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To estimate the energy involved in a shockwave similar to that which I reported on, consider a 2D square array of identically charged objects (charge q) with regular spacing of d. To make this tractable, consider a small segment of such an array (3X3), centered on the origin.
Prompt
1 2 3
4 5 6
7 8 9
What is the electrostatic potential energy per unit area for this arrangement (again in terms of q and d)?
Part B
From Part (a) we know that electrostatic potential energy for the system is J so we divide this by the area (which we can tell is 4)
= J) *( =
1 2 3
4 5 6
7 8 9
How much does the electrostatic energy per area change when the spacing is changed from d to d(1+δd)?
Part C
Electrostatic Potential Energy per unit area:
By changing to we would see a final result of:
This result means that the change in distance is being cubed. To give this additional meaning, if our original d=2 and our change is 1, instead of dividing by 8, we are now dividing by 27. That’s a big difference from a small change.
Part C
Is it dangerous to scale things in the way suggested for a long-range force like the coulomb force. Nonetheless, given the typical spacing and change in space in the shockwave in the microgravity paper I reported on, provide a numerical estimate of the total energy associated with the shock wave, compared to the unshocked lattice.
Part D
Since we have the equation , we know that K is the same for the shocked and the particles are the same so the is assumed constant, the only difference in the energies will be due to the distance. Simply by looking at the systems, we can see that the distance is smaller in the unshocked lattice therefore it will have the higher energy.
Conceptual Answer
In the conceptual answer we determined that the unshocked lattice should have higher energy due to the particles being close and we found that it was indeed higher it was higher by a large degree of energy per unit area for such a small change in distance.
Does it match?