$A$ and $B$ are two concentric conducting spherical shells. $A$ is given a positive charge $+Q$ while $B$ is earthed. Then:

$A$ hollow closed conductor of irregular shape is given some charge. Which of the following statements are correct?

$A$ metallic sphere $A$ isolated from the ground is charged to $+50 \mu C$. This sphere is brought into contact with another isolated metallic sphere $B$ of half the radius of sphere $A$. Then the charge on the two isolated spheres $A$ and $B$ are in the ratio:

$(a)$ $A$ conductor $A$ with a cavity as shown in Figure $(a)$ is given a charge $Q$. Show that the entire charge must appear on the outer surface of the conductor.
$(b)$ Another conductor $B$ with charge $q$ is inserted into the cavity keeping $B$ insulated from $A$. Show that the total charge on the outside surface of $A$ is $Q+q$ [Figure $(b)$].
$(c)$ $A$ sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way.

$A$ dipole consisting of two charges $\pm q$ separated by a distance $2a$ is placed with its centre at a distance $D$ from the center of a grounded conducting sphere of radius $R$ $(D \gg a)$. When the dipole moment vector is perpendicular to the line joining the two centres (those of the dipole and sphere),the total charge induced on the sphere is:

Two metal spheres,one of radius $R$ and the other of radius $2R$,respectively,have the same surface charge density $\sigma$. They are brought in contact and separated. What will be the new surface charge densities on them?