Let a total charge \(2Q\) be distributed in a sphere of radius \(R\), with the charge density given by \(\rho(r)=kr \), where \(r\) is the distance from the centre. Two charges \(A\) and \(B\), of \(-Q \) each, are placed on diametrically opposite points, at equal distance, a from the centre. If \(A\) and \(B\) do not experience any force, then:
1. \(a=8^{-1/4}R\)
2. \(a=2^{-1/4}R\)
3. \(a=\frac{3R}{2^{1/4}}\)
4. \(a=R/\sqrt{3}\)

Consider the force \(F\) on a charge \(q\)  due to a uniformly charged spherical shell of radius \(R\) carrying charge \(Q\) distributed uniformly over it. Which one of the following statements is true for \(F\), if \(q\) is placed at a distance \(r\) from the centre of the shell? 

Two electrons are fixed at a distance of \(2d\) apart. A proton is placed at the midpoint between them and is displaced slightly by a distance \(x\) \((x\ll d)\) perpendicular to the line joining the two fixed electrons. The proton will undergo simple harmonic motion with an angular frequency given by:
(here, \(m\) is the mass of the proton, \(q\) is the magnitude of the charge, and \(\varepsilon_0\)​ is the permittivity of free space)

1. \(\sqrt{\left( {\dfrac{2 q^2}{\pi \varepsilon_0 m d^3}}\right)}\)

2. \(\sqrt{\left (\dfrac{\pi \varepsilon_0 {md}^3}{2 {q}^2} \right )}\)

3. \(\sqrt{\left ({\dfrac{ q^2}{2\pi \varepsilon_0 m d^3}}\right)}\)

4. \(\sqrt{\left( \dfrac{2\pi \varepsilon_0 {md}^3}{m{q}^2}\right)}\)

Two small spheres each of mass \(10~\text{mg}\) are suspended from a point by threads \(0.5~\text{m}\) long. They are equally charged and repel each other to a distance of \(0.20~\text m.\). The charge on each of the spheres is \(\frac{a}{21} \times 10^{-8} \mathrm{~C}.\) The value of '\(a\)' will be:
(given \(g=10~\mathrm{ms^{-2}}\))

Two identical conducting spheres, each with negligible volume, carry initial charges of \(2.1~\text{nC}\) and \(-0.1~\text{nC},\) respectively. The spheres are brought into contact, allowing charge to redistribute, and are then separated by a distance of \(0.5~\text m.\) The electrostatic force acting between the spheres is: 
\(\left ( 4 \pi \varepsilon_0=\frac{1}{9 \times 10^9}~\text{in SI units} \right )\) 
1. \(36 \times 10^{-7}~\mathrm{N}\)
2. \(36 \times 10^{-9}~\mathrm{N}\)
3. \(18 \times 10^{-7}~\mathrm{N}\)
4. \(18 \times 10^{-9}~\mathrm{N}\)