What is the formula of Coulomb potential?

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Solution 1

The top equation is electric potential energy while the bottom is electric potential. They use that coulomb potential energy term for hydrogen like atoms in quantum mechanics where Z is the number of protons in the nucleus.

Solution 2

Yes, using the integral $V = -\int \mathbf{E} \cdot \mathbf{dr}$ to calculate the potential is correct, but the expression - $V(r) = \dfrac{-Ze^2}{r}$ is for the potential energy of an electron in Bohr's classical model of an atom. $Z$ is just the number of protons in the atom.

You could relate the Coulombic force with the centripetal force for an electron in a hydrogen atom, and get the relation,

$$ E_{kinetic} = \dfrac{1}2 mv^2 = \dfrac{1}2 \dfrac{kZe^2}{r}\tag*{(1)} $$ And by the Virial Theorem for a spherical system ($n = -1$), $$ 2\langle T \rangle = -1\langle U \rangle\tag*{(2)} $$ Where $\langle T \rangle$ and $\langle U \rangle$ are the total kinetic and potential energies of the system.

Therefore, substituting $(1)$ in $(2)$ , we have, $$ U = -\dfrac{kZe^2}{r} $$

Solution 3

Usually when we see $Z$ in this equation, we are relating the potential of electrical forces between electron and protons. Hence, $Z$ is a scalar for the number of particles with $+e$, i.e. protons that interact with the $-e$ electron, rendering $Ze^2.$ This comes up in the Bohr model of the hydrogen atom, for example.

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Updated on June 27, 2020

Comments

  • ynn
    ynn over 3 years

    According to "Lectures on Quantum Mechanics" by Steven Weinberg, the formula of Coulomb potential is $$V(r) = - \frac{Z e^2}{r}.$$

    But it this true? I calculated the integral $$V = - \int _\infty ^r \vec{E} \cdot d\vec{r} = \frac{q}{4 \pi \epsilon _0} \frac{1}{r}.$$

    I don't know what $Z$ is but I'm unfamiliar with the formula in the book.

  • Noon36
    Noon36 over 6 years
    This webpage also gives a good explanation on the difference. hyperphysics.phy-astr.gsu.edu/hbase/electric/elepe.html
  • ynn
    ynn over 6 years
    Thank you very much. So the potential energy $U = qV = (-e) V = - \frac{Ze^2}{r}$. Now I understand the difference.
  • Noon36
    Noon36 over 6 years
    From Wikipedia: An electric potential (also called the electric field potential or the electrostatic potential) is the amount of work needed to move a unit positive charge from a reference point to a specific point inside the field without producing any acceleration.
  • Noon36
    Noon36 over 6 years
    Also From Wiki: Electric potential energy, or electrostatic potential energy, is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. So one is Joules/coulomb and the other is just Joules (energy)
  • lalala
    lalala over 6 years
    Maybe to add here: isnt the first formula using cgs units, while the second one using SI.