Consider a cube made up of eight charges, each with equal magnitude q but alternating sign as shown in the figure on the right. This assembly is called an electric octopole, and is also the basis for simple cubic crystal lattices formed of opposite ions, such as NaCl (table salt). At this scale we may neglect gravity. a) What's the potential energy Udip required to assemble just two opposite charges into a single edge of this cube? This forms an electric dipole. (Hint: It doesn't take any energy to "assemble" the first charge.) +9 +q b) What's the potential energy Uquad required to assemble four charges into a single face of this cube? This forms an electric quadrupole. -9 +9 -9 +9 c) What's the potential energy Uoet required to assemble all eight charges into the octopole? (Hint: For a cube there are a lot of pairings between charges, specifically (2) = 28, comprised of the shown edges, as well as several diagonals. Instead of adding the contribution to the potential for bringing in one charge at a time, it may be easier to consider the overall cube and count the edges and diagonals with same lengths and put the total potential energy together from there.)

Consider a cube made up of eight charges, each with equal magnitude q but alternating sign as shown in the figure on the right. This assembly is called an electric octopole, and is also the basis for simple cubic crystal lattices formed of opposite ions, such as NaCl (table salt). At this scale we may neglect gravity. a) What's the potential energy Udip required to assemble just two opposite charges into a single edge of this cube? This forms an electric dipole. (Hint: It doesn't take any energy to "assemble" the first charge.) +9 +q b) What's the potential energy Uquad required to assemble four charges into a single face of this cube? This forms an electric quadrupole. -9 +9 -9 +9 c) What's the potential energy Uoet required to assemble all eight charges into the octopole? (Hint: For a cube there are a lot of pairings between charges, specifically (2) = 28, comprised of the shown edges, as well as several diagonals. Instead of adding the contribution to the potential for bringing in one charge at a time, it may be easier to consider the overall cube and count the edges and diagonals with same lengths and put the total potential energy together from there.)