Chapter 13

The charge per unit length for a very long straight wire is \(\lambda\). The electric field at points near the wire (but outside it) and far from the ends varies with distance \(r\) as (A) \(r\) (B) \(1 / r\) (C) \(1 / r^{2}\) (D) \(1 / r^{3}\)

A body has a charge of one coulomb. The number of excess (or lesser) electrons on it from its normal state will be (A) \(\infty\) (B) \(1.6 \times 10^{-19}\) (C) \(1.6 \times 10^{19}\) (D) \(6.25 \times 10^{18}\)

The net charge on a condenser is (A) Infinity (B) \(q / 2\) (C) \(2 q\) (D) Zero

A conducting hollow sphere of radius \(0.1 \mathrm{~m}\) is given a charge of \(10 \mu \mathrm{C}\). The electric potential on the surface of sphere will be (A) Zero (B) \(3 \times 10^{5} \mathrm{~V}\) (C) \(9 \times 10^{5} \mathrm{~V}\) (D) \(9 \times 10^{9} \mathrm{~V}\)

Two equal positive charges \(+q\) each are fixed a certain distance apart. A third equal positive charge \(+q\) is placed exactly midway between them. Then the third charge will (A) move at an angle of \(45^{\circ}\) to the line joining the two charges. (B) move at an angle of \(90^{\circ}\) to the line joining the two charges. (C) move along the line joining the two charges. (D) stay at rest.

Two equal negative charges \(-q\) are fixed at points \((0, a)\) and \((0,-a)\) on the \(y\)-axis. A positive charge \(Q\) is released from rest at a point \((2 a, 0)\) on the \(x\)-axis. The charge \(Q\) will (A) execute SHM about the origin. (B) move to the origin and remain at rest there. (C) move to infinity. (D) execute oscillatory but not SHM.

Two point charges \(+4 q\) and \(+q\) are placed \(30 \mathrm{~cm}\) apart. At what point on the line joining them is the electric field zero? (A) \(15 \mathrm{~cm}\) from charge \(4 q\) (B) \(20 \mathrm{~cm}\) from charge \(4 q\) (C) \(7.5 \mathrm{~cm}\) from charge \(q\) (D) \(5 \mathrm{~cm}\) from charge \(q\)

The effective capacitance of two capacitors of capacitances \(C_{1}\) and \(C_{2}\) (with \(C_{2}>C_{1}\) ) connected in parallel is \(\frac{25}{6}\) times the effective capacitance when they are connected in series. The ratio \(C_{2} / C_{1}\) is (A) \(\frac{3}{2}\) (B) \(\frac{4}{3}\) (C) \(\frac{5}{3}\) (D) \(\frac{25}{6}\)

A capacitor of capacitance \(4 \mu \mathrm{F}\) is charged to \(80 \mathrm{~V}\) and another capacitor of capacitance \(6 \mu \mathrm{F}\) is charged to \(30 \mathrm{~V}\). When they are connected together, the energy lost by the \(4 \mu \mathrm{F}\) capacitor is (A) \(7.8 \mathrm{~mJ}\) (B) \(4.6 \mathrm{~mJ}\) (C) \(3.2 \mathrm{~mJ}\) (D) \(2.5 \mathrm{~mJ}\)

Two concentric spheres are of radii \(r_{1}\) and \(r_{2}\). The outer sphere is given a charge \(q\). The charge \(q^{\prime}\) on the inner sphere will be (inner sphere is grounded) (A) \(q\) (B) \(-q\) (C) \(-q \frac{r_{1}}{r_{2}}\) (D) Zero

A parallel plate capacitor of capacitance \(C\) is connected to a battery of \(e m f V\). If a dielectric slab is completely inserted between the plates of the capacitor and battery remains connected, then electric field between plates (A) decreases. (B) increases. (C) remains constant. (D) may be increase or may be decrease.

A uniform electric field \(E=E_{0}(\hat{i}+\hat{j})\) exists in the region. The potential difference \(\left(V_{Q}-V_{P}\right)\) between point \(P(0,0)\) and \(Q(a, 0)\) is (A) \(-E_{0} a\) (B) \(E_{0} \sqrt{2} a\) (C) \(+E_{0} a\) (D) \(-E_{0} \sqrt{2} a\)

A long string with a charge of \(\lambda\) per unit length passes through an imaginary cube of edge \(a\). The maximum flux of the electric field through the cube will be (A) \(\lambda a / \varepsilon_{0}\) (B) \(\frac{\sqrt{2} \lambda a}{\varepsilon_{0}}\) (C) \(\frac{6 \lambda a^{2}}{\varepsilon_{0}}\) (D) \(\frac{\sqrt{3} \lambda a}{\varepsilon_{0}}\)

The electric potential \(V\) at any point \(x, y, z\) (all in metres) in space is given by \(V=4 x^{2}\) volts. The electric field (in \(\mathrm{V} / \mathrm{m}\) ) at the point ( \(1 \mathrm{~m}, 0,2 \mathrm{~m}\) ) (A) \(-8 \hat{i}\) (B) \(8 \hat{i}\) (C) \(-16 \hat{i}\) (D) \(8 \sqrt{5} \hat{i}\)

In a parallel plate capacitor of capacitance \(\mathrm{C}\), a metal sheet is inserted between the plates, parallel to them. The thickness of the sheet is half of the separation between the plate. The capacitance now becomes (A) \(4 \mathrm{C}\) (B) \(2 \mathrm{C}\) (C) \(\mathrm{C} / 2\) (D) \(\mathrm{C} / 4\)

Which one of the following statement is incorrect? (A) A moving charged particle produced electric and magnetic field both. (B) Equipotential surface is always perpendicular to electric field. (C) Kirchhoff's junction law follows conservation of charge. (D) Electric field inside the conductor is always zero.

Two capacitors of capacitance \(3 \mu \mathrm{F}\) and \(6 \mu \mathrm{F}\) are charged to a potential of \(12 \mathrm{~V}\) each. They are now connected to each other, with the positive plate of one to the negative plate of the other. Then (A) the potential difference across \(3 \mu \mathrm{F}\) is zero. (B) the potential difference across \(3 \mu \mathrm{F}\) is \(4 \mathrm{~V}\). (C) the charge on \(3 \mu \mathrm{F}\) is zero. (D) the charge on \(3 \mu \mathrm{F}\) is \(10 \mu \mathrm{C}\).

Four charges \(+2 q,-2 q,-3 q\), and \(+3 q\) are kept in the corners of a square of side \(a\). The total field at the centre \(O\) is, (A) Zero (B) \(\frac{2 \sqrt{2} q}{4 \pi \varepsilon_{0} a^{2}}\) (C) \(\frac{\sqrt{2} q}{4 \pi \varepsilon_{0} a^{2}}\) (D) \(\frac{10 \sqrt{2} q}{4 \pi \varepsilon_{0} a^{2}}\)

Two spheres of radii \(r\) and \(R\) carry charges \(q\) and \(Q\), respectively. When they are connected by a wire, there will be no loss of energy of the system if (A) \(q r=Q R\) (B) \(q R=Q r\) (C) \(q r^{2}=Q R^{2}\) (D) \(q R^{2}=O r^{2}\)

If three moles of monatomic gas is mixed with 1 moles diatomic gas, the resultant value of \(\gamma\) for the mixture is (A) \(1.66\) (B) \(1.50\) (C) \(1.40\) (D) \(1.57\)

Capacitance of a capacitor becomes \(\frac{4}{3}\) times its original value if a dielectric slab of thickness \(t=d / 2\) is inserted between the plates \((d=\) separation between the plates). The dielectric constant of the slab is (A) 2 (B) 4 (C) 6 (D) 8

Three point charges \(q,-2 q\), and \(-2 q\) are placed at the vertices of an equilateral triangle of side \(a\). The work done by some external force to slowly increase their separation to \(2 a\) will be (A) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{2 q^{2}}{a}\) (B) \(\frac{q^{2}}{4 \pi \varepsilon_{0} a}\) (C) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{3 q^{2}}{3 R}\) (D) Zero

A uniform electric field \(E_{0}\) exists in a region at angle \(45^{\circ}\) with \(x\)-axis. There are two point \(A(a, 0)\) and \(B(0, b)\) having potential \(V_{A}\) and \(V_{B}\), respectively, then (A) \(V_{A}>V_{B}\) if \(a>b\) (B) \(V_{A}=V_{B}\) if \(a=b\) (C) \(V_{A}>V_{B}\) if \(ab\)

A non-conducting solid sphere of radius \(R\) is uniformly charged. The magnitude of the electric field due to the sphere at a distance \(r\) form its centre (A) increases as \(r\) increases for \(r

A point charge \(q\) is placed at origin. Let \(\vec{E}_{A}, \vec{E}_{B}\) and \(\vec{E}_{C}\) be the electric field at three points \(A(1,2,3)\), \(B(1,1,-1)\), and \(C(2,2,2)\) due to charge \(q\). Then (A) \(\vec{E}_{A} \perp \vec{E}_{B}\) (B) \(\vec{E}_{A} \| \vec{E}_{B}\) (C) \(\left|\vec{E}_{B}\right|=4\left|\vec{E}_{C}\right|\) (D) \(\left|\vec{E}_{B}\right|=16\left|\vec{E}_{C}\right|\)

Charges \(Q_{1}\) and \(Q_{2}\) lie inside and outside, respectively, of a closed Gaussian surface \(S .\) Let \(E\) be the field at any point on \(S\) and \(\phi\) be the flux of \(E\) over \(S\), (A) If \(Q_{1}\) changes, both \(E\) and \(\phi\) will change. (B) If \(Q_{2}\) changes, \(E\) will change. (C) If \(Q_{1}=0\) and \(Q_{2} \neq 0\) then \(E \neq 0\) but \(\phi=0\). (D) If \(Q_{1} \neq 0\) and \(Q_{2}=0\) then \(E=0\) but \(\phi \neq 0\).

Four charges, all of the same magnitude, are placed at the four corners of a square. At the centre of the square, the potential is \(V\) and the field is \(E\). By suitable choices of the signs of the four charges, which of the following can be obtained? (A) \(V=0, E=0\) (B) \(V=0, E \neq 0\) (C) \(V \neq 0, E=0\) (D) \(V \neq 0, E \neq 0\)

Three points \(A, B\), and \(C\) are at a distance of \(1 \mathrm{~m}, 2 \mathrm{~m}\), and \(\mathrm{lm}\) from an infinitely long charged wire of linear charge density \(\lambda \mathrm{C} / \mathrm{m}\). A charge \(q\) is taken from \(A\) to \(B, B\) to \(C\), and finally \(C\) to \(A\). Which of the following is/are correct about the work done in the above process? (A) \(W_{A B}=2 W_{B C}\) (B) \(W_{A B}=-W_{B C}\) (C) \(W_{B C}=0\) (D) \(W_{C A}=0\)

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

Choose the correct statements from the following: (A) If the electric field is zero at a point, the electric potential must also be zero at that point. (B) If electric potential is constant in a given region of space, the electric field must be zero in that region. (C) Two different equipotential surfaces can never intersect. (D) Electrons move from a region of lower potential to a region of higher potential.

Total charge on plate (2) initially is (A) \(\frac{\varepsilon_{0} A}{2 d} V\) (B) \(\frac{2 \varepsilon_{0} A}{d} V\) (C) \(\frac{\varepsilon_{0} A}{d} V\) (D) \(\frac{3 \varepsilon_{0} A}{2 d} V\)

Work done by electric field, when particle moves from \(C_{A}\) to \(C_{B}\) is (A) \(-1.2 \mathrm{~J}\) (B) \(1.2 \mathrm{~J}\) (C) \(-3.6 \mathrm{~J}\) (D) \(3.6 \mathrm{~J}\)

The energy density in the electric field created by a point charge falls off with distance from the point charge as (A) \(\frac{1}{r}\) (B) \(\frac{1}{r^{2}}\) (C) \(\frac{1}{r^{3}}\) (D) \(\frac{1}{r^{4}}\)

A charge \(q_{1}\) is placed at the centre of a spherical conducting shell of radius \(R\). Conducting shell has a total charge \(q_{2} .\) Electrostatic potential energy of the system (A) \(\frac{q_{1}^{2}+2 q_{1} q_{2}}{8 \pi \varepsilon_{0} R}\) (B) \(\frac{q_{2}^{2}+2 q_{1} q_{2}}{8 \pi \varepsilon_{0} R}\) (C) \(\frac{q_{1}^{2}+q_{1} q_{2}}{4 \pi \varepsilon_{0} R}\) (D) \(\frac{q_{2}^{2}+q_{1} q_{2}}{4 \pi \varepsilon_{0} R}\)

Let \(u_{a}\) and \(u_{d}\) represent the energy density in air and in a dielectric, respectively, for the same field in both. Let \(K=\) dielectric constant. Then (A) \(u_{a}=u_{d}\) (B) \(u_{a}=K u_{d}\) (C) \(u_{d}=K u_{a}\) (D) \(u_{a}=(K-1) u_{d}\)

A parallel plate capacitor is connected to a battery. The plates are pulled apart with a uniform speed. If \(x\) is the separation between the plates, then the rate of change of electrostatic energy of the capacitor is proportional to (A) \(x\) (B) \(x^{2}\) (C) \(\frac{1}{x}\) (D) \(\frac{1}{x^{2}}\)

A position charge \(Q\) is uniformly distributed over the ring of radius \(R=1 \mathrm{~m}\), and potential at infinity is assumed to be zero. Column-I (A) Electric field at centre of ring is (B) Electric field at the axis of ring at a distance \(x=\sqrt{3} \mathrm{~m}\) from centre is (C) Electric potential at centre of ring is (D) Electric potential on axis of ring at a distance \(\sqrt{3} \mathrm{~m}\) from centre of ring is Column-II 1\. \(\frac{K Q}{2}\) 2\. \(K Q\) 3\. \(\frac{K Q \sqrt{3}}{8}\) 4\. Zero 5\. \(\frac{K Q \sqrt{2}}{7}\)

Column-I (A) During the charging of capacitor (B) Terminal potential of battery is (C) When tre charge flows from higher potential to lower potential through a resistance (D) Work done to carry unit tre charge from negative terminal to positive terminal Column-II 1\. Heat loss will be take place 2\. emf of battery 3\. emf of battery when current through it is zero. 4\. inside the battery electron transfer from -ve terminal of battery to +re terminal of battery.

Assertion: When an uncharged capacitor of capacitance \(C\) is charged by a cell of emf \(V\), the energy stored by capacitor is \(\frac{1}{2} C V^{2}\), and energy supplied by battery is \(C V^{2}\). Reason: In charging an uncharged capacitor, energy is lost in the form of heat. (A) A (B) \(\mathrm{B}\) (C) \(\mathrm{C}\) (D) \(\mathrm{D}\)

Assertion: In an uniform electric field, equipotential surfaces must be plane surface. Reason: Electrons move from a region of lower potential to a region of higher potential if electrons start from rest. (A) \(\mathrm{A}\) (B) \(\mathrm{B}\) (C) \(\underline{\mathrm{C}}\) (D) D

Assertion: A point charge \(q\) is rotated along a circle in the electric field generated by another point change \(Q\). The work done by electric field on the rotating charge in one complete revolution is zero if the change \(Q\) is at centre and not zero otherwise. Reason: Work done by conservative force in closed loop is always zero. (A) \(\mathrm{A}\) (B) \(\mathrm{B}\) (C) \(\bar{C}\) (D) D

Assertion: Gaussian surface chosen should not pass through a discrete charge. Reason: Electric field due to a system of discrete charges is well-defined at the location of charge. (A) \(\mathrm{A}\) (B) \(\mathrm{B}\) (C) \(\overline{\mathrm{C}}\) (D) D

Electric potential \(V\) in volt in a region is given by \(V=a x^{2}+a y^{2}+2 a z^{2}\), where \(a\) is a constant. Work done by the field when a \(2 \mu \mathrm{C}\) charge moves from point \((0,0,0.1 \mathrm{~m})\) to origin is \(5 \times 10^{-5} \mathrm{~J}\). Find \(a ?\) (in \(\mathrm{V} / \mathrm{m}^{2}\) )

A capacitor of capacitance \(\mathrm{C}\) is fully charged by a \(200 \mathrm{~V}\) supply. It is then discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat \(2.5 \times 10^{2} \mathrm{~J} / \mathrm{kg}-\mathrm{K}\) and of mass \(0.1 \mathrm{~kg}\). If temperature of the block rises by \(0.4 \mathrm{~K}\). Find the value of \(C\).

Two identical particles each having charge \(q\) are very far apart. They are given velocity \(v_{0}\) parallel to each other such that initial perpendicular separation between them is \(d\). If the subsequent minimum separation between them is \(2 d\), find the initial velocity \(v_{0}\) and the loss in their total kinetic energies.

A spherical shell of radius \(R_{1}=10 \mathrm{~cm}\) with a uniform charge \(q=6 \mu \mathrm{C}\) has a point charge \(q_{0}=3 \mu \mathrm{C}\) at its centre. Find the work performed by electric forces in milli joules during the shell expansion from radius \(R_{1}\) to radius \(R_{2}=20 \mathrm{~cm}\). Take \(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\).

In the diagram, there are four conducting plates \(A, B\), \(C\), and \(D\) placed parallel to each other at equal separation \(L\). If plate \(C\) starts moving towards plate \(B\) with velocity \(v\). Find the current (in mA) flowing in the wire connecting \(A\) and \(D\) (assume all other plates to be fixed). \(\left(\right.\) Given, \(\left.q_{2}=2 \mu \mathrm{C}, q_{3}=3 \mu \mathrm{C}, v=10 \mathrm{~m} / \mathrm{s}, L=0.05 \mathrm{~m}\right)\)

On moving a charge of \(20 \mathrm{C}\) by \(2 \mathrm{~cm}, 2 \mathrm{~J}\) of work is done. The potential difference between the points (A) \(0.1 \mathrm{~V}\) (B) \(8 \mathrm{~V}\) (C) \(2 \mathrm{~V}\) (D) \(0.5 \mathrm{~V}\)

If a charge \(q\) is placed at the centre of the line joining two equal charges \(Q\) such that the system is in equilibrium, then the value of \(q\) is (A) \(Q / 2\) (B) \(-Q / 2\) (C) \(Q / 4\) (D) \(-Q / 4\)

If the electric flux entering and leaving an enclosed surface, respectively, is \(\phi_{1}\) and \(\phi_{2}\), the electric charge inside the surface will be (A) \(\left(\phi_{2}-\phi_{1}\right) / \varepsilon_{0}\) (B) \(\left(\phi_{1}+\phi_{2}\right) / \varepsilon_{0}\) (C) \(\left(\phi_{2}-\phi_{1}\right) \varepsilon_{0}\) (D) \(\left(\phi_{1}+\phi_{2}\right) \varepsilon_{0}\)