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Electrostatic Potential Energy

David Like

Overview of the problem Refresher of useful equations Problem Questions

Outline

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

Useful Equations

Given: Q=q and spacing between 2 dots is d

𝑈=𝐾𝑄𝑞𝑟

=9

𝑈= 𝐾 𝑞2

𝑟

What is the electrostatic potential energy of this arrangement (in terms of q and d)?

Part A

3 parts: Center Corners (4) Between Corners (4)

Electrostatic Potential Energy

)+4()=

Center Potential Energy

𝑑√2

𝑑

𝑑

1 2 3

4 5 6

7 8 9

𝑈= 𝐾 𝑞2

𝑟

= = =

= = =

Nm

J

Corner Potential Energy

𝑑√2

𝑑

𝑑

𝑑

2𝑑

2𝑑

2𝑑

𝑑√5

2𝑑 √21 2 3

4 5 6

7 8 9

(1) (2) (2)

(2)

(1)

)+1()+2()+2()+1()=

)+3()+2()=

(4)* = J

Corner Potential Energy Cont’d

Between Potential Energy

𝑑√2

𝑑

𝑑

𝑑

2𝑑𝑑√51 2 3

4 5 6

7 8 9

(2) (2) (1)

(3)

)+2()+2()+1()=

)+2()+2()=

(4)* = J

Between Potential Energy Cont’d

J + J + J =

J

Total Electrostatic Potential Energy

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

It is stated in the article that the number density went from 4.0 to 4.8

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

Say q=1

=

Numerical AnswerUnshocked Lattice Shocked Lattice

Say q=1

=

VS.

Difference of

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?

Questions???

Samsonov, D. "Shock Melting of a Two-Dimensional Complex (Dusty) Plasma." The American Physical Society 92.25 (2004): n. pag. Print.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elepe.html

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