Embibe Experts Solutions for Chapter: Electrostatics, Exercise 1: JEE Main - 1 February 2023 Shift 2

As shown in figure, a cuboid lies in a region with electric field $E=2x^2\hat{\mathrm{i}}-4y\hat{\mathrm{j}}+6\hat{\mathrm{k}} \mathrm{N} {\mathrm{C}}^{-1}$. The magnitude of charge within the cuboid is $n{\epsilon }_0 \mathrm{C}$.

The value of $n$ is ______ (if dimension of cuboid is $1\times 2\times 3 {\mathrm{m}}^3$)

Expression for an electric field is given by $\overrightarrow{E}=4000 x^2 \hat{\mathrm{i}} \mathrm{V} {\mathrm{m}}^{-1}$. The electric flux through the cube of side $20 \mathrm{cm}$ when placed in electric field (as shown in the figure) is ______ $\mathrm{V} \mathrm{cm}$.

Considering a group of positive charges, which of the following statements is correct?

Two parallel plate capacitors $C_1$ and $C_2$ each having capacitance of $10 \mathrm{\mu F}$ are individually charged by a $100 \mathrm{V}$ D.C. source. Capacitor $C_1$ is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor $C_2$ is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor $C_1$ is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be _____ $\mathrm{V}$.

(Assuming Dielectric constant $=10$)

Let $\sigma$ be the uniform surface charge density of two infinite thin plane sheets shown in figure. Then the electric fields in three different region $E_I , E_{II}$ and $E_{III}$

Two equal positive point charges are separated by a distance $2a$. The distance of a point from the centre of the line joining two charges on the equatorial line (perpendicular bisector) at which force experienced by a test charge $q_0$ becomes maximum is $\frac{a}{\sqrt{x}}$. The value of $x$ is ______.

Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Two metallic spheres are charged to the same potential. One of them is hollow and another is solid, and both have the same radii. Solid sphere will have lower charge than the hollow one.

Reason R : Capacitance of metallic spheres depend on the radii of spheres.

In the light of the above statements, choose the correct answer from the options given below.

A cubical volume is bounded by the surfaces $x=0, x=a, y=0, y=a, z=0, z=a$. The electric field in the region is given by $\overrightarrow{\mathit{E}}=E_{0}x\hat{\mathit{i}}$. Where $E_0=4\times {10}^4 {\mathrm{NC}}^{-1} {\mathrm{m}}^{-1}$. If $a=2 \mathrm{cm}$, the charge contained in the cubical volume is $Q\times {10}^{–14} \mathrm{C}$. The value of $Q$ is ______.

(Take ${ϵ}_0=9\times {10}^{-12} {\mathrm{C}}^2 {\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$)