How Close Must Two Electrons Be to Make Their Combined Mass Sevenfold?

[onikishi]

How close would two stationary electrons have to be positioned so that their total mass is 7 times what it is when the electrons are very far apart?


Here's my setup:

2mC^2 is the energy of the 2 electrons when they're far apart.
2mC^2 + kqq/r^2 is the energy of the 2 electrons when they're close.
Since the mass is 7 times the far apart mass:


14mC^2= 2mC^2 + kqq/r^2

m = mass of electron = 9.11e-31
k = 8.99e9
q = 1.6e-19
C = 3e8

at the end I got r = 1.53e-8 meters... but that's wrong...
can anyone tell me where I messed up?

Thanks

 

[anathema]

for sharing your setup and calculations! Here are some corrections that might help you find where you went wrong:

1. The equation for the total energy of two electrons at a distance r is E = 2mC^2 + kqq/r. The 2mC^2 term only accounts for the rest energy of the two electrons, while the second term takes into account their electrostatic potential energy.

2. Since we are looking for the distance at which the total mass is 7 times the rest mass, we can set the two energies equal to each other: 2mC^2 + kqq/r = 7(2mC^2). This gives us kqq/r = 6mC^2.

3. Next, we can substitute the values for k, q, and C, and solve for r: (8.99e9)(1.6e-19)^2/r = 6(9.11e-31)(3e8)^2. This gives us r = 1.38e-11 meters, which is much smaller than your previous answer of 1.53e-8 meters.

I hope this helps! Just remember to double check your equations and units to make sure everything is consistent. Good luck!

 

[m2003]

for reaching out for help with your relativity problem. It seems like you have set up the equations correctly and have the correct values for the mass of the electron, the Coulomb constant, the charge of the electron, and the speed of light. However, it is important to note that the equation you used, 2mC^2 + kqq/r^2, is the total energy of the system, not the total mass. In order to find the total mass, you would need to use the famous equation E=mc^2 and solve for m.

So, the equation you should be using is m = (2m + kqq/r^2)/c^2. Substituting the values, we get:

m = (2*9.11e-31 + 8.99e9*1.6e-19/(1.53e-8)^2)/ (3e8)^2

m = 2.020e-31 kg

Therefore, the total mass of the two electrons when they are close together would be 2.020e-31 kg, which is approximately 0.22 times the mass of a single electron. This means that the two electrons would have to be positioned approximately 1.53e-8 meters apart in order for their total mass to be 7 times what it is when they are far apart.

I hope this helps clarify where you may have made a mistake. Keep up the good work with your calculations!