Electric Field Lines, Flux and Gauss's Law

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LEVELJEE Main

In a uniformly charged sphere of total charge $Q$ and radius $R$ , the electric field $E$ is plotted as function of distance from the centre. The graph which would correspond to the above will be

Consider the force $F$ on a charge $q$ due to a uniformly charged spherical shell of radius $R$ carrying charge $Q$ distributed uniformly over it. Which one of the following statement is true for $F$ , if $q$ is placed at distance $r$ from the centre of the shell ?

(A)

F = \frac{1}{4 π ε _{0}} \frac{Qq}{R ^{2}}

(for

r < R

)

(B)

\frac{1}{4 π ε _{0}} \frac{Qq}{R ^{2}} > F > 0

(for

r < R

)

(C)

F = \frac{1}{4 π ε _{0}} \frac{Qq}{r ^{2}}

(for

r > R

)

(D)

F = \frac{1}{4 π ε _{0}} \frac{Qq}{R ^{2}}

(for all

r

)

JEE Main 2020

LEVELJEE Main

In finding the electric field using Gauss law the formula $∣ E ∣ = \frac{q _{e n c}}{ε _{0} ∣ A ∣}$ is applicable. In the formula, $ε_{0}$ is permittivity of free space, $A$ is the area of Gaussian surface and $q_{e n c}$ is charge enclosed by the Gaussian surface. This equation can be used in which of the following situation?

(A)

Only when the Gaussian surface is an equipotential surface and

∣ E ∣

is constant on the surface.

(B)

Only when the Gaussian surface is an equipotential surface.

(C)

For any choice of Gaussian surface.

(D)

Only when

∣ E ∣ = constant

on the surface.

JEE Main 2021

LEVELJEE Main

The electric field in a region is given $E = (\frac{3}{5} E_{0} \hat{i} + \frac{4}{5} E_{0} \hat{j}) \frac{N}{C}$ . The ratio of flux of reported field through the rectangular surface of area $0.2 m^{2}$ (parallel to YZ-plane) to that of the surface of area $0.3 m^{2}$ (parallel to XZ- plane) is $a : b$ , where $a = \dots\dots\dots$ . [Here $\hat{i}$ , $\hat{j}$ and $\hat{k}$ are unit vectors along X, Y and Z-axes, respectively]

JEE Main 2019

LEVELJEE Advanced

Let a total charge $2 Q$ be distributed in a sphere of radius $R$ , with the charge density given by $ρ (r) = k r$ , where $r$ is the distance from the centre. Two charges $A$ and $B$ , of $- Q$ each, are placed on diametrically opposite points, at equal distance $a$ , from the centre. If $A$ and $B$ do not experience any force, then

(A)

a = 8^{- 1/4} R

(B)

a = \frac{3 R}{2 ^{1/4}}

(C)

a = 2^{- 1/4} R

(D)

a = R / 3

JEE Main 2021

LEVELJEE Advanced

Find out the surface charge density at the intersection of point $X = 3 m$ plane and X-axis, in the region of uniform line charge of $8 nC/m$ lying along the Z-axis in free space.

(A)

0.424 nCm^{- 2}

(B)

47.88 nCm^{- 2}

(C)

0.07 nCm^{- 2}

(D)

4.0 nCm^{- 2}

JEE Main 2020

LEVELJEE Advanced

Consider a sphere of radius $R$ which carries a uniform charge density $ρ$ . If a sphere of radius $\frac{R}{2}$ is carved out of it, as shown in the figure the ratio $\frac{∣ E _{A} ∣}{∣ E _{B} ∣}$ of magnitude of electric field $E_{A}$ and $E_{B}$ respectively, at points $A$ and $B$ due to the remaining portion is

(A)

\frac{21}{34}

(B)

\frac{18}{54}

(C)

\frac{17}{54}

(D)

\frac{18}{34}

LEVELJEE Advanced

Let $ρ (r) = \frac{Q}{π R ^{4}} r$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q$ . For a point $P$ inside the sphere at distance $r_{1}$ from the centre of the sphere, the magnitude of electric field is

(A)

zero

(B)

\frac{Q}{4 π ϵ _{0} r _{1}^{2}}

(C)

\frac{Q r _{1}^{2}}{4 π ϵ _{0} R ^{4}}

(D)

\frac{Q r _{1}^{2}}{3 π ϵ _{0} R ^{4}}

JEE Main 2020

LEVELJEE Main

An electric field $E = 4 x \hat{i} - (y^{2} + 1) \hat{j}$ N/C passes through the box shown in figure. The flux of the electric field through surfaces $A B C D$ and $B C GF$ are marked as $ϕ_{I}$ and $ϕ_{I I}$ , respectively. The difference between $(ϕ_{I} - ϕ_{I I})$ is (in $N-m^{2} / C$ ) ...... .

LEVELJEE Advanced

Let there be a spherically symmetric charge distribution with charge density varying as $ρ (r) = ρ_{0} (\frac{5}{4} - \frac{r}{R})$ upto $r = R$ and $ρ (r) = 0$ for $r > R$ , where $r$ is the distance from the origin. The electric field at a distance $r, (r < R)$ from the origin is given by

(A)

\frac{4 π ρ _{0} r}{3 ε _{0}} (\frac{5}{3} - \frac{r}{R})

(B)

\frac{ρ _{0} r}{4 ε _{0}} (\frac{5}{3} - \frac{r}{R})

(C)

\frac{4 ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})

(D)

\frac{ρ _{0} r}{3 ε _{0}} (\frac{5}{4} - \frac{r}{R})

LEVELJEE Main

A thin spherical shell of radius $R$ has charge $Q$ spread uniformly over its surface. Which of the following graphs most closely represents the electric field $E (r)$ produced by the shell in the range $0 \leq r < \infty$ , where $r$ is the distance from the centre of the shell?

A charged ball $B$ hangs from a silk thread $S$ , which makes an angle $θ$ with a large charged conducting sheet $P$ , as shown in the figure. The surface charge density $σ$ of the sheet is proportional to

(A)

cos θ

(B)

cot θ

(C)

sin θ

(D)

tan θ

LEVELJEE Main

A charged particle $q$ is placed at the centre $O$ of cube of length $L (A B C D E F G H)$ . Another same charge $q$ is placed at a distance $L$ from $O$ . Then, the electric flux through $A B C D$ is

(A)

\frac{q}{4 π ε _{0} L}

(B)

\frac{q}{2 π ε _{0} L}

(C)

\frac{q}{2 π ε _{0} L}

(D)

None of these

LEVELJEE Main

The electrostatic potential inside a charged spherical ball is given by $ϕ = a r^{2} + b$ , where $r$ is the distance from the centre and $a, b$ are constants. Then, the charge density inside the ball is

(A)

- 6 a ε_{0} r

(B)

- 24 π a ε_{0}

(C)

- 6 a ε_{0}

(D)

- 24 π a ε_{0} r

JEE Main 2012

LEVELJEE Advanced

This question has Statement I and Statement II. Of the four choices given after the statements, choose the one that best describes the two statements. An insulating solid sphere of radius $R$ has a uniform positive charge density $ρ$ . As a result of this uniform charge distribution, there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at a point outside the sphere. The electric potential at infinite is zero. Statement I When a charge $q$ is taken from the centre of the surface of the sphere, its potential energy changes by $\frac{q ρ}{3 ε _{0}}$ . Statement II The electric field at a distance $r (r < R)$ from the centre of the sphere is $\frac{ρ r}{3 ε _{0}}$ .

(A)

Statement I is false, Statement II is true

(B)

Statement I is true, Statement II is false

(C)

Statement I is true, Statement II is true; Statement II the correct explanation for Statement I

(D)

Statement I is true, Statement II is true; Statement II is not the correct explanation of Statement I

JEE Main 2015

LEVELJEE Main

A long cylindrical shell carries positive surface charge $σ$ in the upper half and negative surface charge $- σ$ in the lower half. The electric field lines around the cylinder will look like figure given in (figures are schematic and not drawn to scale)

Charge is distributed within a sphere of radius $R$ with a volume charge density $ρ (r) = \frac{A}{r ^{2}} e^{- 2 r / a}$ , where $A$ and $a$ are constants. If $Q$ is the total charge of this charge distribution, the radius $R$ is

(A)

a lo g (\frac{1}{1 - \frac{Q}{2 π a A}})

(B)

a lo g (1 - \frac{Q}{2 π a A})

(C)

\frac{a}{2} lo g (1 - \frac{Q}{2 π a A})

(D)

\frac{a}{2} lo g (\frac{1}{1 - \frac{Q}{2 π a A}})

JEE Main 2020

LEVELJEE Advanced

Two infinite planes each with uniform surface charged density $+ σ$ are kept in such a way that the angle between them is $3 0^{\circ}$ . The electric field in the region shown between them is given by

(A)

\frac{σ}{2 ε _{0}} [(1 + 3) \overset{y}{^} - \frac{x ^}{2}]

(B)

\frac{σ}{2 ε _{0}} [(1 - \frac{3}{2}) \overset{y}{^} - \frac{x ^}{2}]

(C)

\frac{σ}{2 ε _{0}} [(1 + 3) \overset{y}{^} + \frac{x ^}{2}]

(D)

\frac{σ}{ε _{0}} [(1 + \frac{3}{2}) \overset{y}{^} + \frac{x ^}{2}]

JEE Main 2020

LEVELJEE Advanced

Two charged thin infinite plane sheets of uniform surface charge densities $σ_{+}$ and $σ_{-}$ , where $∣ σ_{+} ∣ > ∣ σ_{-} ∣$ , intersect at right angle. Which of the following best represents the electric field lines for this system?

The electric field in a region is given by $E = \frac{2}{5} E_{0} \hat{i} + \frac{3}{5} E_{0} \hat{j}$ with $E_{0} = 4.0 \times 1 0^{3} N/C$ . The flux of this field through a rectangular surface area $0.4 m^{2}$ parallel to the yz-plane is ............ $N-m^{2} C^{- 1}$ .

LEVELJEE Main

If the electric flux entering and leaving an enclosed surface respectively is $ϕ_{1}$ and $ϕ_{2}$ , the electric charge inside the surface will be

(A)

(ϕ_{2} - ϕ_{1}) ε_{0}

(B)

\frac{( ϕ _{1} + ϕ _{2} )}{ε _{0}}

(C)

\frac{( ϕ _{2} - ϕ _{1} )}{ε _{0}}

(D)

(ϕ_{1} + ϕ_{2}) ε_{0}

JEE Main 2016

LEVELJEE Advanced

The region between two concentric spheres of radii $a$ and $b$ , respectively (see the figure), has volume charge density $ρ = \frac{A}{r}$ , where $A$ is a constant and $r$ is the distance from the centre. At the centre of the spheres is a point charge $Q$ . The value of $A$ such that the electric field in the region between the spheres will be constant is

(A)

\frac{Q}{2 π a ^{2}}

(B)

\frac{Q}{2 π ( b ^{2} - a ^{2} )}

(C)

\frac{2 Q}{π ( a ^{2} - b ^{2} )}

(D)

\frac{2 Q}{π a ^{2}}

JEE Main 2021

LEVELJEE Advanced

The total charge enclosed in an incremental volume of $2 \times 1 0^{- 9} m^{3}$ located at the origin is ...... nC, if electric flux density of its field is found as $D = e^{- x} sin y \hat{i} - e^{- x} cos y \hat{j} + 2 z \hat{k} C/m^{2}$ .

JEE Main 2021

LEVELJEE Main

Given below are two statements: Statement I An electric dipole is placed at the centre of a hollow sphere. The flux of electric field through the sphere is zero but the electric field is not zero anywhere in the sphere. Statement II If $R$ is the radius of a solid metallic sphere and $Q$ be the total charge on it. The electric field at any point on the spherical surface of radius $r$ ( $r < R$ ) is zero but the electric flux passing through this closed spherical surface of radius $r$ is not zero. In the light of the above statements, choose the correct answer from the options given below.

(A)

Both Statement I and Statement II are true.

(B)

Statement I is true but Statement II is false.

(C)

Both Statement I and Statement II are false.

(D)

Statement I is false but Statement II is true.

JEE Main 2021

LEVELJEE Main

A point charge of $+ 12 μ C$ is at a distance $6 cm$ vertically above the centre of a square of side $12 cm$ as shown in figure. The magnitude of the electric flux through the square will be ......... $\times 1 0^{3} N-m^{2} /C$ .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find out the surface charge density at the intersection of point X=3 m plane and X-axis, in the region of uniform line charge of 8 nC/m lying along the Z-axis in free space.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A charged particle q is placed at the centre O of cube of length L(ABCDEFGH). Another same charge q is placed at a distance L from O. Then, the electric flux through ABCD is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the electric flux entering and leaving an enclosed surface respectively is ϕ1​ and ϕ2​, the electric charge inside the surface will be

Find out the surface charge density at the intersection of point $X = 3 m$ plane and X-axis, in the region of uniform line charge of $8 nC/m$ lying along the Z-axis in free space.

A charged particle $q$ is placed at the centre $O$ of cube of length $L (A B C D E F G H)$ . Another same charge $q$ is placed at a distance $L$ from $O$ . Then, the electric flux through $A B C D$ is

If the electric flux entering and leaving an enclosed surface respectively is $ϕ_{1}$ and $ϕ_{2}$ , the electric charge inside the surface will be