A spherical volume having a \(2-\mu \mathrm{m}\) radius contains a uniform volume charge density of \(10^{15} \mathrm{C} / \mathrm{m}^{3}\). (a) What total charge is enclosed in the spherical volume? (b) Now assume that a large region contains one of these little spheres at every corner of a cubical grid \(3 \mathrm{~mm}\) on a side and that there is no charge between the spheres. What is the average volume charge density throughout this large region?

To find the total charge inside the sphere, we'll use the formula \(Q = \rho V\), where \(\rho = 10^{15} C/m^3\) is the volume charge density and V is the volume of the sphere. The volume of a sphere can be calculated using the formula \(V = \frac{4}{3} \pi r^3\), where r is the radius of the sphere. In this case, the radius is given as \(r = 2 \times 10^{-6}m\). First, let's find the volume of the sphere: \(V = \frac{4}{3} \pi (2 \times 10^{-6})^3 = 33.51 \times 10^{-18} m^3\) Now, we can find the total charge enclosed in the sphere: \(Q = \rho V = (10^{15} C/m^3)(33.51 \times 10^{-18} m^3) = 33.51 \times 10^{-3} C\) The total charge enclosed in the sphere is \(33.51 \times 10^{-3} C\).

We are given a cubical grid with side lengths of \(3 mm\), and the spheres are located at each corner. There are 8 corners in a cube, and hence, there are 8 spheres in total. First, let's find the volume of the large cubical grid: \(V_{cube} = (3 \times 10^{-3} m)^3 = 27 \times 10^{-9} m^3\) To find the average volume charge density of the large region, we can divide the total charge enclosed by these 8 spheres by the volume of the cubical grid. Let \(\rho_{avg}\) be the average volume charge density. From part (a), we know the total charge enclosed in one sphere, but we have 8 spheres, so the total charge for all spheres will be \(Q_{total} = 8 \times (33.51 \times 10^{-3} C)\). Now, we can find the average volume charge density: \(\rho_{avg} = \frac{Q_{total}}{V_{cube}} = \frac{8 \times (33.51 \times 10^{-3} C)}{27 \times 10^{-9} m^3} = 9.93 \times 10^{11} C/m^3\) The average volume charge density throughout the large region is \(9.93 \times 10^{11} C/m^3\).