Chapter 17

The electric potential \(V\) at any point \(x, y, z\) (all in metre) in space is given by \(V=4 x^{2}\) volt. The electric field at the point \((1 \mathrm{~m}, 0,2 \mathrm{~m})\) in \(\mathrm{Vm}^{-1}\) is (a) \(-8 \hat{\mathbf{i}}\) (b) \(+8 \hat{i}\) (c) \(-16 \hat{i}\) (d) \(16 \hat{\mathrm{k}}\)

Two particles \(A\) and \(B\) having charges \(8 \times 10^{-6} \mathrm{C}\) and \(-2 \times 10^{-6} \mathrm{C}\) respectively, are held fixed with a separation \(20 \mathrm{~cm} .\) Where should a third charged particle be placed so that it does not experience a net electric force? (a) \(0.2 \mathrm{~m}\) (b) \(0.5 \mathrm{~m}\) (c) \(0.6 \mathrm{~m}\) (d) \(0.1 \mathrm{~m}\)

Three concentric spherical shells have radii \(a, b\) and \(c\) ( \(a

A long charged cylinder of linear charged density \(\lambda\) is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders? [NCERT] (a) \(\frac{\lambda}{2 \pi \varepsilon_{0} r}\) (b) \(\frac{\lambda r}{\sqrt{2} \pi \varepsilon_{0}}\) (c) \(\frac{\lambda}{\sqrt{2} \pi \varepsilon_{0} r}\) (d) None of these

In the electric field shown in figure, the electric lines in the left have twice the separation as that between those on right. If the magnitude of the field at point \(A\) is \(40 \mathrm{NC}^{-1}\). The force experienced by a proton placed at point \(A\) is (a) \(6.4 \times 10^{-18} \mathrm{~N}\) (b) \(3.2 \times 10^{-15} \mathrm{~N}\) (c) \(5.0 \times 10^{-12} \mathrm{~N}\) (d) \(1.2 \times 10^{-18} \mathrm{~N}\)

Two insulated metallic sphere of \(3 \mu \mathrm{F}\) and \(5 \mu \mathrm{F}\) capacitances are charged to \(300 \mathrm{~V}\) and \(500 \mathrm{~V}\), respectively. The energy loss, when they are connected by a wire, is (a) \(0.0375 \mathrm{~J}\) (b) \(0.235 \mathrm{~J}\) (c) \(0.375 \mathrm{~J}\) (d) \(375 \mathrm{~J}\)

Two charges \(5 \times 10^{-8} \mathrm{C}\) and \(-3 \times 10^{-8} \mathrm{C}\) are located \(16 \mathrm{~cm}\) apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero. [NCERT] (a) \(6 \mathrm{~cm}\) from the charge \(-3 \times 10^{-8} \mathrm{C}\) (b) \(6 \mathrm{~cm}\) from the charge \(5 \times 10^{-8} \mathrm{C}\) (c) \(9 \mathrm{~cm}\) from the charge \(-3 \times 10^{-8} \mathrm{C}\) (d) \(9 \mathrm{~cm}\) from the charge \(5 \times 10^{-8} \mathrm{C}\)

Two identical spheres carrying charges \(-9 \mu \mathrm{C}\) and \(5 \mu \mathrm{C}\), respectively are kept in contract and then separated from each other. Point out true statement from the following in each sphere. (a) \(1.25 \times 10^{13}\) electrons are in excess (b) \(1.25 \times 10^{13}\) electrons are in deficit (c) \(4.15 \times 10^{12}\) electrons are in excess (d) None of the above

A \(4 \mu \mathrm{F}\) capacitor and a resistance of \(2.5 \Omega\) are in series with \(12 \mathrm{~V}\) battery. Find the time after which potential difference across the capacitor in 3 times the potential difference across the resher. [Given, \(\log (2)=0.693]\) (a) \(14 \mathrm{~s}\) (b) \(16.93 \mathrm{~s}\) (c) \(13.86 \mathrm{~s}\) (d) \(8 \mathrm{~s}\)

A charge \(5 \mu \mathrm{C}\) is placed at a point. What is the work required to carry \(1 \mathrm{C}\) of charge once round it in circle of \(12 \mathrm{~cm}\) radius? (a) 100 (b) 0 (c) 1 (d) \(\infty\)

Two metallic spheres \(A\) and \(B\) of same radii one solid and other hollow are charged to the same potential. Which of the two will hold more charge? (a) Sphere \(A\) (b) Sphere \(B\) (c) Both spheres (d) None of these

A semi circular arc of radius \(a\) in charged uniformly and the charge per unit length is \(\lambda .\) The electric field at the centre is (a) \(\frac{\lambda^{2}}{2 \pi \varepsilon_{0} a}\) (b) \(\frac{\lambda}{2 \pi \varepsilon_{0} a}\) (c) \(\frac{\lambda}{2 \pi \varepsilon_{0} a^{2}}\) (d) \(\frac{\lambda}{4 \varepsilon_{0} a}\)

The tangential component of electrostatic field is continuous from one side of a charged surface to another is [NCERT] (a) \(\frac{1}{4 \pi \varepsilon_{0}}\left(\frac{1}{r_{A}}+\frac{1}{r_{B}}-\frac{1}{r_{C}}\right)\) (b) Zero (c) \(\frac{1}{4 \pi \varepsilon_{0}}\left(\frac{1}{r_{A}}-\frac{1}{r_{B}}+\frac{1}{r_{C}}\right)\) (d) \(\frac{1}{4 \pi \varepsilon_{0}}\left(\frac{1}{r_{A}}+\frac{1}{r_{B}}+\frac{1}{r_{C}}\right)\)

An infinite number of charges, each \(1 \mu \mathrm{C}\) are placed on the \(x\)-axis with coordinates \(x=1,2,4,8 \ldots \infty\). If a charge of \(1 \mathrm{C}\) is kept at the origin, then what is the net force acting on \(1 \mathrm{C}\) charge (a) \(10000 \mathrm{~N}\) (b) \(32000 \mathrm{~N}\) (c) \(12000 \mathrm{~N}\) (d) \(18000 \mathrm{~N}\)

Two plates are \(1 \mathrm{~cm}\) apart, and potential difference between them is 10 volt. The electric field between the plates is (a) \(10 \mathrm{~N} / \mathrm{C}\) (b) \(500 \mathrm{~N} / \mathrm{C}\) (c) \(10^{3} \mathrm{~N} / \mathrm{C}\) (d) \(250 \mathrm{~N} / \mathrm{C}\)

Two particles of equal mass \(m\) and charge \(q\) are placed at a distance of \(16 \mathrm{~cm}\). They do not experience any force. The value of \(\frac{q}{m}\) is (a) \(\sqrt{\frac{\pi \varepsilon_{0}}{G}}\) (b) \(\sqrt{\frac{G}{\pi \varepsilon_{0}}}\) (c) \(\sqrt{4 \pi \varepsilon_{0}} G\) (d) \(l\)

A positively charged ball hangs from a silk thread. We put a positive test charge \(q_{0}\) at a point and measure \(\frac{F}{q_{0}}\) then it can be predicted that the electric field strength \(E\) is (a) \(>F / q_{\mathrm{o}}\) (b) \(=\frac{F}{q_{0}}\) (c) \(<\frac{F}{q_{0}}\) (d) Cannot be estimated

A semi-circular arc of radius \(a\) is charged uniformly and the charge per unit length is \(\lambda\). The electric field at its centre is (a) \(\frac{\lambda}{2 \pi \varepsilon_{0}+^{2}}\) (b) \(\frac{\lambda}{4 \varepsilon_{0} a}\) (c) \(\frac{\lambda^{2}}{2 \pi e_{0} a}\) (d) \(\frac{\lambda}{2 \pi \varepsilon_{0} a}\)

Three charges each of magnitude \(q\) are placed at the corners of an equilateral triangle. The electrostatic force on the charge placed at the centre is (a) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{q^{2}}{L^{2}}\) (b) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{3 q^{2}}{L^{2}}\) (c) \(\frac{1}{12 \pi \varepsilon_{0}} \frac{q^{2}}{L^{2}}\) (d) zero

An electric dipole is placed at the centre of a hollow conducting sphere. Which of the following option is correct? (a) Electric field is non-zero anywhere on the sphere (b) The flux of electric field is zero through the sphere (c) Options (a) and (b) both are true (d) No option is correct

Two point charges placed at a certain distance \(r\) in air exert a force \(F\) on each other. Then the distance \(r\) at which these charges will exert the same force in a medium of dielectric constant \(K\) is given by (a) \(r / K\) (b) \(r / \sqrt{K}\) (c) \(r \sqrt{K}\) (d) \(r\)

A glass rod rubbed with silk is used to charged a gold leaf electroscope and the leaves are observed to diverse. The electroscope thin, charged is exposed to X-rays for short period. Then, (a) the leaves will diverge further (b) the leaves will melt (c) the leaves will not be affected (d) None of the above

Five balls numbered 1 to 5 are suspended using separate threads. Pairs \((1,2),(2,4)\) and \((4,1)\) show electrostatic attraction, while pair \((2,3)\) and \((4,5)\) show repulsion, therefore ball 1 must be (a) neutral (b) positively charged (c) negatively charged (d) None of these

A uniform electric field of \(100 \mathrm{~N} / \mathrm{C}\) exist in the vertically downward direction. The increase in the electric potential as one goes up the through a height of \(50 \mathrm{~cm}\) is (a) \(10 \mathrm{~V}\) (b) \(5 \mathrm{~V}\) (c) \(0 \mathrm{~V}\) (d) \(0.5 \mathrm{~V}\)

A solid metallic sphere has a charge \(+3 Q .\) Concentric with this sphere is a conducting spherical shell having charge \(-Q\). The radius of the sphere is \(a\) and that of the spherical shell is \(b(b>a)\). What is the electric field at a distance \(R(a

A charge \(+q\) is fixed at each of the points \(x=x_{0}\) \(x=3 x_{0}, x=5 x_{0} \ldots \infty\), on the \(x\)-axis and a charge \(-q\) is fixed at each of the points \(x=2 x_{0}, x=4 x_{0} x=6 x_{0} \ldots \infty .\) Here, \(x_{0} \quad\) is the constant. Take the electric potential at a point due to a charge \(Q\) at a distance \(\mathrm{r}\) from it to be \(Q / 4 \pi \varepsilon_{0} r\). Then, the potential at the origin due to the above system of charges is (a) \(\frac{q}{4 \pi \varepsilon_{0} x_{0}} \log _{c} 2\) (b) \(\frac{q}{8 \pi \varepsilon_{0} x_{0}} \log _{c} 2\) (c) 0 (d) \(\infty\)

Two equal charges \(q\) of opposite sign separated by a distance \(2 a\) constitute an electric dipole of dipole moment \(p\). If \(P\) is a point at a distance \(r\) from the centre of the dipole and the line joining the centre of the dipole to this point makes and angle \(\theta\) with the axis of the dipole, then the potential at \(P\) is given by \((r>>2 a)\) (where, \(p=2 q a)\) (a) \(V=\frac{p \cos \theta}{2 \pi \varepsilon_{0} r^{2}}\) (b) \(V=\frac{p \sin \theta}{4 \pi \varepsilon_{0} r}\) (c) \(V=\frac{p \cos \theta}{4 \pi \varepsilon_{0} r}\) (d) \(V=\frac{p \cos \theta}{4 \pi \varepsilon_{0} r^{2}}\)

The charge of \(+\frac{10}{3} \times 10^{-9} \mathrm{C}\) are placed at each of the four corners of a square of side \(8 \mathrm{~cm}\). The potential at the point of intersection of the diagonals, is (a) \(1500 \sqrt{2} \mathrm{~V}\) (b) \(1800 \sqrt{2} \mathrm{~V}(\mathrm{c}) 600 \sqrt{2} \mathrm{~V}\) (d) \(900 \sqrt{2} \mathrm{~V}\)

Two infinitely long parallel wires having linear charge densities \(\lambda_{1}\) and \(\lambda_{2}\) respectively are placed at a distance of \(R\) metres. The force per unit length on either wire will be \(\left(K=\frac{1}{4 \pi \varepsilon_{0}}\right)\) (a) \(K \frac{2 \lambda_{1} \lambda_{2}}{R^{2}}\) (b) \(K \frac{2 \lambda_{1} \lambda_{2}}{R}\) (c) \(K \frac{\lambda_{1} \lambda_{2}}{R^{2}}\) (d) \(K \frac{\lambda_{1} \lambda_{2}}{R}\)

Suppose, an imaginary cube is with a charge situated at the centre of it. The total electric flux passing through each of the faces of the cube will be (a) \(\frac{q}{6 \varepsilon_{0}}\) (b) \(\frac{q}{2 \varepsilon_{0}}\) (c) \(\frac{q}{12 \varepsilon_{0}}\) (c) None of these

An alpha particle of energy \(5 \mathrm{MeV}\) is scattered through \(180^{\circ}\) by a fixed uranium nucleus. The distance of closest approach is of the order of (a) \(1 \dot{A}\) (b) \(10^{-10} \mathrm{~cm}\) (c) \(10^{-12} \mathrm{~cm}\) (d) \(10^{-15} \mathrm{~cm}\)

Two equal negative charges \(-q\) are fixed at the points \((0, a)\) and \((0,-a)\) on the \(Y\)-axis. A positive charge \(Q\) is released from rest at the point \((2 a, 0)\) on the \(X\)-axis. The charge \(Q\) will (a) execute simple harmonic motion about the origin (b) move to the origin and remain at rest (c) move to infinity (d) execute oscillatory but not simple harmonic motion

A cylinder of radius, \(R\) and length, \(L\) is placed in a uniform electric field, \(E\) parallel to the cylinder axis. The total flux for the surface of the cylinder is given by (a) zero (b) \(\pi R^{2} / E\) (c) \(2 \pi R^{2} E\) (d) None of these

An electric line of force in the \(x y\)-plane is given by equation \(x^{2}+y^{2}=1 .\) A particle with unit positive charge, initially at rest at the point \(x=0, y=0\) in the \(x y\)-plane (a) not move at all (b) will move along straight line (c) will move along the circular line of force (d) information is insufficient to draw any conclusion

Figure given below shows two identical parallel plate capacitors connected to a battery with switch \(S\) closed. The switch is now opened and the free space between the plate of capacitors is filled with a dielectric constant 3 . What will be the ratio of total electrostatic energy stored in both capacitors before and after the introduction of the dielectric? (a) \(1: 2\) (b) \(1: 5\) (c) \(3: 5\) (d) \(5: 2\)

A parallel plate capacitor of capacitance \(C\) is connected to a battery and is charged to a potential difference \(V\). Another capacitor of capacitance \(2 C\) is connected to another battery and is charged to potential difference \(2 V\). The charging batteries are now disconnected and the capacitors are connected in parallel to each other in such a way that the positive terminal of one is connected to the negative terminal of the other. The final energy of the configuration is (a) infinite (b) zero (c) \(\frac{3 C v^{2}}{2}\) (d) \(\frac{6 \mathrm{C} \mathrm{V}^{2}}{2}\)

An electric dipole is placed at an angle of \(60^{\circ}\) with an electric field of intensity \(10^{5} \mathrm{NC}^{-1}\). It experiences a torque equal to \(8 \sqrt{3} \mathrm{~N}-\mathrm{m} .\) Calculate the charge on the dipole, if the dipole length is \(2 \mathrm{~cm}\). (a) \(-8 \times 10^{3} \mathrm{C}\) (b) \(8.54 \times 10^{-4} \mathrm{C}\) (c) \(8 \times 10^{-3} \mathrm{C}\) (d) \(0.85 \times 10^{-6} \mathrm{C}\)

The electrostatic potential inside a charged spherical ball is given by \(\phi=a r^{2}+b\), where, \(r\) is the distance from the centre, \(a\) and \(b\) are constants. Then the charge density inside the ball is (a) \(-24 \pi a \varepsilon_{0} r\) (b) \(-6 a \varepsilon_{0}\) (c) \(-24 \pi \varepsilon_{0}\) (d) \(-6 a \varepsilon_{0} r\)

An electric dipole consists of two opposite charges of magnitude \(q=1 \times 10^{-6} \mathrm{C}\) separated by \(2.0 \mathrm{~cm}\). The dipole is placed in an external field of \(1 \times 10^{5} \mathrm{NC}^{-1}\). What maximum torque does the field exert on the dipole? How much work must an external agent do to turn the dipole end, starting from position of alignment \(\left(\theta=0^{\circ}\right)\) ? (a) \(4.4 \times 10^{6} \mathrm{~N}-\mathrm{m}, 32 \times 10^{-4} \mathrm{~J}\) (b) \(2 \times 10^{3} \mathrm{~N}-\mathrm{m},-4 \times 10^{-3} \mathrm{~J}\) (c) \(4 \times 10^{3} \mathrm{~N}-\mathrm{m}, 2 \times 10^{-3} \mathrm{~J}\) (d) \(2 \times 10^{-3} \mathrm{~N}-\mathrm{m}, 4 \times 10^{-26} \mathrm{~J}\)

Two insulated charged conducting spheres of radii \(20 \mathrm{~cm}\) and \(15 \mathrm{~cm}\), respectively and having an equal charge of \(10 \mu \mathrm{C}\) are connected by a copper wire and then they are separated. Then (a) both spheres will have equal charges (b) surface charge density on the \(20 \mathrm{~cm}\) sphere will be greater than that on the \(15 \mathrm{~cm}\) sphere (c) surface charge density on the \(15 \mathrm{~cm}\) sphere will be greater than that on the \(20 \mathrm{~cm}\) sphere (d) surface charge density on the two spheres will be equal

Two electric dipoles of moment \(P\) and \(64 P\) are placed in opposite direction on a line at a distance of \(25 \mathrm{~cm}\). The electric field will be zero at point between the dipoles whose distance from dipole of moment \(P\) is (a) \(10 \mathrm{~cm}\) (b) \(5 \mathrm{~cm}\) (c) \(8 \mathrm{~cm}\) (d) \(20 \mathrm{~cm}\)

A point charge \(q\) moves from point \(P\) to point \(S\) along the path \(P Q R S\) in a uniform electric field \(\mathbf{E}\) pointing parallel to the positive direction of the \(x\)-axis as shown in figure. The coordinates of the points \(P, Q, R \quad\) and \(S\) are \((a, b, 0)(2 a, 0,0),(a,-b, 0)\) and \((0,0,0)\) respectively. The work done by the field in the above process is given by the expression (a) \(q E\) (b) \(-q a E\) (c) \(q\left(\sqrt{a^{2}+b^{2}}\right)+E\) (d) \(3 q E\left(\sqrt{a^{2}+b^{2}}\right)\)

The electrostatic potential on the surface of a charged conducting sphere is \(100 \mathrm{~V}\). Two statements are made in this regard. \(S_{1}\) : At any point inside the sphere, electric intensity is zero. \(S_{2}:\) At any point inside the sphere, the electrostatic potential is \(100 \mathrm{~V}\). Which of the following is a correct statements. [NCERT Exemplar] (a) \(S_{1}\) is true but \(S_{2}\) is false. (b) Both \(S_{1}\) and \(S_{2}\) are false (c) \(S_{1}\) is true, \(S_{2}\) is also true and \(S_{1}\) is the cause of \(S_{2}\) (d) \(S_{1}\) is true, \(S_{2}\) is also true but the statements are independent.

Two insulated metal spheres of radii \(10 \mathrm{~cm}\) and \(15 \mathrm{~cm}\) charged to a potential of \(150 \mathrm{~V}\) and \(100 \mathrm{~V}\) respectively, are connected by means of a metallic wire. What is the charge on the first sphere? (a) 2 esu (b) 4 esu (c) 6 esu (d) 8 esu

The electric potential \(\mathrm{V}\) at any point \((\mathrm{x}, \mathrm{y}, \mathrm{z})\) in space is given by \(V=4 x^{2}\). The electric field at \((1,0,2) \mathrm{m}\) in \(\mathrm{Vm}^{-1}\) is (a) 8 , along negative \(X\)-axis (b) 8 , along positive \(X\)-axis (c) 16, along negative \(X\)-axis (d) 16, along positive \(Z\)-axis

A hollow conducting sphere of radius, \(R\) has a charge \((+Q)\) on its surface. What is the electric potential within the sphere at a distance, \(r=R / 3\) from its centre? (a) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r}\) (b) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{2}}\) (c) \(\frac{1}{4 \pi E_{0}} \cdot \frac{Q}{R}\) (d) Zero

A parallel plate capacitor has the space between its plates filled by two slabs of thickness \(\frac{d}{2}\) each and dielectric constants \(K_{1}\) and \(K_{2} . d\) is the plate separation of the capacitor. The capacity of the capacitor is (a) \(\frac{2 \varepsilon_{0} d}{A}\left(\frac{K_{1}+K_{2}}{K_{1} K_{2}}\right)\) (b) \(\frac{2 \varepsilon_{0} A}{d}\left(\frac{K_{1} K_{2}}{K_{1}+K_{2}}\right)\) (c) \(\frac{2 \varepsilon_{0} d}{A}\left(K_{1}+K_{2}\right)\) (d) \(\frac{2 \varepsilon_{\mathrm{o}} A}{d}\left(\frac{K_{1}+K_{2}}{K_{1} K_{2}}\right)\)

Two point charges \(-q\) and \(+q / 2\) are situated at the origin and at the point \((a, 0,0)\), respectively. The point along the X-axis, whereas the electric field vanished, is (a) \(x=\frac{\sqrt{2} a}{\sqrt{2}-1}\) (b) \(X=\sqrt{2} a-\sqrt{2}-1\) (c) \(x=(\sqrt{2}-1) \sqrt{2} a\) (d) None of these

A positively charged particle is released from rest in an uniform electric field. The electric potential energy of the charge [NCERT Exemplar] (a) remains a constant because the electric field is uniform (b) increases because the charge moves along the electric field (c) decreases because the charge moves along the electric field (d) decreases because the charge moves opposite to the electric field

A ball of mass \(1 \mathrm{~kg}\) carrying a charge \(10^{-8} \mathrm{C}\) moves from a point \(A\) at potential \(600 \mathrm{~V}\) to a point \(B\) at zero potential. The change in its kinctic energy is (a) \(-6 \times 10^{-6} \mathrm{erg}\) (b) \(-6 \times 10^{-6} \mathrm{~J}\) (c) \(6 \times 10^{-6} \mathrm{~J}\) (d) \(6 \times 10^{-6} \mathrm{erg}\)