Chapter 13

A positively charged thin metal ring of radius \(R\) is fixed in \(x-y\) plane with its centre at the origin \(O\). A negatively charged particle \(P\) is released from rest at the point \(\left(0,0, Z_{0}\right)\). Then the motion of \(P\) is (A) periodic for all values of \(Z_{0}\) - (B) SHM for all values of \(Z_{0}\) satisfying \(0

Two spherical conductors \(A\) and \(B\) of radii \(1 \mathrm{~mm}\) and \(2 \mathrm{~mm}\) are separated by a distance of \(5 \mathrm{~cm}\) and are uniformly charged. If the spheres are connected by a conducting wire then in equilibrium condition, the ratio of the magnitude of the electric fields at the surface of spheres \(A\) and \(B\) is (A) \(1: 4\) (B) \(4: 1\) (C) \(1: 2\) (D) \(2: 1\)

A point charge \(+q\) is fixed at point \(B\). Another point charge \(+q\) at \(A\) of mass \(m\) vertically above \(B\) at height \(h\) is dropped from rest. Choose the correct statement: (A) It will collide with \(B\) (B) It will execute SHM (C) It will go down only if \(\frac{q^{2}}{4 \pi \varepsilon_{0}}

The electric field intensity at a point at a distance \(2 \mathrm{~m}\) from a charge \(q\) is \(E\). The amount of work done in bringing a charge of 2 coulomb from infinity to this point will be (A) \(2 E \mathrm{~J}\) (B) \(4 E \mathrm{~J}\) (C) \(\frac{E}{2} \mathrm{~J}\) (D) \(\frac{E}{4} \mathrm{~J}\)

A simple pendulum of length \(l\) has a bob of mass \(m\), with a charge \(q\) on it. A vertical sheet of charge, with surface charge density \(\sigma\) passes through the point of suspension. At equilibrium, the string makes an angle \(\theta\) with the vertical, then (A) \(\tan \theta=\frac{\sigma q}{2 \varepsilon_{0} m g}\) (B) \(\tan \theta=\frac{\sigma q}{\varepsilon_{0} m g}\) (C) \(\cot \theta=\frac{\sigma q}{2 \varepsilon_{0} m g}\) (D) \(\cot \theta=\frac{\sigma q}{\varepsilon_{0} m g}\)

A charged particle of mass \(m\) and charge \(q\) is released from rest in an electric field of constant magnitude \(E\). The kinetic energy of the particle after time \(t\) will be (A) \(\frac{2 E^{2} t^{2}}{m q}\) (B) \(\frac{E q^{2} m}{2 t^{2}}\) (C) \(\frac{E^{2} q^{2} t^{2}}{2 m}\) (D) \(\frac{E q m}{2 t}\)

A table tennis ball which has been covered with a conducting paint is suspended by a silk thread so that it hangs between two metal plates. One plate is earthed. When the other plate is connected to a high voltage generator, the ball (A) is attracted to the high voltage plate and stays there. (B) hangs without moving. (C) swings backward and forward hitting each plate in turn. (D) is repelled to the earthed plate and stays there.

A spring block system undergoes vertical oscillations above a large horizontal metal sheet with uniform positive charge. The time period of the oscillation is \(T\). If the block is given a charge \(Q\), its time period of oscillation (A) remains same. (B) increases. (C) decreases. (D) increases if \(Q\) is positive and decreases if \(Q\) is negative.

There is an electric field \(E\) in \(x\)-direction. If work done in moving a charge \(0.2 \mathrm{C}\) through a distance of \(2 \mathrm{~m}\) along a line making an angle of \(60^{\circ}\) with \(x\)-axis is 4.0 J. The value of \(E\) is (A) \(\sqrt{3} \mathrm{~N} / \mathrm{C}\) (B) \(4 \mathrm{~N} / \mathrm{C}\) (C) \(5 \mathrm{~N} / \mathrm{C}\) (D) \(20 \mathrm{~N} / \mathrm{C}\)

In electrolysis, the amount of mass deposited or liberated at an electrode is directly proportional to (A) amount of charge. (B) square of current. (C) concentration of electrolyte. (D) square of electric charge.

The ratio of the forces between two small spheres with same charges when they are in air to when they are in a medium of dielectric constant \(K\) is (A) \(1: K\) (B) \(K: 1\) (C) \(1: K^{2}\) (D) \(K^{2}: 1\)

A charge \(Q\) is divided into two parts of magnitude \(q\) and \(Q-q\). If the coulomb repulsion between them when they are separated at some distance is to be maximum, the ratio of \(\frac{Q}{q}\) should be (A) 2 (B) \(1 / 2\) (C) 4 (D) \(1 / 4\)

There are two charges \(+1 \mu \mathrm{C}\) and \(+5 \mu \mathrm{C}\). The ratio of the forces acting on them will be (A) \(1: 5\) (B) \(1: 1\) (C) \(5: 1\) (D) \(1: 25\)

The electric potential \(V\) is given as a function of distance \(x\) (metre) by \(V=\left(5 x^{2}+10 x-9\right)\) v. Magnitude of electric field at \(x=1\) is (A) \(20 \mathrm{~V} / \mathrm{m}\) (B) \(6 \mathrm{~V} / \mathrm{m}\) (C) \(11 \mathrm{~V} / \mathrm{m}\) (D) \(-23 \mathrm{~V} / \mathrm{m}\)

The magnitude of electric field intensity \(E\) is such that, an electron of mass \(m\) and charge \(e\) placed in it would experience an electrical force equal to its weight is given by (A) \(m g e\) (B) \(\frac{m g}{e}\) (C) \(\frac{e}{m g}\) (D) \(\frac{e^{2}}{m^{2}} g\)

\(A B C\) is an equilateral triangle. Charges \(+q\) are placed at each corner. The electric field intensity at centroid \(O\) will be (A) \(\frac{1}{2 \pi \varepsilon_{0}} \times \frac{q}{r^{2}}\) (B) \(\frac{1}{2 \pi \varepsilon_{0}} \times \frac{3 q}{r^{2}}\) (C) \(\frac{1}{2 \pi \varepsilon_{0}} \times \frac{\sqrt{3} q}{r^{2}}\) (D) Zero

A charged particle of mass \(m\) and charge \(q\) is released from rest in an electric field of constant magnitude \(E\). The kinetic energy of the particle after a time \(t\) is (A) \(\frac{E^{2} q^{2} t^{2}}{m}\) (B) \(\frac{2 E^{2} q^{2} t^{2}}{m}\) (C) \(\frac{E^{2} q^{2} t^{2}}{2 m}\) (D) \(\frac{4 E^{2} q^{2} t^{2}}{m}\)

Three charges \(Q,+q\), and \(+q\) are placed at the vertices of a right angle triangle (isosceles triangle) as shown. The net electrostatic energy of the configuration is zero, if \(Q\) is equal to (A) \(\frac{-q}{1+\sqrt{2}}\) (B) \(\frac{-2 q}{2+\sqrt{2}}\) (C) \(-2 q\) (D) \(+q\)

A spherical conductor \(A\) of radius \(r\) is placed concentrically inside a conducting shell \(B\) of radius \(R(R>r)\). A charge \(Q\) is given to \(A\), and then \(A\) is joined to \(B\) by a metal wire. The charge flowing from \(A\) to \(B\) will be (A) \(Q\left(\frac{R}{R+r}\right)\) (B) \(Q\left(\frac{r}{R+r}\right)\) (C) \(Q\) (D) Zero

An electron moves in a circular orbit at a distance from a proton with kinetic energy \(E\). To escape to infinity, the energy which must be supplied to the electron is (A) \(E\) (B) \(2 E\) (C) \(0.5 E\) (D) \(\sqrt{2} E\)

A capacitor is connected to a battery. The force of attraction between the plates when the separation between them is halved (A) remains the same. (B) becomes eight times. (C) becomes four times. (D) becomes double.

The electric \(\vec{E}\) is given by \(\vec{E}=a \hat{i}+b \hat{j}\) (where \(a\) and \(b\) is constant and \(\hat{i}, \hat{j}\) are unit vector along \(x\) and \(y\) axis, respectively), the flux passing through a square area of side \(l\) and parallel to \(y=z\) plane is (A) \(b l^{2}\) (B) \(a l^{2}\) (C) \(\sqrt{\left(a^{2}+b^{2}\right)} l^{2}\) (D) \(\sqrt{\left(a^{2}-b^{2}\right)} l^{2}\)

A cylinder of radius \(R\) and length \(L\) is placed in a uniform electric field \(E\) parallel to the axis of cylinder. The total flux through the curved surface of the cylinder is given by (A) \(2 \pi R^{2} E\) (B) \(2 \pi R^{2} / E\) (C) \(E 2 \pi R L\) (D) Zero

An uniform electric field in positive \(x\)-direction exists in a region. Let \(A\) be the origin, \(B\) be the point on the \(x\)-axis at \(x=+1 \mathrm{~cm}\) and \(C\) be the point on the \(y\)-axis at \(y=+1 \mathrm{~cm} .\) The potential at the points \(A, B\), and \(C\) are \(V_{A}, V_{B}\), and \(V_{C}\), respectively, then (A) \(V_{A}V_{B}\) (C) \(V_{A}V_{C}\)

The electric field at the origin is along the positive \(X\)-axis. A small circle is drawn with the centre at the origin cutting the axes at points \(A, B, C\), and \(D\) having coordinates \((a, 0)(0, a),(-a, 0),(0,-a)\), respectively. Out of the points on the periphery of the circle, the potential is minimum at (A) \(A\) (B) \(B\) (C) \(C\) (D) \(D\)

Charge \(Q\) is given a displacement \(\vec{r}=a \hat{i}+b \hat{j}\) in an electric field \(\vec{E}=E_{1} \hat{i}+E_{2} \hat{j}\). The work done is (A) \(Q\left(E_{1} a+E_{2} b\right)\) (B) \(Q \sqrt{\left(E_{1} a\right)^{2}+\left(E_{2} b\right)^{2}}\) (C) \(Q\left(E_{1}+E_{2}\right) \sqrt{a^{2}+b^{2}}\) (D) \(Q \sqrt{ \left.E_{1}^{2}+E_{2}^{2}\right)} \sqrt{a^{2}+b^{2}}\)

A non-conducting ring of radius \(R\) has charge \(Q\) distributed uniformly over it. If it rotates with an angular velocity \(\omega\), the equivalent current will be (A) Zero (B) \(Q \omega\) (C) \(Q \frac{\omega}{2 \pi}\) (D) \(Q \frac{\omega}{2 \pi R}\)

A half ring of radius \(R\) has a charge of \(\lambda\) per unit length. The potential at the centre of the half ring is (A) \(k \frac{\lambda}{R}\) (B) \(k \frac{\lambda}{\pi R}\) (C) \(k \frac{\pi \lambda}{R}\) (D) \(k \pi \lambda\)

If electric field is given by \(\vec{E}=\left(\frac{1}{x^{2}}\right) \hat{i} \mathrm{~V} / \mathrm{m}\), the magnitude of potential difference between points \(x=10 \mathrm{~cm}\) and \(x=20 \mathrm{~cm}\) is (A) \(\mathbb{V}\) (B) \(2 \mathrm{~V}\) (C) \(5 \mathrm{~V}\) (D) \(10 \mathrm{~V}\)

A cube of side \(b\) has a charge \(q\) at each of its vertices. The electric potential at the centre of the cube is (A) \(\frac{4 q}{\sqrt{3} \pi \varepsilon_{0} b}\) (B) \(\frac{\sqrt{3} q}{\pi \varepsilon_{0} b}\) (C) \(\frac{2 q}{\pi \varepsilon_{0} b}\) (D) Zero

Four equal charges \(Q\) are placed at the four corners of a square of side \(a\). The work done in removing a charge \(-Q\) from the centre of the square to infinity is (A) Zero (B) \(\frac{\sqrt{2} Q^{2}}{4 \pi \varepsilon_{0} a}\) (C) \(\frac{\sqrt{2} Q^{2}}{\pi \varepsilon_{0} a}\) (D) \(\frac{Q^{2}}{2 \pi \varepsilon_{0} a}\)

In a regular polygon of \(n\) sides, each corner is at a distance \(r\) from the centre. Identical charges of magnitude \(Q\) are placed at \((n-1)\) corners. The field at the centre is (A) \(k \frac{Q}{r^{2}}\) (B) \((n-1) k \frac{Q}{r^{2}}\) (C) \(\frac{n}{n-1} k \frac{Q}{r^{2}}\) (D) \(\frac{n-1}{n} k \frac{Q}{r^{2}}\)

Seven point charges each of charge \(q\) is placed at the seven corners of a cube of side \(a\) (one corner is empty). Find the magnitude of electric field at centre of cube. (A) Zero (B) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{a^{2}}\) (C) \(\frac{1}{3 \pi \varepsilon_{0}} \frac{q}{a^{2}}\) (D) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{7 q}{a^{2}}\)

A charged sphere of diameter \(4 \mathrm{~cm}\) has a charge density of \(10^{-4}\) coulombs/cm \(^{2}\). The work done in joules when a charge of 40 nano-coulombs is moved from infinity to a point which is at a distance of \(2 \mathrm{~cm}\) from the surface of the sphere is (A) \(14.4 \pi\) (B) \(28.8 \pi\) (C) \(144 \pi\) (D) \(288 \pi\)

The magnitude of electric intensity at a distance \(x\) from a charge \(q\) is \(E\). An identical charge is placed at a distance \(2 x\) from it. Then the magnitude of the force it experiences is (A) \(q E\) (B) \(2 q E\) (C) \(\frac{q E}{2}\) (D) \(\frac{q E}{4}\)

A conductor of resistance \(3 \Omega\) is stretched uniformly till its length is doubled. The wire is now bent in the form of an equilateral triangle. The effective resistance between the ends of any side of the triangle in ohms is (A) \(\frac{9}{2}\) (B) \(\frac{8}{3}\) (C) 2 (D) 1

Eight dipoles of charges of magnitude \(e\) are placed inside a cube. The total flux coming out of the cube equals to (A) \(\frac{8 e}{\varepsilon_{0}}\) (B) \(\frac{16 e}{\varepsilon_{0}}\) (C) \(\frac{e}{\varepsilon_{0}}\) (D) Zero

Four point charges \(q_{1}, q_{2}, q_{3}\), and \(q_{4}\) are placed at the corners of the square of side \(\mathrm{a}\), as shown in Fig. 13.36. The potential at the centre of the square is (Given: \(q_{1}=1 \times 10^{-8} \mathrm{C}, q_{2}=-2 \times 10^{-8} \mathrm{C}\), \(\left.q_{3}=3 \times 10^{-8} \mathrm{C}, q_{4}=2 \times 10^{-8} \mathrm{C}, a=1 \mathrm{~m}\right)\) (A) \(507 \mathrm{~V}\) (B) \(607 \mathrm{~V}\) (C) \(550 \mathrm{~V}\) (D) \(650 \mathrm{~V}\)

If there are \(n\) capacitors in parallel connected to \(\mathrm{V}\) volt source, then total energy stored is equal to (A) \(\mathrm{CV}\) (B) \(\frac{1}{2} n \mathrm{CV}^{2}\) (C) \(\mathrm{CV}^{2}\) (D) \(\frac{1}{2 n} \mathrm{CV}^{2}\)

There is an air-filled 1 pF parallel plate capacitor. When the plate separation is doubled and the space is filled with wax, the capacitance increases to \(2 \mathrm{pF}\). The dielectric constant of wax is (A) 2 (B) 4 (C) 6 (D) 8

Two conducting spheres of radii \(r_{1}\) and \(r_{2}\) are at the same potential. The ratio of their charges is (A) \(\left(\frac{r_{1}^{2}}{r_{2}^{2}}\right)\) (B) \(\left(\frac{r_{2}^{2}}{r_{1}^{2}}\right)\) (C) \(\frac{r_{1}}{r_{2}}\) (D) \(\frac{r_{2}}{r_{1}}\)

Two large plates separated by a distance \(d\) in vertical plane and connected to battery as shown. An electron of charge \(e\) and mass \(m\) is at rest between the plates. Find the value of potential difference of battery. (A) \(\frac{m g e}{d}\) (B) \(\frac{2 m g e}{d}\) (C) \(\frac{m g d}{e}\) (D) \(\frac{2 m g d}{e}\)

Two plates are \(2 \mathrm{~cm}\) apart. A potential difference of \(10 \mathrm{~V}\) is applied between them, the electric field between the plates is (A) \(20 \mathrm{~N} / \mathrm{C}\) (B) \(500 \mathrm{~N} / \mathrm{C}\) (C) \(5 \mathrm{~N} / \mathrm{C}\) (D) \(250 \mathrm{~N} / \mathrm{C}\)

Suppose the electrostatic potential at some points in space are given by \(V=\left(x^{2}-2 x\right)\). The electrostatic field strength at \(x=1\) is (A) Zero (B) \(-2\) (C) 2 (D) 4

An electric dipole is placed at an angle of \(30^{\circ}\) to a non-uniform electric field. The dipole will experience (A) a translational force only in the direction of the field. (B) a translational force only in a direction normal to the direction of the field. (C) a torque as well as a translational force. (D) a torque only.

Coulomb's law is applicable to, (A) Point charges (B) Spherical charges (C) Like charges (D) All of these

A point charge \(q\) and a charge \(-q\) are placed at \(x=-a\) and \(x=+a\), respectively. Which of the following represents a part of \(E-x\) graph?

The electric potential \(V\) (in volt) varies with \(x\) (in metre) according to the relation \(V=5+4 x^{2}\). The force experienced by a negative charge of \(2 \times 10^{-6} \mathrm{C}\) located at \(x=0.5 \mathrm{~m}\) is (A) \(2 \times 10^{-6} \mathrm{~N}\) (B) \(4 \times 10^{-6} \mathrm{~N}\) (C) \(6 \times 10^{-6} \mathrm{~N}\) (D) \(8 \times 10^{-6} \mathrm{~N}\)

A charge \(q\) is placed at the centre of the line joining two equal charges \(Q .\) The system of the three charges will be in equilibrium if \(q\) is equal to (A) \(-(Q / 4)\) (B) \(-(Q / 2)\) (C) \((Q / 2)\) (D) \((Q / 4)\)

The electric potential at a point situated at a distance \(r\) on the axis of a short electric dipole of moment \(p\) will be \(1 / 4\left(\pi \varepsilon_{0}\right)\) times (A) \(p / r^{3}\) (B) \(p / r^{2}\) (C) \(p / r\) (D) None of the above