Mass difference due to electrical potential energy

[Adams2020]

Homework Statement

The masses of protons and neutrons are different. Suppose a proton is a sphere with a uniformly positive charge distribution. Can the mass difference between protons and neutrons be due to the electrical potential energy of the protons? Justify your answer with a simple calculation.

Relevant Equations

I'm not sure

I do not really know the relationship between potential energy and mass difference.
Isn't the difference in mass of protons and neutrons due to their quarks? (the neutron is made of two down quarks and an up quark and the proton of two up quarks and a down quark.)
Please help.

 

[BvU]

I do not really know the relationship between potential energy and mass difference.

When in doubt, use $E=mc^2\quad$

Suppose a proton is a sphere with a uniformly positive charge distribution.

How much energy does that give you ?

 

[PeroK]

Isn't the difference in mass of protons and neutrons due to their quarks? (the neutron is made of two down quarks and an up quark and the proton of two up quarks and a down quark.)

Up and down quarks have masses of a few $MeV$; whereas, neutrons and protons have a mass of about $1GeV$.

 

[Adams2020]

How much energy does that give you ?

i don't know

 

[DrClaude]

i don't know

As per the rules, you'll have to put in some effort.

 

[Adams2020]

As per the rules, you'll have to put in some effort.

I asked for guidance, not a complete answer.
If someone can not guide, no problem.

 

[DrClaude]

I asked for guidance, not a complete answer.
If someone can not guide, no problem.

But you are getting guidance. @BvU asked you to calculate the potential energy of a sphere with a uniformly positive charge distribution. Can you do that?

 

[Adams2020]

But you are getting guidance. @BvU asked you to calculate the potential energy of a sphere with a uniformly positive charge distribution. Can you do that?

Regarding the potential energy of a charged sphere, I know the following relation:
U = k(3/5)((Q^2)/R)
But I do not know how to use this to solve.

 

[vela]

What’s your background in physics?

 

[Adams2020]

What’s your background in physics?

Im a physics student and this exercise was given to us by a professor of particle physics.

 

[vela]

So you've already taken upper-division quantum mechanics and electromagnetism?

 

[Adams2020]

So you've already taken upper-division quantum mechanics and electromagnetism?

yes

 

[vela]

OK. The reason I'm asking is because your initiative and level of engagement seems unusually low for the typical student at your level of education. It just feels like something is off here. I thought perhaps you were taking this class before you were properly prepared for it, but you seemed to have taken the usual prerequisites.

I'll refer you again to BvU's reply above for the next step.

 

[Adams2020]

How much energy does that give you ?

Regarding the potential energy of a charged sphere, I know the following relation:
U = k(3/5)((Q^2)/R)
But I do not know how to use this to solve.

 

[BvU]

how to use this to solve

You simply fill in the values and the dimensions to calculate a result: the amount of energy associated with the charge configuration. And, via $E= mc^2$, a mass equivalent.

From the way the problem statement is formulated, you can already suspect that this isn't going to explain the mass difference (which, of course, you have already looked up ? )

 

[etotheipi]

If I'm not missing something, the neutron is heavier than the proton, so even without doing any calculations we should know that the positive electrostatic self energy of the model of the proton as a uniformly charged sphere actually does the opposite of explain the mass difference!

 

[Adams2020]

You simply fill in the values and the dimensions to calculate a result: the amount of energy associated with the charge configuration. And, via $E= mc^2$, a mass equivalent.

From the way the problem statement is formulated, you can already suspect that this isn't going to explain the mass difference (which, of course, you have already looked up ? )

Suppose I get a number for potential energy this way. How can this number explain the difference in mass to me?

 

[Adams2020]

If I'm not missing something, the neutron is heavier than the proton, so even without doing any calculations we should know that the positive electrostatic self energy of the model of the proton as a uniformly charged sphere actually does the opposite of explain the mass difference!

So we can not explain the mass difference in this way?

 

[etotheipi]

Well you'd need to ask someone who knows a bit of particle physics to get a proper answer, but I think a simple explanation is that the quark compositions are different, and u has a mass of 2.3MeV whilst d has a mass of 4.8MeV. In any case, most of the mass of a proton or neutron (over 900MeV) is due to the strong interaction (see here).

(N.B. that we take zero electric potential energy to be at infinite separation [where there is zero electric interaction], so bound states have reduced mass whilst configurations with high electric potential energy have increased mass. On the other hand, the zero potential energy of strong interaction between quarks and gluons is taken at zero separation [e.g. like a spring], so you end up with positive potential energy associated with this bound state, and the strong force actually increases the mass here).

 

[BvU]

Justify your answer with a simple calculation.

The exercicse wants you to do a calculation. What did you actually calculate so far ?

 

[Adams2020]

The exercicse wants you to do a calculation. What did you actually calculate so far ?

Nothing. I do not know exactly what to calculate. All I know is a relationship to potential energy. Suppose I got it, what does it have to say and how does it explain the mass difference?

 

[BvU]

See #14. Do not reply with "suppose I get a number", but do the calculation

 

[Adams2020]

See #14. Do not reply with "suppose I get a number", but do the calculation

Using the relation sent here and the following data, I obtained a value for the electric potential energy of the proton, which is:
1.6 * 10^ (-13) J

R= 8.7 * 10 ^ (-16) m
k= 9 * 10 ^ (19) Nm^2/c^2
Q= 1.6 * 10 (-19) c

What does this number say?

 

[BvU]

If this number is somewhat equal to the mass difference between proton and neutron (with the correct sign and in the same units), you can answer the question

Can the mass difference between protons and neutrons be due to the electrical potential energy of the protons?

affirmativeley Side nite:

U = k(3/5)((Q^2)/R

Others find $1\over 2$ ; where does your $3\over 5$ come from ?

 

[vela]

Others find $1\over 2$ ; where does your $3\over 5$ come from ?

Feynman agrees with the factor 3/5 for a uniform sphere of charge.

 

[vanhees71]

So you have to calculate the energy yourself carefully. In physics an argument that the one or other authority made a statement, doesn't count ;-).

Hint: The energy density of an electrostatic field is
$$u=\frac{\epsilon_0}{2} \vec{E}^2.$$
You can simplify the task to evaluate the total energy by using
$$\vec{E}=-\vec{\nabla} \Phi,$$
plugging this into the integral and then use Gauss's Law
$$\vec{\nabla} \cdot \vec{E}=-\frac{1}{\epsilon_0} \rho.$$

 

[etotheipi]

And if in doubt, you can always try doing it explicitly, $$
E_r(r) = \begin{cases}
\frac{kQr}{R^3} & r \leq R \\
\frac{kQ}{r^2} & r> R
\end{cases}$$to find$$U = \frac{\varepsilon_0}{2} \int_{\mathbb{R}^3} E^2 dV = \frac{\varepsilon_0}{2} \int_0^{4\pi} \int_0^{\infty} E^2 r^2 dr d\Omega = 2\pi \varepsilon_0 \int_0^\infty E^2 r^2 dr$$

 

[Adams2020]

If this number is somewhat equal to the mass difference between proton and neutron (with the correct sign and in the same units), you can answer the question

After equalizing the units, the mass difference is almost the same as this number. Thanks for your guidance

 

[etotheipi]

If this number is somewhat equal to the mass difference between proton and neutron (with the correct sign and in the same units), you can answer the question...
affirmativeley

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?

 

[Adams2020]

And if in doubt, you can always try doing it explicitly, $$
E_r(r) = \begin{cases}
\frac{kQr}{R^3} & r \leq R \\
\frac{kQ}{r^2} & r> R
\end{cases}$$to find$$U = \frac{\varepsilon_0}{2} \int_{\mathbb{R}^3} E^2 dV = \frac{\varepsilon_0}{2} \int_0^{4\pi} \int_0^{\infty} E^2 r^2 dr d\Omega = 2\pi \varepsilon_0 \int_0^\infty E^2 r^2 dr$$

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 ?