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Electric chagre

  1. Jun 25, 2006 #1

    tony873004

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    Science Advisor
    Gold Member

    3. Suppose the charge q2 in Figure 19-30 can be moved left or right along the line connecting the charges q1 and q3, rather than sitting at a distance d from each. Given that q = 15 µC, find the distance from q1 where q2 experiences a net electrostatic force of zero. (The charges q1 and q3 remain separated by a fixed distance of 2d=38 cm.)
    [​IMG]

    As I stare at this, it seems like there should be an easier way than the way I did it:

    [tex]
    \begin{array}{l}
    k\frac{{q_{1} \,q_2 }}{{d_{1,2}^2 }} = k\frac{{q_{2} \,q_3 }}{{d_{2,3}^2 }} \\
    \\
    \rlap{--} k\frac{{q_{1} \,q_2 }}{{d_{1,2}^2 }} = \rlap{--} k\frac{{q_{2} \,q_3 }}{{d_{2,3}^2 }} \\
    \\
    \frac{{q \times 2q}}{{d_{1,2}^2 }} = \frac{{2q \times 3q}}{{d_{2,3}^2 }} \\
    \\
    \frac{{2q^2 }}{{d_{1,2}^2 }} = \frac{{6q^2 }}{{d_{2,3}^2 }} \\
    \\
    \frac{{2\rlap{--} q^{\rlap{--} 2} }}{{d_{1,2}^2 }} = \frac{{6\rlap{--} q^{\rlap{--} 2} }}{{d_{2,3}^2 }} \\
    \\
    \frac{2}{{d_{1,2}^2 }} = \frac{6}{{d_{2,3}^2 }} \\
    \\
    d_{2,3} = 0.38 - d_{1,2} \\
    \\
    \frac{2}{{d_{1,2}^2 }} = \frac{6}{{\left( {0.38 - d_{1,2} } \right)^2 }} \\
    \\
    \frac{2}{{d_{1,2}^2 }} = \frac{6}{{\left( {0.38 - d_{1,2} } \right)\left( {0.38 - d_{1,2} } \right)}} \\
    \\
    \frac{2}{{d_{1,2}^2 }} = \frac{6}{{0.38^2 - 0.76d_{1,2} + d_{1,2}^2 }} \\
    \\
    2\left( {0.38^2 - 0.76d_{1,2} + d_{1,2}^2 } \right) = 6d_{1,2}^2 \\
    \end{array}
    [/tex]

    [tex]
    \[
    \begin{array}{l}
    0.38^2 - 0.76d_{1,2} = 2d_{1,2}^2 \\
    \\
    - 2d_{1,2}^2 - 0.76d_{1,2} - 0.38^2 = 0 \\
    \\
    d_{1,2} = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
    \\
    d_{1,2} = \frac{{ - \left( { - 0.76} \right) \pm \sqrt {\left( { - 0.76} \right)^2 - 4 \times - 2 \times - 0.38^2 } }}{{2 \times - 2}} \\
    \\
    d_{1,2} = - 0.51908965 \\
    d_{1,2} = 0.13908965 \\
    \end{array}
    [/tex]
    With 0.139 m or (13.9 cm) being the correct answer
     
  2. jcsd
  3. Jun 25, 2006 #2
    Ahh, the same old story of knowing too much math.
    From [tex]\frac{2}{{d_{1,2}}^2}=\frac{6}{(0.38-{d_{1,2}})^2}[/tex]

    you could have simply divided both sides by 2 and taken the square root.
    No quadratics involved.
     
  4. Jun 25, 2006 #3

    tony873004

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    Science Advisor
    Gold Member

    Thanks. I've never been accused of knowing too much math :).

    Your way was simpler.
     
  5. Jun 26, 2006 #4
    You're welcome ;)
     
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