Find electric potential of field inside and outside nucleus

In summary, to derive the expression for the electrostatic potential energy of an electron in the field of a finite nucleus of charge, ##+Ze##, and radius, ##R=r_0A^{1/3}##, we first approximate the nucleus as being point-like for ##r>R## and use the standard Coulomb potential. For ##r<R##, we use Gauss' law and integrate from infinity to R and then from R to r to get the derived potential. The potential must also be continuous at R.
  • #1
vbrasic
73
3

Homework Statement


Derive following expression for the electrostatic potential energy of an electron in the field of a finite nucleus of charge, ##+Ze##, and radius, ##R=r_0A^{1/3}##, where ##r_0## is a constant. (Charge density is constant.)

The potential we are asked to derive is:
$$
V(r) = \begin{cases}
\frac{-Ze^2}{r} & \text{if } r>R\\
\frac{Ze^2}{R}(\frac{r^2}{2R^2}-\frac{3}{2}) & \text{if } r<R
\end{cases}.
$$

Homework Equations


Gauss' law.

The Attempt at a Solution


Naturally, for ##r>R##, we approximate the nucleus as being point-like, with electric field of magnitude, $$\frac{Ze}{4\pi\epsilon_0r^2}.$$ I'm not sure how to find potential, though physical intuition suggests it's just the standard Coulomb potential.

However, inside the nucleus we use Gauss' law. We have that the charge enclosed is ##Q_{enc}=Ze\frac{r^3}{R^3}##, for a uniform charge density. Then using Gauss' law, we have, $$\int |E|da=Ze\frac{r^3}{R^3\epsilon_0}\rightarrow |E|=Ze\frac{r}{4\pi\epsilon_0R^3}$$ directed radially, such that, $$E=Ze\frac{r}{4\pi\epsilon_0R^3}\hat{r}.$$ I'm not sure how to get potential from here, or exactly what integral I'm supposed to use to get the derived potential above.
 
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  • #2
Same integral as outside the sphere ... :rolleyes:
And at R the potential has to be continuous.
 
  • #3
BvU said:
Same integral as outside the sphere ... :rolleyes:
And at R the potential has to be continuous.
So essentially all I have to do is first integrate from infinity to R, and then from R to r?
 
  • #4
I am surprised you have to ask. Yes ! Exactly !
 

Related to Find electric potential of field inside and outside nucleus

1. What is electric potential?

Electric potential is a measure of the electric potential energy per unit charge at a specific point in an electric field.

2. How is electric potential related to electric field?

Electric potential is related to electric field by the equation V = -∫E•dl, where V is the electric potential, E is the electric field, and dl is the infinitesimal displacement along a path in the electric field. In other words, the electric potential is the negative of the line integral of the electric field along a path.

3. What is the electric potential inside the nucleus?

The electric potential inside the nucleus is extremely high, due to the strong electric field generated by the positively charged protons. This potential is often on the order of millions of volts per meter.

4. How is the electric potential outside the nucleus determined?

The electric potential outside the nucleus can be determined using the equation V = kQ/r, where V is the electric potential, k is the Coulomb constant, Q is the charge of the nucleus, and r is the distance from the nucleus. This equation follows the inverse square law, meaning that the potential decreases as the distance from the nucleus increases.

5. Is the electric potential inside and outside the nucleus the same for all elements?

No, the electric potential inside and outside the nucleus can vary for different elements depending on the number of protons and neutrons in the nucleus. This is because the electric potential is directly related to the charge of the nucleus, and different elements have different numbers of protons and neutrons.

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