Electrostatics · Physics · JEE Advanced
MCQ (More than One Correct Answer)
A small electric dipole $\vec{p}_0$, having a moment of inertia $I$ about its center, is kept at a distance $r$ from the center of a spherical shell of radius $R$. The surface charge density $\sigma$ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle $\theta$ as shown in the figure. While staying at a distance $r$, the dipole is free to rotate about its center.
If released from rest, then which of the following statement(s) is(are) correct?
[ $\varepsilon_0$ is the permittivity of free space.]
In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_{A}=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as $\rho_{A}=k r$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_{B}$, the electrostatic charge density varies as $\rho_{B}=$ $\frac{2 k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.
Which of the following statement(s) is(are) correct?
$$ V(z)=\frac{\sigma}{2 \epsilon_{0}}\left(\sqrt{R^{2}+z^{2}}-z\right) . $$
A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z=z_{0}$ and $z_{0}>0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec{F}=-c \hat{k}$ with $c>0$. Let $\beta=\frac{2 c \epsilon_{0}}{q \sigma}$.
Which of the following statement(s) is(are) correct?
Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center 0 of the hexagon and is bisected by the side.
Which of the following statement(s) is(are) correct in SI units?
Take $${q \over m}$$ = 1010 Ckg−1 . Then
If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are, ($$ \in $$0 is the permittivity of the free space, R >> dipole size)
[$$ \in $$0 is the permittivity of free space]
Numerical
A charge is kept at the central point $\mathrm{P}$ of a cylindrical region. The two edges subtend a half-angle $\theta$ at $\mathrm{P}$, as shown in the figure. When $\theta=30^{\circ}$, then the electric flux through the curved surface of the cylinder is $\Phi$. If $\theta=60^{\circ}$, then the electric flux through the curved surface becomes $\Phi / \sqrt{n}$, where the value of $n$ is ___.
An infinitely long thin wire, having a uniform charge density per unit length of $5 \mathrm{nC} / \mathrm{m}$, is passing through a spherical shell of radius $1 \mathrm{~m}$, as shown in the figure. A $10 \mathrm{nC}$ charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points $\mathrm{P}$ and $\mathrm{R}$, in Volt, is ________.
[Given: In SI units $\frac{1}{4 \pi \epsilon_0}=9 \times 10^9, \ln 2=0.7$. Ignore the area pierced by the wire.]
A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is $\frac{n q}{6 \epsilon_{0}}$ (in SI units). The value of $n$ is _______.
The value of R is __________ meter.
The value of b is __________ meter.
(Note that for three coplanar forces keeping a point mass in equilibrium, $${F \over {\sin \theta }}$$ is the same for all forces, where F is any one of the forces and $$\theta $$ is the angle between the other two forces)
(neglect the buoyancy force, take acceleration due to gravity = 10 ms−2 and charge on an electron (e) = 1.6 $$ \times $$ 10–19 C)
$$\sigma \left( r \right) = {\sigma _0}\left( {1 - {r \over R}} \right)$$, where $$\sigma $$0 is a constant and r is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is $$\phi $$0. Electric flux through another spherical surface of radius $${R \over 4}$$ and concentric with the disc is $$\phi $$. Then the ratio $${{{\phi _0}} \over \phi }$$ is ________.
An infinitely long solid cylinder of radius R has a uniform volume charge density $$\rho$$. It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression $${{23\rho R} \over {16k{\varepsilon _0}}}$$. The value of k is _____________.
Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap film of side a. The surface tension of the soap film is $$\gamma$$. The system of charges and planar film are in equilibrium, and $$a = k{\left[ {{{{q^2}} \over \gamma }} \right]^{1/N}}$$, where k is a constant. Then N is __________.
A solid sphere of radius R has a charge Q distributed in its volume with a charge density $$\rho = K{r^a}$$, where K and a are constants and r is the distance from its centre. If the electric field at $$r = R/2$$ is 1/8 times than at $$r = R$$, find the value of $$a$$.
MCQ (Single Correct Answer)
Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by
[ $\varepsilon_0$ is the permittivity of free space.]
| LIST - I | LIST - II | ||
|---|---|---|---|
| P. | $$E$$ is independent of $$d$$ | 1. | A point charge Q at the origin |
| Q. | $$E\, \propto \,1/d$$ | 2. | A small dipole with point charges $$Q$$ at $$\left( {0,0,l} \right)$$ and $$-Q$$ at $$\left( {0,0, - l} \right).$$ Take $$2l < < d$$ |
| R. | $$E\, \propto \,1/{d^2}$$ | 3. | An infinite line charge coincident with the x-axis, with uniform linear charge density $$\lambda $$ |
| S. | $$E\, \propto \,1/{d^3}$$ | 4. | Two infinite wires carrying uniform linear charge density parallel to the $$x$$-axis. The one along $$\left( {y = 0,z = l} \right)$$ has a charge density $$ + \lambda $$ and the one along $$\left( {y = 0,z = - l} \right)$$ has a charge density Take |
| 5. | Infinite plane charge coincident
with the $$xy$$-plane with uniform surface charge density |
||
Which one of the following statement is correct?
The average current in the steady state registered by the ammeter in the circuit will be
Four charges Q1, Q2, Q3 and Q4 of same magnitude are fixed along the x axis at x = $$-$$2a, $$-$$a, +a and +2a, respectively. A positive charge q is placed on the positive y axis at a distance b > 0. Four options of the signs of these charges are given in List I. The direction of the forces on the charge q is given in List II. Match List I with List II and select the correct answer using the code given below the lists.
| List I | List II | ||
|---|---|---|---|
| P. | Q$$_1$$, Q$$_2$$, Q$$_3$$, Q$$_4$$ all positive | 1. | +x |
| Q. | Q$$_1$$, Q$$_2$$ positive; Q$$_3$$, Q$$_4$$ negative | 2. | $$ - $$x |
| R. | Q$$_1$$, Q$$_4$$ positive; Q$$_2$$, Q$$_3$$ negative | 3. | +y |
| S. | Q$$_1$$, Q$$_3$$ positive; Q$$_2$$, Q$$_4$$ negative | 4. | $$ - $$y |
A disk of radius $${a \over 4}$$ having a uniformly distributed charge 6C is placed in the xy-plane with its centre at ($$-$$a/2, 0, 0). A rod of length a carrying a uniformly distributed charge 8C is placed on the x-axis from x = a/4 to x = 5a/4. Two points charges $$-$$7C and 3C are placed at (a/4, $$-$$a/4, 0) and ($$-$$3a/4, 3a/4, 0), respectively. Consider a cubical surface formed by six surfaces $$x=\pm a/2,y=\pm a/2,z=\pm a/2$$. The electric flux through this cubical surface is
Three concentric metallic spherical shells of radii $$R,2R,3R$$ are given charges $$Q_1,Q_2,Q_3$$, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, $$Q_1:Q_2:Q_3$$, is
Six point charges, each of the same magnitude q, are arranged in different manners as shown in Column II. In each case, a point M and a line PQ passing through M are shown. Let E be the electric field and V be the electric potential at M (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line PQ. Let B be the magnetic field at M and $$\mu$$ be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.
| Column I | Column II | ||
|---|---|---|---|
| (A) | $$E=0$$ | (P) |  Charge are at the corners of a regular hexagon. M is at the centre of the hexagon. PQ is perpendicular to the plane of the hexagon. |
| (B) | $$V\ne 0$$ | (Q) |  Charges are on a line perpendicular to PQ at equal intervals. M is the midpoint between the two innermost charges. |
| (C) | $$B=0$$ | (R) |  Charges are placed on two coplanar insulating rings at equal intervals. M is the common centre of the rings. PQ is perpendicular to the plane of the rings. |
| (D) | $$\mu \ne 0$$ | (S) |  Charges are placed at the corners of a rectangle of sides a and 2a and at the mid points of the longer sides. M is at the centre of the rectangle. PQ is parallel to the longer sides. |
| (T) |  Charges are placed on two coplanar, identical insulating rings are equal intervals. M is the midpoint between the centres of the rings. PQ is perpendicular to the line joining the centres and coplanar to the rings. |
Consider a system of three charges $${q \over 3},{q \over 3}$$ and $$ - {{2q} \over 3}$$ placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius R and angle CAB = 60$$^\circ$$
A parallel plate capacitor C with plates of unit area and separation d is filled with a liquid of dielectric constant K = 2. The level of liquid is $$\frac{d}{3}$$ initially. Suppose the liquid level decreases at a constant speed V, the time constant as a function of time t is:
STATEMENT 1 : For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and
STATEMENT 2 : The electrical potential of a sphere of radius R with charge Q uniformly distributed on the surface is given by $${Q \over {4\pi {\varepsilon _0}R}}$$
The electric field at r = R is :
For a = 0, the value of d (maximum value of $$\rho$$ as shown in the figure) is
The electric field within the nucleus is generally observed to be linearly dependent on r. This implies
A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the figure. The electric field inside the emptied space is
Positive and negative point charges of equal magnitude are kept at $$\left(0,0, \frac{a}{2}\right)$$ and $$\left(0,0, \frac{-a}{2}\right)$$, respectively. The work done by the electric field when another positive point charge is moved from $$(-a, 0,0)$$ to $$(0, a, 0)$$ is