In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these potentials vary like \(\frac{1}{r}\), so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:
Bohr radius: \(a_0\approx 0.0529~\textrm{nm}\)
Proton charge: \(Q\approx 1.60\times 10^{-19}~\textrm{C}\)
\begin{equation*} \frac{1}{4\pi\epsilon_0}\approx 9.0\times 10^{9}~\textrm{Nm}^2/\textrm{C}^2 \end{equation*}
Gravitational constant: \(G\approx 6.67\times 10^{-11}~\textrm{Nm}^2/\textrm{kg}^2\)
Proton mass: \(M\approx 1.67\times 10^{-27}~\textrm{kg}\) Electron charge: \(q\approx -1.60\times 10^{-19}~\textrm{C}\)
Electron mass: \(m\approx 9.11\times 10^{-31}~\textrm{kg}\)
Therefore:\begin{align*} U_{\textrm elec} &= qV_{\textrm elec}\\ &= \frac{1}{4\pi\epsilon_0}\frac{qQ}{a_0}\\ &\approx -4.35\times 10^{-18}~J \\ U_{\textrm grav} &= mV_{\textrm grav}\\ &= -G \frac{mM}{a_0}\\ &\approx -1.92\times 10^{-57}~J \end{align*}
The potential energy due to gravity in an atom is 28 orders of magnitude smaller than the electromagnetic force! (Notice that \(qV_{\textrm elec}\) is not the binding energy of the electron. The electron in the Bohr model also has kinetic energy.)
Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
See constants above. Therefore:
\begin{align*} V_{\textrm elec} &= \frac{1}{4\pi\epsilon_0}\frac{Q}{a_0}\approx 27.2~\textrm{J}/\textrm{C}\\ V_{\textrm grav} &= -G \frac{M}{a_0} \approx -2.11\times 10^{-27}~\textrm{J}/\textrm{kg} \end{align*}
Even though these are measured in the same system of units, they are not in the same units and cannot be compared.
Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?
One difference is that charges of both signs exist, so that electromagnetic forces can cancel each other. The earth is essentially electromagnetically neutral, so that we do not feel an electromagnetic attraction to the earth. (Be careful, the earth is essentially electromagnetically neutral ON AVERAGE. However, there can be largish charge differences locally--this is what causes lightening, for example.) Gravitational forces, however, can only add, so that the total force due to lots of mass can be very large.
Another difference is that masses always attract, so that the gravitational potential is negative. Whereas, positive charges repel each other so that the electrostatic potential of a positive charge is positive.