# Electric charge inside a cubical volume

1. Sep 3, 2006

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?

Last edited: Sep 3, 2006
2. Sep 4, 2006

### Tomsk

Hi, it looks like a units problem, the numbers look fine.

$$\nabla \cdot \vec E$$ is measured in V/m2, not V/m. Then you multiply by m3, and get Vm. The units of $${\epsilon_0}$$ are C/Vm, so Vm cancels and you get C.

3. Sep 4, 2006