Electric Potential Energy of a Cube of 8 protons

[acedeno]

Homework Statement

Suppose we there is a cube (a x a x a) made up of 8 protons. 1 at each point
What is the Electric Potential Energy

Homework Equations

U=q1*q1/4*pi*epsilon_0*r

The Attempt at a Solution

*U12 = Potential Energy to bring 2 to 1*
I'm aware of how to do the question but I'm not sure if I'm expressing it correctly. Remember that the potential energy is the work done configure this system in space.
W1=0
W2=U12
W3=U12+U13
W4=W3+U14
W5=W4+U15
W6=W5+U16
W7=W6+U17
W8=W7+U18

Then the Electric Potential Energy would be = Wtotal = W1+W2 . . . + W8

I am also aware that when I show my equation fully that some of the distances will be sqrt(2a^2)

 

[Redbelly98]

Homework Statement

Suppose we there is a cube (a x a x a) made up of 8 protons. 1 at each point
What is the Electric Potential Energy

Homework Equations

U=q1*q1/4*pi*epsilon_0*r

The Attempt at a Solution

*U12 = Potential Energy to bring 2 to 1*
I'm aware of how to do the question but I'm not sure if I'm expressing it correctly. Remember that the potential energy is the work done configure this system in space.
W1=0
W2=U12
W3=U12+U13

The work to bring in the 3rd charge is U13 + U23

W4=W3+U14

And for the 4th charge it's U14 + U24 + U34

W5=W4+U15
W6=W5+U16
W7=W6+U17
W8=W7+U18

Then the Electric Potential Energy would be = Wtotal = W1+W2 . . . + W8

Yes.

I am also aware that when I show my equation fully that some of the distances will be sqrt(2a^2)

 

[acedeno]

Thanks a lot! That really helps!

 

[SammyS]

Some pairs of protons have a separation distance of (√3)a .

 

[rwisz]

and some will be sqrt(3a^2).

Your attempt at a solution is on the right track. To calculate the electric potential energy of a cube of 8 protons, we first need to determine the distance between each proton. Since the cube has a side length of "a", the distance between each proton will be either sqrt(2a^2) or sqrt(3a^2), depending on the specific configuration.

Next, we can use the formula U=q1*q2/4*pi*epsilon_0*r to calculate the potential energy between each pair of protons. We can then add up all of these potential energies to get the total electric potential energy of the cube.

In this case, since we have 8 protons, we will have 28 total pairs (since we don't need to calculate the potential energy between a proton and itself). So, the final equation would be:

U = U12 + U13 + U14 + U15 + U16 + U17 + U18 + U23 + U24 + U25 + U26 + U27 + U28 + U34 + U35 + U36 + U37 + U38 + U45 + U46 + U47 + U48 + U56 + U57 + U58 + U67 + U68 + U78

Note that this may seem like a lot of calculations, but since we are dealing with protons, all of the charges are the same (q1 = q2 = 1.6 x 10^-19 C), and we can simplify the equation to:

U = (1.6 x 10^-19 C)^2 * (1/4*pi*epsilon_0) * (U12 + U13 + U14 + U15 + U16 + U17 + U18 + U23 + U24 + U25 + U26 + U27 + U28 + U34 + U35 + U36 + U37 + U38 + U45 + U46 + U47 + U48 + U56 + U57 + U58 + U67 + U68 + U78)

Overall, the electric potential energy of this cube of 8 protons will be a large positive value, as the protons repel each other due to their positive charges.