Two uniformly charged non-conducting hemispherical shells each having uniform charge density $\sigma$ and radius $R$ form a complete sphere (not stuck together) and surround a concentric spherical conducting shell of radius $\frac{R}{2}$. If hemispherical parts are in equilibrium then minimum surface charge density of inner conducting shell is:

A charge $q$ is placed at the centre of a conducting spherical shell of radius $R$, which is given a charge $Q$. An external charge $Q^{\text{'}}$ is also present at distance $R^{\text{'}}(R^{\text{'}}>R)$ from $q$. Then the resultant field will be best represented for region $r<R$ by: [ where $r$ is the distance of the point from $q$ ]

A charge $q$ is placed at the centre of a conducting spherical shell of radius $R$, which is given a charge $Q$. An external charge $Q\text{'}$ is also present at distance $R\text{'}(R\text{'}>R)$ from $q$. Then the resultant field will be best represented for region $r<R$ by, [where $r$ is the distance of the point from $q$]

In the above question, if $Q\text{'}$ is removed then which option is correct.

Two identical charged spheres suspended from a common point by two light strings of length $l$, are initially at a distance $d\left(\ll l\right)$ apart due to their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity $v$. If $x$ denotes the distance between the spheres, the $v$ varies as

Acidified water from a certain reservoir kept at a potential $V$ falls in the form of small droplets each of radius $r$ through a hole into a hollow conducting sphere of radius $a$. The sphere is insulated and is initially at zero potential. If the drops continue to fall until the sphere is half full, the potential acquired by the sphere is