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FrogPad
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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 \vec E = \hat x 100 x \,\,(V/m), find the total electric charge contained inside a cubical volume 100 \,\, (mm) on a side centered symmetrically at the orgin.
My Work)
Recall:
\oint \vec E \cdot d\vec s = \frac{Q}{\epsilon_0} (Gauss's Law)
\int_V \nabla \cdot \vec A \, dv = \oint_S \vec A \cdot d\vec s (Divergence Thm)
Thus,
\oint_S \vec E \cdot d\vec s = \int_V \nabla \cdot \vec E \, dv = \frac{Q}{\epsilon_0}
\nabla \cdot \vec E = 100 \,\, (V/m)
100 (V/m) \int_V \, dv = 100\, (V/m)(100\times 10^{-3})^3(m^3) = \frac{1}{10} \,\, (V/m^2)
\frac{1}{10} \,\, (V/m^2) = \frac{Q}{\epsilon_0}
Thus,
Q = \frac{\epsilon_0}{10} \,\, (v/m^2) = 8.854\times 10^{-12} \frac{coul}{m^3}
I thought the units for Q should be in coul? Why am I getting coul per unit volume? Am I not doing this right?
Q) Assuming that the electric field intensity is \vec E = \hat x 100 x \,\,(V/m), find the total electric charge contained inside a cubical volume 100 \,\, (mm) on a side centered symmetrically at the orgin.
My Work)
Recall:
\oint \vec E \cdot d\vec s = \frac{Q}{\epsilon_0} (Gauss's Law)
\int_V \nabla \cdot \vec A \, dv = \oint_S \vec A \cdot d\vec s (Divergence Thm)
Thus,
\oint_S \vec E \cdot d\vec s = \int_V \nabla \cdot \vec E \, dv = \frac{Q}{\epsilon_0}
\nabla \cdot \vec E = 100 \,\, (V/m)
100 (V/m) \int_V \, dv = 100\, (V/m)(100\times 10^{-3})^3(m^3) = \frac{1}{10} \,\, (V/m^2)
\frac{1}{10} \,\, (V/m^2) = \frac{Q}{\epsilon_0}
Thus,
Q = \frac{\epsilon_0}{10} \,\, (v/m^2) = 8.854\times 10^{-12} \frac{coul}{m^3}
I thought the units for Q should be in coul? Why am I getting coul per unit volume? Am I not doing this right?
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