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Homework Help: Coulomb's Law to find the net force

  1. Jul 8, 2007 #1
    1. The problem statement, all variables and given/known data
    Charge 8e-18 C is on the y axis a distance 2 m from the origin and charge
    9e-18 C is on the x axis a distance d from the origin. The Coulomb constant is 8.98755e9 Nm^2/C^2.

    What is the value of d for which the x component of the force on 9e-18 C is the greatest?


    2. Relevant equations

    Coulomb's law: F = kq1q2/r^2

    3. The attempt at a solution
    I tried to use Coulomb's law to find the net force and then to find the force in the x direction but I became very stuck.

    F = 8.98755e9 Nm^2/C^2 * 8e-18 C * 9e-18 C / (4 + d^2)

    My problem is I have two unknowns and I can't find the value of d. Please help if you can.
     
  2. jcsd
  3. Jul 8, 2007 #2

    Astronuc

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    d is the only unknown.

    One has F, from which one finds Fx = F cos (theta). What is cos (theta) in terms of 'd'?

    How would one find the maximum of Fx as a function of d?
     
    Last edited: Jul 8, 2007
  4. Jul 8, 2007 #3
    theta = adjacent/hypotenuse


    How do you already know what F is?
     
  5. Jul 8, 2007 #4

    Astronuc

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    My apology - I should have asked - What is cos (theta) in terms of 'd'?

    Coulomb's law: F = kq1q2/r^2

    which one then writes

    F = 8.98755e9 Nm^2/C^2 * 8e-18 C * 9e-18 C / (4 + d^2)
     
  6. Jul 8, 2007 #5
    cos theta would equal d/ sqrt(4 + d^2)?

    Could you clarify a little bit more about how to solve this problem? Sorry I'm a bit confused.
     
  7. Jul 9, 2007 #6
    Hey,

    Let,

    [itex]q_{1} = 8{\textcolor[rgb]{1.00,1.00,1.00}{.}}x{\textcolor[rgb]{1.00,1.00,1.00}{.}}10^{-18}{\textcolor[rgb]{1.00,1.00,1.00}{.}}C[/itex]

    [itex]q_{2} = 9{\textcolor[rgb]{1.00,1.00,1.00}{.}}x{\textcolor[rgb]{1.00,1.00,1.00}{.}}10^{-18}{\textcolor[rgb]{1.00,1.00,1.00}{.}}C[/itex]

    Also let the distance between [itex]q_{1}[/itex] and [itex]q_{2}[/itex] be [itex]r_{12}[/itex] (read as: distance r from 1 to 2) instead of plain r, makes the problem clearer.

    First, draw a picture, makes the problem much easier.

    Second, consider what you already know.

    You know Coulomb's Law:

    Vector Form:

    [tex]
    \vec{F}_{12} = \frac{k_{e}q_{1}q_{2}}{{r_{12}}^2}\hat{r}_{21}
    [/tex]

    Scalar Form:

    [tex]
    |\vec{F}_{12}| = \frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}
    [/tex]

    Now, you also know that,

    [tex]
    F_{21}_{x} = |\vec{F}_{21}|cos{\theta}
    [/tex]

    And you need to find the value of d that would maximize [itex]
    F_{21}_{x}[/itex], therefore consider rewriting as,

    [tex]
    F_{21}_{x}(d) = |\vec{F}_{21}|\left(\frac{d}{\sqrt{d^2+2^2}}\right)
    [/tex]

    [tex]
    F_{21}_{x}(d) = \left(\frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}\right)\left(\frac{d}{\sqrt{d^2+2^2}}\right)
    [/tex]

    [tex]
    F_{21}_{x}(d) = \left(\frac{k_{e}|q_{1}||q_{2}|}{{(\sqrt{d^2+2^2})}^2}\right)\left(\frac{d}{\sqrt{d^2+2^2}}\right)
    [/tex]

    Now ask yourself, "Given a function of a single variable, how do you maximize that function? (hint: think calculus)".

    Also remember d is a variable, not a constant.

    Best,

    -PFStudent
     
    Last edited: Jul 9, 2007
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