An electric field $\overrightarrow{E} = 4x \hat{i} - (y^2 + 1) \hat{j} \text{ N/C}$ passes through the box shown in the figure. The flux of the electric field through surfaces $ABCD$ and $BCGF$ are marked as $\phi_I$ and $\phi_{II}$ respectively. The difference $(\phi_I - \phi_{II})$ is (in $\text{Nm}^2/C$):

Obtain Gauss's law from the flux associated with a sphere of radius $r$ and charge $q$ at its centre.

$A$ thin spherical shell encloses a concentric solid sphere. The radius of the shell is $(0.060)^{1/2} \ m$ and its surface charge density is $-10^{-5} \ C/m^2$. The radius of the solid sphere is $(0.01)^{1/3} \ m$ and its volumetric charge density is $3 \times 10^{-5} \ C/m^3$. $\varepsilon_0$ is the permittivity of free space in $C^2/Nm^2$. The electric flux through a spherical surface concentric with the spherical shell and of radius greater than that of the shell in $V-m$ is:

$A$ metallic solid sphere is placed in a uniform electric field. The lines of force follow the path$(s)$ shown in the figure as:

If an electric charge $q$ is placed at the centre of a cube,then the flux associated with each surface of the cube is . . . . . . .

If $\oint_s \vec{E} \cdot \overrightarrow{d S}=0$ over a surface,then: