Question 24.101: In the quark model of fundamental particles, a proton is com......
In the quark model of fundamental particles, a proton is composed of three quarks: two “up” quarks, each having charge +2e/3, and one “down” quark, having charge -e/3. Suppose that the three quarks are equidistant from one another. Take that separation distance to be 1.32 × 10^{-15} m and calculate the electric potential energy of the system of (a) only the two up quarks and (b) all three quarks.
(a) Let the quark-quark separation be r. To “naturally” obtain the eV unit, we only plug in for one of the e values involved in the computation:
\begin{aligned} U_{\text {up -up }} & =\frac{1}{4 \pi \varepsilon_0} \frac{(2 e / 3)(2 e / 3)}{r}=\frac{4 \,k e}{9 r} e=\frac{4\left(8.99 \times 10^9\, N \cdot m ^2 / C ^2\right)\left(1.60 \times 10^{-19} \,C \right)}{9\left(1.32 \times 10^{-15}\, m \right)} e \\ & =4.84 \times 10^5 \,eV =0.484 \,MeV . \end{aligned}
(b) The total consists of all pair-wise terms:
U=\frac{1}{4 \pi \varepsilon_0}\left[\frac{(2 e / 3)(2 e / 3)}{r}+\frac{(-e / 3)(2 e / 3)}{r}+\frac{(-e / 3)(2 e / 3)}{r}\right]=0 .