Two conducting spheres $A$ and $B$ have radii $1 \, mm$ and $2 \, mm$ respectively and are charged. They are placed at a distance of $5 \, cm$. When they are connected by a conducting wire,the ratio of the electric field intensities on their surfaces at equilibrium is:

$A$ sphere '$1$' with radius $R$ has charge $q$. Sphere '$2$' with radius $3R$ is far from sphere '$1$' and is initially uncharged. If the two spheres are now connected with a thin conducting wire,then the ratio $\frac{\sigma_1}{\sigma_2}$ of the surface charge densities is

Consider two conducting spheres of radii $R_1$ and $R_2$ with $(R_1 > R_2)$. If the two are at the same potential,the larger sphere has more charge than the smaller sphere. State whether the charge density of the smaller sphere is more or less than that of the larger one.

Consider an initially neutral hollow conducting spherical shell with inner radius $r$ and outer radius $2r$. $A$ point charge $+Q$ is now placed inside the shell at a distance $r/2$ from the centre. The shell is then grounded by connecting the outer surface to the earth. $P$ is an external point at a distance $2r$ from the point charge $+Q$ on the line passing through the centre and the point charge $+Q$ as shown in the figure. The magnitude of the force on a test charge $+q$ placed at $P$ will be

Four very large metal plates are given the charges as shown in the figure. The middle two plates are then connected through a wire. Find the charge that will flow through the wire.

Which of the following statements is always true for a charged conductor?
$(1)$ The electric field just outside the surface is parallel to the surface.
$(2)$ $E_{in} = 0$
$(3)$ Electric field lines are perpendicular to the equipotential surface.