Mass difference due to electrical potential energy

[etotheipi]

Do I have to use both electric fields? That is, I have to use both the electric field inside the sphere and outside the sphere ?

Yeah, you would need to split the integral into$$U = 2\pi \varepsilon_0 \int_0^\infty E^2 r^2 dr = 2\pi \varepsilon_0 \int_0^R \frac{k^2 Q^2 r^4}{R^6} dr + 2\pi \varepsilon_0 \int_R^\infty \frac{k^2 Q^2}{r^2} dr$$

 

[Adams2020]

Yeah, you would need to split the integral into$$U = 2\pi \varepsilon_0 \int_0^\infty E^2 r^2 dr = 2\pi \varepsilon_0 \int_0^R \frac{k^2 Q^2 r^4}{R^6} dr + 2\pi \varepsilon_0 \int_R^\infty \frac{k^2 Q^2}{r^2} dr$$

The results of this way should be the same as the previous way, right?

 

[vanhees71]

I've not seen andy previous way in this thread, only guesses about factors, which haven't been before either ;-)).

 

[Adams2020]

The results of this way should be the same as the previous way, right?

Yes, the result is exactly the same. I got the previous formula.
Thank you and those who guided me in this exercise.

 

[BvU]

The sign is incorrect, so this certainly cannot account for the mass difference. The protons are actually less heavy than the neutrons, not the other way around. Did I miss something?

You did not. But it's not your exercise !

 

[etotheipi]

You did not. But it's not your exercise !

Well no, but I thought you had endorsed the statement "Can the mass difference between protons and neutrons be due to the electrical potential energy of the protons?" as being plausible by consideration of $E=mc^2$. I had pointed out in #16 that there was no need to perform the calculation, since we already know the electric potential energy would increase, rather than decrease, the mass!

 

[BvU]

You and I already knew. OP was supposed to discover $-$1.3 MeV/c² mass difference can not be explained with a $+$1 MeV/c² from electrostatic energy

 

[etotheipi]

Whoops... now I realized you had said "can you answer... affirmatively", instead of saying 'affirmative' to the quote. Please forgive my naivety, I'm not good at interpreting subtext

 

[Adams2020]

I came back. I still have a knot in understanding this exercise.
Now the values are almost equal. That is, the potential difference is equal to the mass difference. What exactly does this mean? That is, how do you analyze this?

 

[BvU]

I came back. I still have a knot in understanding this exercise.

No problem

Now the values are almost equal.

What about the sign ?

That is, the potential difference is equal to the mass difference.

What about the sign ?

What exactly does this mean? That is, how do you analyze this?

Re-read the thread at leisure
The notion that mass and energy are "interchangeable" is not easy to grasp (It took an Einstein to find out )

 

[Adams2020]

What about the sign ?

Yes. I did not pay attention to it. I thought the exercise was solved!☺

The notion that mass and energy are "interchangeable" is not easy to grasp (It took an Einstein to find out )