A circular disc of radius $\mathrm{R}$ carries surface charge density $\sigma \left(\mathrm{r}\right)={\sigma }_0\left(1-\frac{\mathrm{r}}{\mathrm{R}}\right),$ where ${\sigma }_0$ is a constant and $\mathrm{r}$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is ${\varphi }_0.$Electric flux through another spherical surface of radius $\frac{\mathrm{R}}{4}$ and concentric with the disc is $\varphi .$ Then the ratio is $\frac{{\varphi }_0}{\varphi }=\frac{x}{5}$, the value of $x$ is___

A uniformly charged and infinitely long line having a linear charge density $\lambda$ is placed at a normal distance $\mathrm{y}$ from a point $\mathrm{O}$ . Consider a sphere of radius $\mathrm{R}$ with $\mathrm{O}$ as centre and $R>y$ . Electric flux through the surface of the sphere is:

The switch $S$ shown in figure is kept closed for a long time and then opened at $t=0,$ then the current in the middle $20 \mathrm{\Omega }$ resistor at $t=0.25 \mathrm{ms}$ is

There are two conducting spheres of radius $a$ and $b$ $(b>a)$ carrying equal and opposite charges. They are placed at a separation $d(>>>a\mathit{ }\mathit{\&} b).$ The capacitance of the system is

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance $C$. The switch $S_1$ is pressed first to fully charge the capacitor $C_1$ and then released. The switch $S_2$ is then pressed to charge the capacitor $C_2$. After some time, $S_2$ is released and then $S_3$ is pressed. After some time,

In the given circuit diagram, $E=12 \mathrm{V}, C_1=4 \mathrm{\mu F}, C_2=2 \mathrm{\mu F}$, $C_3=6 \mathrm{\mu F}$ and $C_4=3 \mathrm{\mu F}$. Find the heat produced in the circuit (in $\mathrm{\mu J}$) after switch $S$ is shorted.

In the figure shown below, the initial status of capacitors and their connection is shown. Which of the following is incorrect about this circuit?

Three capacitors of capacitance $C, C \& 3C$ are connected to a $14 \mathrm{V}$ battery and grounded, as shown in figure. Find potential (in volt) of point $A$.

Four plates of the same area of cross-section are joined as shown in the figure. The distance between each plate is $d$. The equivalent capacity across $A$ and $B$ will be: