ sqrt( (0.20m)2 - (0.10m)2 ) / 3 ]2 + (0.1m)2
    r2 = 0.0133 m

    Thank you very much, ehild! :-D
     
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