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Homework Help: Electric charge inside a cubical volume

  1. Sep 3, 2006 #1
    Would someone be so kind to check if I am doing this properly. I'm confused on the units, as it doesn't seem to be coming out properly.

    Q) Assuming that the electric field intensity is [itex] \vec E = \hat x 100 x \,\,(V/m) [/itex], find the total electric charge contained inside a cubical volume [itex] 100 \,\, (mm) [/itex] on a side centered symmetrically at the orgin.

    My Work)
    Recall:

    [tex] \oint \vec E \cdot d\vec s = \frac{Q}{\epsilon_0} [/tex] (Gauss's Law)
    [tex] \int_V \nabla \cdot \vec A \, dv = \oint_S \vec A \cdot d\vec s [/tex] (Divergence Thm)

    Thus,
    [tex] \oint_S \vec E \cdot d\vec s = \int_V \nabla \cdot \vec E \, dv = \frac{Q}{\epsilon_0} [/tex]

    [tex] \nabla \cdot \vec E = 100 \,\, (V/m) [/tex]
    [tex] 100 (V/m) \int_V \, dv = 100\, (V/m)(100\times 10^{-3})^3(m^3) = \frac{1}{10} \,\, (V/m^2) [/tex]

    [tex] \frac{1}{10} \,\, (V/m^2) = \frac{Q}{\epsilon_0} [/tex]

    Thus,
    [tex] Q = \frac{\epsilon_0}{10} \,\, (v/m^2) = 8.854\times 10^{-12} \frac{coul}{m^3} [/tex]

    I thought the units for [itex] Q [/itex] should be in coul? Why am I getting coul per unit volume? Am I not doing this right?
     
    Last edited: Sep 3, 2006
  2. jcsd
  3. Sep 4, 2006 #2
    Hi, it looks like a units problem, the numbers look fine.

    [tex] \nabla \cdot \vec E[/tex] is measured in V/m2, not V/m. Then you multiply by m3, and get Vm. The units of [tex]{\epsilon_0}[/tex] are C/Vm, so Vm cancels and you get C.
     
  4. Sep 4, 2006 #3
    Mighty appreciated my man :smile:

    thanks
     
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