Question

3 Neutron Atomic nuclei are made up of protons and neutrons. Protons and neutrons are made up of quarks. In this problem, we will investigate the electric field of a neutron. The neutron consists of three quarks: an up quark with a charge of +2e/3 and two down quarks, each with a charge of-c/3. (Here, e-1.6-10 19 C is the charge of an electron) In one model of a neutron, the up quark occupies a radius of about a = 0.3 fm. The two down quarks occupy the distance between α and the radius of the neutron. R=2fm. For this problem, you may consider the neutron to be a uniformly charged sphere of charge +2e/3 and radius a surrounded by a uniformly charged spherical shell of inner radius a and outer radius R and charge -2e/3, as shown in the figure. (a) Use Gauss' law to find the electrie field as a function of distance from the center of the neutr Analyze all three regions of interest: r<a, a<r< R, and r>R (b) Make a plot of electric field strength as a function of radius for 0 SrS fm. Your plot should be neat and accurate. Use graph paper plotting software. Be sure to include units, label your axes, and clearly indicate the location of a and R on your plot

Answer 1

Integral form of Gauss Law

$\int \vec{E}\cdot d\vec{S}=\frac{Q_{enc}}{\varepsilon_0}$

For spherically symmetric geometry this becomes,

$\vec{E}=\frac{Q_{enc}}{4\pi\varepsilon_0r^2}\hat{r}$

at a radius r.

For simplicity, I'll use

$K=\frac{1}{4\pi\varepsilon_0}$

(a) For r<a,

$E_1=\frac{2eKr}{3a^3}$

for a<r<R,

$E_2=\frac{2eK}{3r^2}-\frac{2eK(r^3-a^3)}{3r^2(R^3-a^3)}=\frac{2eK(R^3-r^3)}{3r^2(R^3-a^3)}=\frac{2eK(8a^3-r^3)}{21r^2a^3}$

For r>R,

$Q_{enc}=0$

$\Rightarrow E=0$