Three identical charges are placed on three vertices of a square. If the force acting between $q_1$ and $q_2$ is $F_{12}$ and between $q_1$ and $q_3$ is $F_{13}$,then $\frac{F_{13}}{F_{12}} = $ . . . . . . .

$A$ small uncharged conducting sphere is placed in contact with an identical sphere having a charge of $4 \times 10^{-8} \text{ C}$. They are then separated to a distance such that the force of repulsion between them is $9 \times 10^{-3} \text{ N}$. The distance between them is (Take $\frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \text{ N m}^2 \text{ C}^{-2}$ in $SI$ units). (in $\text{ cm}$)

Two unit negative charges are placed on a straight line. $A$ positive charge $q$ is placed exactly at the midpoint between these unit charges. If the system of these three charges is in equilibrium,the value of $q$ (in $C$) is

$(a)$ In a quark model of elementary particles,a neutron is made of one up quark [ 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 the electrostatic potential energy of the neutron and compare it with its mass $939 \ MeV$. $(b)$ Repeat the above exercise for a proton which is made of two up and one down quark.

Two similar tiny balls of mass $m$,each carrying charge $q$,are hung from silk threads of length $l$ as shown in the figure. These are separated by a distance $x$ and the angle $2\theta$ is small. Then,for equilibrium:

Three charges $+3q$,$Q$,and $+q$ are placed in a straight line of length $\ell$ at points at distances $0$,$\frac{\ell}{2}$,and $\ell$ respectively. The value of $Q$ in order to have the net force on $+q$ to be zero is $Q = xq$. The value of $x$ is: