Two conducting hollow sphere of radius $R$ and $3R$ are found to have $Q$ charge on outer surface when both are connected with a long wire and $q'$ charge is kept at the centre of bigger sphere. Then which one is true

Describe schematically the equipotential surfaces corresponding to

$(a)$ a constant electric field in the $z-$direction,

$(b)$ a field that uniformly increases in magnitude but remains in a constant (say, $z$) direction,

$(c)$ a single positive charge at the origin, and

$(d)$ a uniform grid consisting of long equally spaced parallel charged wires in a plane

Three equal charges are placed at the corners of an equilateral triangle. Which of the graphs below correctly depicts the equally-spaced equipotential surfaces in the plane of the triangle? (All graphs have the same scale.)

The equation of an equipotential line in an electric field is $y = 2x$, then the electric field strength vector at $(1, 2)$ may be

Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.

A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius $R$ as shown in the figure. Which of the following statements is/are correct?

(IMAGE)

$[A]$ The electric flux passing through the curved surface of the hemisphere is $-\frac{\mathrm{Q}}{2 \varepsilon_0}\left(1-\frac{1}{\sqrt{2}}\right)$

$[B]$ Total flux through the curved and the flat surfaces is $\frac{Q}{\varepsilon_0}$

$[C]$ The component of the electric field normal to the flat surface is constant over the surface

$[D]$ The circumference of the flat surface is an equipotential