$(a)$ In a quark model of elementary particles, a neutron is made of one up quarks [ charge $\frac{2}{3}e$ ] and two down quarks [ charges $ - \frac{1}{3}e$ ]. Assume that they have a triangle configuration with side length of the order of ${10^{ - 15}}$ $m$. Calculate electrostatic potential energy of neutron and compare it with its mass $939$ $Me\,V$. $(b)$ Repeat above exercise for a proton which is made of two up and one down quark.
$(a)$ In a quark model of elementary particles, a neutron is made of one up quarks [ charge $\frac{2}{3}e$ ] and two down quarks [ charges $ - \frac{1}{3}e$ ]. Assume that they have a triangle configuration with side length of the order of ${10^{ - 15}}$ $m$. Calculate electrostatic potential energy of neutron and compare it with its mass $939$ $Me\,V$. $(b)$ Repeat above exercise for a proton which is made of two up and one down quark.
$(a)$ Electrostatic potential energy $\mathrm{U}=\frac{k q_{1} q_{2}}{r}$ The system of three charges is as below,
Electrostatic potential energy,
$\mathrm{U}=k\left(\frac{q_{d} q_{d}}{r}+\frac{q_{u} q_{d}}{r}+\frac{q_{u} q_{d}}{r}\right)$
$\mathrm{U}=k\left[\frac{\left[\left(-\frac{1}{3} e\right)\left(-\frac{1}{3} e\right)\right.}{r}+\frac{\left(\frac{2}{3} e\right)\left(-\frac{1}{3} e\right)}{r}+\frac{\left(\frac{2}{3} e\right)\left(-\frac{1}{3} e\right)}{r}\right]$
$\mathrm{U}=\frac{k}{r}\left[\frac{1}{9} e^{2}-\frac{2}{9} e^{2}-\frac{2}{9} e^{2}\right]$
$=\frac{k e^{2}}{q r}[1-2-2]$
$=\frac{k e^{2}}{q r} \times(-3)$
$=\frac{9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2} \times(-3)}{9 \times 10^{-15}}$
$=-7.68 \times 10^{-14} \mathrm{~J}$
$\therefore \quad \mathrm{U}=\frac{-7.68 \times 10^{-14}}{-1.6 \times 10^{-19}} \mathrm{eV} $
$\therefore \quad\mathrm{U}=4.8 \times 10^{5} \mathrm{eV}$
$=0.48 \times 10^{6} \mathrm{eV}=0.48 \mathrm{MeV}$
Similar Questions
The electrostatic potential $V$ at a point on the circumference of a thin non-conducting disk of radius $r$ and uniform charge density $\sigma$ is given by equation $V = 4 \sigma r$. Which of the following expression correctly represents electrostatic energy stored in the electric field of a similar charged disk of radius $R$?
The electrostatic potential $V$ at a point on the circumference of a thin non-conducting disk of radius $r$ and uniform charge density $\sigma$ is given by equation $V = 4 \sigma r$. Which of the following expression correctly represents electrostatic energy stored in the electric field of a similar charged disk of radius $R$?
Two particles each of mass $m$ and charge $q$ are separated by distance $r_1$ and the system is left free to move at $t = 0$. At time $t$ both the particles are found to be separated by distance $r_2$. The speed of each particle is
Two particles each of mass $m$ and charge $q$ are separated by distance $r_1$ and the system is left free to move at $t = 0$. At time $t$ both the particles are found to be separated by distance $r_2$. The speed of each particle is
Six charges $+ q ,- q ,+ q ,- q ,+ q$ and $- q$ are fixed at the corners of a hexagon of side $d$ as shown in the figure. The work done in bringing a charge $q _0$ to the centre of the hexagon from infinity is :$\left(\varepsilon_0-\right.$ permittivity of free space)
Six charges $+ q ,- q ,+ q ,- q ,+ q$ and $- q$ are fixed at the corners of a hexagon of side $d$ as shown in the figure. The work done in bringing a charge $q _0$ to the centre of the hexagon from infinity is :$\left(\varepsilon_0-\right.$ permittivity of free space)
- [NEET 2022]
A bullet of mass $m$ and charge $q$ is fired towards a solid uniformly charged sphere of radius $R$ and total charge $+ q$. If it strikes the surface of sphere with speed $u$, find the minimum speed $u$ so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces)
A bullet of mass $m$ and charge $q$ is fired towards a solid uniformly charged sphere of radius $R$ and total charge $+ q$. If it strikes the surface of sphere with speed $u$, find the minimum speed $u$ so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces)
A point chargr $Q$ is fixed A small charge $q$ and mass $m$ is given a velocity $v_0$ from infinity & perpendicular distance $r_0$ as shown. If distance of closest approach is $r_0/2$. The value of $q$ is [Given $mv_0^2 = \frac{{{Q^2}}}{{4\pi { \in _0}\,{r_0}}}$]
A point chargr $Q$ is fixed A small charge $q$ and mass $m$ is given a velocity $v_0$ from infinity & perpendicular distance $r_0$ as shown. If distance of closest approach is $r_0/2$. The value of $q$ is [Given $mv_0^2 = \frac{{{Q^2}}}{{4\pi { \in _0}\,{r_0}}}$]