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1 Question 2.12: A charge of 8 mc is located at the origin. Calculate the work done in taking a small charge of C from a point P (0, 0, 3 cm) to a point Q (0, 4 cm, 0), via a point R (0, 6 cm, 9 cm). Answer 2.12: Charge located at the origin, q = 8 mc= C Magnitude of a small charge, which is taken from a point P to point R to point Q, q 1 = C All the points are represented in the given figure. Point P is at a distance, d 1 = 3 cm, from the origin along z-axis. Point Q is at a distance, d 2 = 4 cm, from the origin along y-axis. Potential at point P, Potential at point Q, Work done (W) by the electrostatic force is independent of the path. Therefore, work done during the process is 1.27 J. Question 2.13: A cube of side b has a charge q at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube. Answer 2.13: Length of the side of a cube = b Charge at each of its vertices = q A cube of side b is shown in the following figure. d = Diagonal of one of the six faces of the cube 1

2 l = Length of the diagonal of the cube The electric potential (V) at the centre of the cube is due to the presence of eight charges at the vertices. 4q Therefore, the potential at the centre of the cube is 3πε 0 b. The electric field at the centre of the cube, due to the eight charges, gets cancelled. This is because the charges are distributed symmetrically with respect to the centre of the cube. Hence, the electric field is zero at the centre. Question 2.14: Two tiny spheres carrying charges 1.5 µc and 2.5 µc are located 30 cm apart. Find the potential and electric field: (a) at the mid-point of the line joining the two charges, and (b) at a point 10 cm from this midpoint in a plane normal to the line and passing through the mid-point. Answer 2.14: Two charges placed at points A and B are represented in the given figure. O is the midpoint of the line joining the two charges. Magnitude of charge located at A, q 1 = 1.5 µc Magnitude of charge located at B, q 2 = 2.5 µc Distance between the two charges, d = 30 cm = 0.3 m (a) Let V 1 and E 1 are the electric potential and electric field respectively at O. V 1 = Potential due to charge at A + Potential due to charge at B Where, 0 = Permittivity of free space 2

3 E 1 = Electric field due to q 2 Electric field due to q 1 Therefore, the potential at mid-point is V and the electric field at mid-point is V m 1. The field is directed from the larger charge to the smaller charge. (b) Consider a point Z such that normal distance OZ = 10 cm = 0.1 m, as shown in the following figure. V 2 and E 2 are the electric potential and electric field respectively at Z. It can be observed from the figure that distance, V 2= Electric potential due to A + Electric Potential due to B Electric field due to q at Z, Electric field due to q 2 at Z, The resultant field intensity at Z, Where, 2θ is the angle, AZ B From the figure, we obtain 3

4 Therefore, the potential at a point 10 cm (perpendicular to the mid-point) is V and electric field is V m 1. Question 2.15: A spherical conducting shell of inner radius r1 and outer radius r2 has a charge Q. (a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell? (b) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain. Answer 2.15: (a) Charge placed at the centre of a shell is +q. Hence, a charge of magnitude q will be induced to the inner surface of the shell. Therefore, total charge on the inner surface of the shell is q. Surface charge density at the inner surface of the shell is given by the relation, A charge of +q is induced on the outer surface of the shell. A charge of magnitude Q is placed on the outer surface of the shell. Therefore, total charge on the outer surface of the shell is Q + q. Surface charge density at the outer surface of the shell, (b) Yes The electric field intensity inside a cavity is zero, even if the shell is not spherical and has any irregular shape. Take a closed loop such that a part of it is inside the cavity along a field line while the rest is inside the conductor. Net work done by the field in carrying a test charge over a closed loop is zero because the field inside the conductor is zero. Hence, electric field is zero, whatever is the shape. Question 2.16: (a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by Where n a unit vector is normal to the surface at a point and σ is the surface charge density at that point. (The direction of n is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is σ = n / 0. (b) Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another. [Hint: For (a), use Gauss s law. For, (b) use the fact that work done by electrostatic field on a closed loop is zero.] Answer 2.16: (a) Electric field on one side of a charged body is E 1 and electric field on the other side of the same body is E 2. If infinite plane charged body has a uniform thickness, then electric field due to one surface of the charged body is given by, 4

5 Where, = Unit vector normal to the surface at a point σ = Surface charge density at that point Electric field due to the other surface of the charged body, Electric field at any point due to the two surfaces, Since inside a closed conductor, = 0, Therefore, the electric field just outside the conductor is σ 0 n. (b) When a charged particle is moved from one point to the other on a closed loop, the work done by the electrostatic field is zero. Hence, the tangential component of electrostatic field is continuous from one side of a charged surface to the other. Question 2.17: A long charged cylinder of linear charged density λ is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders? Answer 2.17: Charge density of the long charged cylinder of length L and radius r is λ. Another cylinder of same length surrounds the pervious cylinder. The radius of this cylinder is R. Let E be the electric field produced in the space between the two cylinders. Electric flux through the Gaussian surface is given by Gauss s theorem as, Where, d = Distance of a point from the common axis of the cylinders Let q be the total charge on the cylinder. It can be written as Where, q = Charge on the inner sphere of the outer cylinder 0 = Permittivity of free space Therefore, the electric field in the space between the two cylinders is 5

6 Question 2.18: In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å: (a) Estimate the potential energy of the system in ev, taking the zero of the potential energy at infinite separation of the electron from proton. (b) What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in (a)? (c) What are the answers to (a) and (b) above if the zero of potential energy is taken at 1.06 Å separation? Answer 2.18: The distance between electron-proton of a hydrogen atom, Charge on an electron, q 1 = C Charge on a proton, q 2 = C (a) Potential at infinity is zero. Potential energy of the system, p-e = Potential energy at infinity Potential energy at distance, d Where, 0 is the permittivity of free space Therefore, the potential energy of the system is 27.2 ev. (b) Kinetic energy is half of the magnitude of potential energy. Total energy = = 13.6 ev Therefore, the minimum work required to free the electron is 13.6 ev. (c) When zero of potential energy is taken, Potential energy of the system = Potential energy at d 1 Potential energy at d 6

7 Question 2.19: If one of the two electrons of a H 2 molecule is removed, we get a hydrogen molecular ion H + 2. In the ground state of an H + 2, the two protons are separated by roughly 1.5 Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy. Answer 2.19: The system of two protons and one electron is represented in the given figure. Charge on proton 1, q 1 = C Charge on proton 2, q 2 = C Charge on electron, q 3 = C Distance between protons 1 and 2, d 1 = m Distance between proton 1 and electron, d 2 = m Distance between proton 2 and electron, d 3 = m The potential energy at infinity is zero. Potential energy of the system, Therefore, the potential energy of the system is 19.2 ev. Question 2.20: Two charged conducting spheres of radii a and b are connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to explain why charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions. Answer 2.20: Let a be the radius of a sphere A, Q A be the charge on the sphere, and C A be the capacitance of the sphere. Let b be the radius of a sphere B, Q B be the charge on the sphere, and C B be the capacitance of the sphere. Since the two spheres are connected with a wire, their potential (V) will become equal. Let E A be the electric field of sphere A and E B be the electric field of sphere B. Therefore, their ratio, Putting the value of (2) in (1), we obtain Therefore, the ratio of electric fields at the surface is b/a. 7

8 Question 2.21: Two charges q and +q are located at points (0, 0, a) and (0, 0, a), respectively. (a) What is the electrostatic potential at the points? (b) Obtain the dependence of potential on the distance r of a point from the origin when r/a >> 1. (c) How much work is done in moving a small test charge from the point (5, 0, 0) to ( 7, 0, 0) along the x- axis? Does the answer change if the path of the test charge between the same points is not along the x- axis? Answer 2.21: (a) Zero at both the points Charge q is located at (0, 0, a) and charge + q is located at (0, 0, a). Hence, they form a dipole. Point (0, 0, z) is on the axis of this dipole and point (x, y, 0) is normal to the axis of the dipole. Hence, electrostatic potential at point (x, y, 0) is zero. Electrostatic potential at point (0, 0, z) is given by, Where, = Permittivity of free space p = Dipole moment of the system of two charges = 2qa (b) Distance r is much greater than half of the distance between the two charges. Hence, the potential (V) at a distance r is inversely proportional to square of the distance. i.e. V 1 (c) Zero The answer does not change if the path of the test is not along the x-axis. A test charge is moved from point (5, 0, 0) to point ( 7, 0, 0) along the x-axis. Electrostatic potential (V 1) at point (5, 0, 0) is given by, r 2 Electrostatic potential, V 2, at point ( 7, 0, 0) is given by, Hence, no work is done in moving a small test charge from point (5, 0, 0) to point ( 7, 0, 0) along the x- axis. The answer does not change because work done by the electrostatic field in moving a test charge between the two points is independent of the path connecting the two points. 8

9 Question 2.22: Figure 2.34 shows a charge array known as an electric quadrupole. For a point on the axis of the quadrupole, obtain the dependence of potential on r for r/a >> 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge). Answer 2.22: Four charges of same magnitude are placed at points X, Y, Y, and Z respectively, as shown in the following figure. A point is located at P, which is r distance away from point Y. The system of charges forms an electric quadrupole. It can be considered that the system of the electric quadrupole has three charges. Charge +q placed at point X Charge 2q placed at point Y Charge +q placed at point Z XY = YZ = a YP = r PX = r + a PZ = r a Electrostatic potential caused by the system of three charges at point P is given by, Since, is taken as negligible. It can be inferred that potential, V 1 r 3 However, it is known that for a dipole, V 1 r 2 And, for a monopole, V 1 r 9

10 Question 2.23: An electrical technician requires a capacitance of 2 µf in a circuit across a potential difference of 1 kv. A large number of 1 µf capacitors are available to him each of which can withstand a potential difference of not more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors. Answer 2.23: Total required capacitance, C = 2 µf Potential difference, V = 1, kv = 1000 V Capacitance of each capacitor, C 1 = 1µF Each capacitor can withstand a potential difference, V 1 = 400 V Suppose a number of capacitors are connected in series and these series circuits are connected in parallel (row) to each other. The potential difference across each row must be 1000 V and potential difference across each capacitor must be 400 V. Hence, number of capacitors in each row is given as = 2.5 Hence, there are three capacitors in each row. Capacitance of each row = = 1 3 μf Let there are n rows, each having three capacitors, which are connected in parallel. Hence, equivalent capacitance of the circuit is given as Hence, 6 rows of three capacitors are present in the circuit. A minimum of 6 3 i.e., 18 capacitors are required for the given arrangement. Question 2.24: What is the area of the plates of a 2 F parallel plate capacitor, given that the separation between the plates is 0.5 cm? [You will realize from your answer why ordinary capacitors are in the range of µf or less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of very minute separation between the conductors.] Answer 2.24: Capacitance of a parallel capacitor, V = 2 F Distance between the two plates, d = 0.5 cm = m Capacitance of a parallel plate capacitor is given by the relation, Where, = Permittivity of free space Hence, the area of the plates is too large. To avoid this situation, the capacitance is taken in the range of µf. 10

11 Question 2.25: Obtain the equivalent capacitance of the network in Figure. For a 300 V supply, determine the charge and voltage across each capacitor. Answer 2.25: Capacitance of capacitor C 1 is 100 pf. Capacitance of capacitor C 2 is 200 pf. Capacitance of capacitor C 3 is 200 pf. Capacitance of capacitor C 4 is 100 pf. Supply potential, V = 300 V Capacitors C 2 and C 3 are connected in series. Let their equivalent capacitance be C. Therefore, 1 C = = = 1 C = 100 pf 100 Capacitors C 1 and C are in parallel. Let their equivalent capacitance be C. C = C + C = = 200 pf C and C 4 are connected in series. Let their equivalent capacitance be C. 1 C = 1 C + 1 = 1 C = C = pf Hence, the equivalent capacitance of the circuit is 200 pf Potential difference across C = V Potential difference across C 4 = V 4 3 Charge on C 4 is given by Q 4= CV Hence, potential difference, V 1, across C 1 is 100 V. Charge on C 1 is given by, C 2 and C 3 having same capacitances have a potential difference of 100 V together. Since C 2 and C 3 are in series, the potential difference across C 2 and C 3 is given by, V 2 = V 3 = 50 V Therefore, charge on C 2 is given by, 11

12 And charge on C 3 is given by, Hence, the equivalent capacitance of the given circuit is 200 pf with 3 Question 2.26: The plates of a parallel plate capacitor have an area of 90 cm 2 each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply. (a) How much electrostatic energy is stored by the capacitor? (b) View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volume u. Hence arrive at a relation between u and the magnitude of electric field E between the plates. Answer 2.26: Area of the plates of a parallel plate capacitor, A = 90 cm 2 = m 2 Distance between the plates, d = 2.5 mm = m Potential difference across the plates, V = 400 V (a) Capacitance of the capacitor is given by the relation, C = 0A Electrostatic energy stored in the capacitor is given by the relation, E 1 = 1 2 CV2 = 1 2 Where, = Permittivity of free space = C 2 N 1 m 2 d 0 A d V2 (b) Volume of the given capacitor, V = A d = = m 3 Energy stored in the capacitor per unit volume is given by, Where, V = Electric intensity = E d Therefore, U = E 2 12

13 Question 2.27: A 4 µf capacitor is charged by a 200 V supply. It is then disconnected from the supply, and is connected to another uncharged 2 µf capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation? Answer 2.27: Capacitance of a charged capacitor, C 1 = 4μF = F Supply voltage, V 1 = 200 V Electrostatic energy stored in C 1 is given by, Capacitance of an uncharged capacitor, C 2 = 2μF = F When C 2 is connected to the circuit, the potential acquired by it is V 2. According to the conservation of charge, initial charge on capacitor C 1 is equal to the final charge on capacitors, C 1 and C 2. Electrostatic energy for the combination of two capacitors is given by, Hence, amount of electrostatic energy lost by capacitor C 1 = E 1 E 2 = = = J Question 2.28: Show that the force on each plate of a parallel plate capacitor has a magnitude equal to (½) QE, where Q is the charge on the capacitor, and E is the magnitude of electric field between the plates. Explain the origin of the factor ½. Answer 2.28: Let F be the force applied to separate the plates of a parallel plate capacitor by a distance of x. Hence, work done by the force to do so = F x As a result, the potential energy of the capacitor increases by an amount given as ua x. Where, u = Energy density, A = Area of each plate, d = Distance between the plates V = Potential difference across the plates The work done will be equal to the increase in the potential energy i.e., 13

14 Electric intensity is given by, However, capacitance, Charge on the capacitor is given by, Q = CV The physical origin of the factor, ½, in the force formula lies in the fact that just outside the conductor, field is E and inside it is zero. Hence, it is the average value, E/2, of the field that contributes to the force. Question 2.29: A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.36). Show that the capacitance of a spherical capacitor is given. Where r 1 and r 2 are the radii of outer and inner by C = 4πε 0 r 1 r 2 r 1 r 2 spheres, respectively. Answer 2.29: Radius of the outer shell = r 1, radius of the inner shell = r 2 The inner surface of the outer shell has charge +Q. The outer surface of the inner shell has induced charge Q. Potential difference between the two shells is given by, Where, = Permittivity of free space Hence, proved. 14

15 Question 2.30: A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 µc. The space between the concentric spheres is filled with a liquid of dielectric constant 32. (a) Determine the capacitance of the capacitor. (b) What is the potential of the inner sphere? (c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller. Answer 2.30: Radius of the inner sphere, r 2 = 12 cm = 0.12 m Radius of the outer sphere, r 1 = 13 cm = 0.13 m Charge on the inner sphere, q = 2.5 μc = C Dielectric constant of a liquid, r = 32 (a) Capacitance of the capacitor is given by the relation, C = 4πε 0ε r r 1 r 2 r 1 r 2 Where, = Permittivity of free space = Hence, the capacitance of the capacitor is approximately F.. (b) Potential of the inner sphere is given by, Hence, the potential of the inner sphere is V. (c) Radius of an isolated sphere, r = m Capacitance of the sphere is given by the relation, The capacitance of the isolated sphere is less in comparison to the concentric spheres. This is because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere. Question 2.31: Answer carefully: (a) Two large conducting spheres carrying charges Q 1 and Q 2 are brought close to each other. Is the magnitude of electrostatic force between them exactly given by Q 1Q 2/4πε 0 r 2, where r is the distance between their centres? (b) If Coulomb s law involved 1/r 3 dependence (instead of 1/r 2 ), would Gauss s law be still true? (c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point? 15

16 (d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical? (e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there? (f) What meaning would you give to the capacitance of a single conductor? (g) Guess a possible reason why water has a much greater dielectric constant (80) than say, mica (6). Answer 2.31: (a) The force between two conducting spheres is not exactly given by the expression, Q 1 Q 2/4πε 0 r 2, because there is a non-uniform charge distribution on the spheres. (b) Gauss s law will not be true, if Coulomb s law involved 1/r 3 dependence, instead of1/r 2, on r. (c) Yes, If a small test charge is released at rest at a point in an electrostatic field configuration, then it will travel along the field lines passing through the point, only if the field lines are straight. This is because the field lines give the direction of acceleration and not of velocity. (d) Whenever the electron completes an orbit, either circular or elliptical, the work done by the field of a nucleus is zero. (e) No Electric field is discontinuous across the surface of a charged conductor. However, electric potential is continuous. (f) The capacitance of a single conductor is considered as a parallel plate capacitor with one of its two plates at infinity. (g) Water has an unsymmetrical space as compared to mica. Since it has a permanent dipole moment, it has a greater dielectric constant than mica. Question 2.32: A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 µc. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends). Answer 2.32: Length of a co-axial cylinder, l = 15 cm = 0.15 m Radius of outer cylinder, r 1 = 1.5 cm = m Radius of inner cylinder, r 2 = 1.4 cm = m Charge on the inner cylinder, q = 3.5 µc = C Capacitance of a co-axial cylinder of radii r 1 and r 2 is given by the relation C = 2πε 0l log e r 1 r2 Where, = Permittivity of free space = Potential difference of the inner cylinder is given by, 16

17 Question 2.33: A parallel plate capacitor is to be designed with a voltage rating 1 kv, using a material of dielectric constant 3 and dielectric strength about 10 7 Vm 1. (Dielectric strength is the maximum electric field a material can tolerate without breakdown, i.e., without starting to conduct electricity through partial ionisation.) For safety, we should like the field never to exceed, say 10% of the dielectric strength. What minimum area of the plates is required to have a capacitance of 50 pf? Answer 2.33: Potential rating of a parallel plate capacitor, V = 1 kv = 1000 V Dielectric constant of a material, ε r =3 Dielectric strength = 10 7 V/m For safety, the field intensity never exceeds 10% of the dielectric strength. Hence, electric field intensity, E = 10% of 10 7 = 10 6 V/m Capacitance of the parallel plate capacitor, C = 50 pf = F Distance between the plates is given by, Where, A = Area of each plate = Permittivity of free space = Hence, the area of each plate is about 19 cm 2. Question 2.34: Describe schematically the equipotential surfaces corresponding to (a) a constant electric field in the z-direction, (b) a field that uniformly increases in magnitude but remains in a constant (say, z) direction, (c) a single positive charge at the origin, and (d) a uniform grid consisting of long equally spaced parallel charged wires in a plane. Answer 2.34: (a) Equidistant planes parallel to the x-y plane are the equipotential surfaces. (b) Planes parallel to the x-y plane are the equipotential surfaces with the exception that when the planes get closer, the field increases. (c) Concentric spheres centered at the origin are equipotential surfaces. (d) A periodically varying shape near the given grid is the equipotential surface. This shape gradually reaches the shape of planes parallel to the grid at a larger distance. 17

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