1 UNIT - 11 ELECTROSTATICS Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 2 SUMMARY 1. Electric Charge : Just as masses of two particles are responsible for the gravitational force, charges are responsible for the electric force. Electric charge is an intrinsic property of a particle. Charges are of two types : (1) Positive charges (2) Nagative charges. The force acting between two like charges is repulsive and two unlike charges it is attractive between . 2. Quantization of Electric Charge : The magnitude of all charges found in nature are an integral multiple of a fundamental charge. , ne Q  where e is the fundamental unit of charge. 3. Conservation of Electric Charge : Irrespective of any process taking place, the algebraic sum of electric charges in an electrically isolated system always remains constant. 4. Coulomb's Law : The electric force between two stationary point charges is directly propor- tional to the product of their charges and inversely proportional to the square of the distance between them. 2 2 1 0 2 2 1 r q q 4 1 r q q k F    If 0 q q 2 1  then there is a repulsion between the two charges and for 0 q q 2 2  , there is a attrac- tion between the charges. 5. Equation for Force using Columb’s Law, when two charges are placed in a medium having dielectric constant k. (1) The electric force   F experienced by a test charge (q0) due to a source charge (q) when both are placed in a medium having dielectric constant k and separated by a dis- tance r, is given by : rˆ kr q q 4 1 F 2 2 1 0   F r P O (q) (qo) Here rˆ is the unit vector directed from q to q0. (2) The equation of coulomb's force may be written as follows :   rˆ r k q q 4 1 F 2 2 1 0   (3) If the source charge and test charge are separated by a number of medium of thickness 1 2 3 d , d , d ........ having dielectric constants ........ k , k , k 3 2 1 respectively, then the electric force on charge q0 due to a charge q is given by   0 2 2 2 0 1 1 2 2 3 3 qq 1 ˆ F r 4 k d k d k d      0 2 0 i i qq 1 ˆ F r 4 k d        OR  In this equation ki is dielectric constant of medium which spreads through the distance di along the line joining q and q0. Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 3 For example, see the figure below : Here, the space between the charges q and q0 is filled with medium (1, 2, 3). The thickness of medium 1 is d1 and its dielectric consant is k1 Similarly the thickness of medium 2 and 3 is d2 and d3 of medium 3 and their dielectric constants are k2 and k3 respectively. 6. Conditions for Equilibrium in Various Cases : Suppose three charges q1, q2 and q are situated on a straight line as shown below : If q1 and q2 are like charges and q is of unlike charge then, (1) Force on q1              2 1 2 2 1 1 0 1 1 r q r r q 4 q F (2) Force on q2              2 2 2 2 1 1 0 2 2 r q r r q 4 q F (3) Force on q =        2 2 2 2 1 1 0 r q r q 4 1 F Now, from above equations, it is clear that various equilibrium conditions can be as follows : (a) Condition for 1F to be zero is,     2 2 1 2 1 2 2 1 2 2 1 r r r q q r r q r q      (b) Condition for 2F to be zero is,     2 2 1 2 2 1 2 2 1 1 2 2 r r r q q r r q r q      (c) Condition for F to be zero is, 2 2 2 2 1 1 r q r q  2 2 2 1 2 1 r r r q   If 2 1 q , q and q are of same type charges in nature, then, (1) Charge q will be in equilibrium, if 0 r q r q 4 q F 2 2 2 2 1 1 0         2 1 2 1 1 2 2 2 1 2 2 2 q q q r r r q r     (2) Charges q1 and q2 will not be in equilibrium. Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 4 7. Electric Field Intensity : The electric force acting on a unit positive charges at a given point in an electric field of a system of charges is called the electric field or the intensity of electric field   E at that point. q F E  The SI unit of E is C N or 1 Vm  . If n 2 1 r ... ,......... r , r are the position vectors of the charges n 2 1 q. ,......... q , q respectively, then the resultant electric field at a point of position vector r is,   j n 1 j 3 j j r r r r q k E      8. Electric Dipole : A system of two equal and opposite charge, separated by a finite distance is called electric dipole. Electric dipole moment   a 2 q p  The direction of p is from the negative electric charge to the positive electric charge. 9. Electric field of a dipole on the axis of the dipole at point z = z     ^ 3 2kp E z p for z a z     Electric field of a dipole on the equator of the dipole at point y = y     ^ 3 kp E y p for y a y     10. The torque acting on the dipole place in an uniform the electric field at an angle  ,       sin E p | |, E p 11. Electric Flux : Electric flux associated with surface of area A , placed in the uniform electric field.      cos EA A E where,  is the angle between A and E , Its SI unit is 2 Nm C or V.m. 12. Gauss's Law : The total electric flux associated with the closed surface, 0 S q E d a          where, q  is the net charge enclosed by the surface. Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 5 13. Electric field due to an infinitely long straight charged wire, ,rˆ r 1 2 E 0    where, r is the perpendicular distance from the charged wire. 14. Electric field due to bending of charged rod, 15. Electric field due to uniformly charged thin spherical shell, (1) Electric field inside the shell 0 E  (2) Electric field at a distance r from the centre outside the shell, 2 2 0 2 r R r q k E     where, R = radius of spherical shell. 16. Electric field due to a uniformly charged density sphere of radius R, (1) Electric field inside the region of the sphere, 3 0 0 Q r r E 4 R 3     Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 6 (2) Electric field outside the sphere,   0 2 3 2 0 r 3 R r r 4 Q r E      where, Q is the total charge inside the sphere. 17. The information about the work done to take an electric charge from one point to the other in a given electric field, obtained from the quantities called electric potential and electric potential energy. 18.   B A dr E is the line-integral of electric field between point A and B and it shows the work done by the electric field in taking a unit positive charge from A and B. Moreover, it does not depend on the path and 0 dr E    . 19. "The work required to be done against the electric field to bring a unit positive charge from infinite distance to the given point in the electric field, is called the electric potential (V) at that point". Electric potential at point P is      P p dr E V It unit is . volt coulomb Joule  Symbolically C J V  Its dimensional formula is 1 3 2 1 A T L M   Absolute value of electric potential has no importance but only the change in it is important. 20. "The work required to be done against the electric field to bring a given change (q) from infinite distance to the given point in the electric field is called the electric potential energy of that electric charge at that point." p P p qV dr E q U       The absolute value of electric potential energy has no importance, only the change in it is impor- tant. 21. Electric potential at point P, lying at a distance r from a point charge q is r kq Vp  22. The electric potential at a point at distance r from an electric dipole is   , r p 4 1 r v 2 0   ( For r > > 2a) Potential on its axis is , r p 4 1 V 2 0    Potential on its equator is 0 V  23. Electric potential at a point r due to a system of point charge n 2 1 q ,......... q , q situated at position at position n 2 1 r ,......... r , r is     n 1 i i i r r kq V Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 7 The electric potential at point r , due to a continuous charge distribution is   0 4 1 r V         volume ir r ' d r The electric potential due to a spherical shell is   0 1 q V For r R 4 r    and   0 1 q V For r R 4 R    24. A surface on which electric potential is equal at all points is called an equipotential surface. The direction of electric field is normal to the equipotential surface. 25. dV E dl   gives the magnitude of electric field in the direction of dl  . To find E from V, in general, we can use the equation V V V ˆ ˆ ˆ E i j k x y z                  The direction of electric field is that in which the rate of decrease of electric potential with distance dV dl  is maximum, and this direction is always normal to the equipotential surface. 26. The electrostatic potential energy of a system of point charges n 2 1 q. ,......... q , q situated at positions n 2 1 r ,....... r , r is n i j i 1 ij i j kq q U r     where i j ij r r r   27. The electrostatic potential energy of an electric dipole in an external electric field E, is U E p Epcos        28. When a metallic conductor is placed in an external electric field, (i) A stationary charge distribution is induced on the surface of the conductor. (ii) The resultant electric field inside the conductor is zero. (ii) The net electric charge inside the conductor is zero. (iv) The electric field at every point on the outer surface of conductor is locally normal to the surface. (v) The electric potential inside the region of conductor is the same every where. (vi) If there is a cavity in the conductor then, even when the conductor is placed in an external electric field, the resultant electric field inside the conductor and also inside the cavity is always zero. This fact is called the electrostatic shielding. Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 8 When electric charge is placed on the metallic conductor : (i) The electric field inside the conductor is zero everywhere. (ii) The charge is distributed only on the outer surface of the conductor. (iii) The electric field on the surface is locally normal, and is equal to nˆ E 0          . (iv) If a charge is placed inside the cavity in the conductor, the electric field in the conductor remains zero. 29. "A device formed by two conductors seprated from each other is called a capacitor." Its capaci- tance is   V Q C constant. The unit of C is volt coulomb which is also called farad. F 10 pF .F 10 F 1 12 6      30. The effective capacitance in series connection is C then, .......... C 1 C 1 C 1 C 1 3 2 1     The effective capacitance in parallel connection is C then, ......... C C C C 3 2 1     31. The capacitance of the parallel plate capacitor is d A C 0  . 32. The energy stored in the capacitor is 2 VQ 2 CV C 2 Q U 2 2    and the energy density = energy stored per unit volume , E 2 1 2 0  where  E electric field. 33. When a dielectric is placed in an external electric field 0 E , polarisation of dielectric occurs due to electrical induction. The electric field produced by these induced charges is in the opposite direction to the direction of external electric field. Hence the resultant electric field E, inside the dielectric is less than the external electric field 0 E . The dipole moment produced per uint volume is called the intensity of polarisation or in short polarisation . nˆ P b   Since E P  , P E 0 e e x x    is called the electric susceptibility of the dielectric medium.   1   0 ex is called the permittivitty  of the dielectric medium.  0 is called the relative permittivity of that medium and it is also called the dielectric constant K. Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 9 i.e. K      r 0 E K 1 , E . K    0 ex Thus in the dielectric the electric field reduces to the th K part. P E D 0    is called the electric displacement. Gauss Law in the presence of dielectric is written as q ds D    , where q is only the net free charge. 34. when there is air (or vacuum) between the plates of a parallel plate capacitor, the capacitance is d A C 0  . On placing a medium of dielectric constant K, the capacitance is . CK ' C  Thus the capacitance becomes K times, due to the presence of the dielectric. 35. With the help of Van-De-Graf generator a potential differance of a few nillion volt can be established. Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 10 CONCEPT MAP Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 11 MCQ For the answer of the following questions choose the correct alternative from among the given ones. 1. When a Piece of Polythene is rubbed with wool, a charge of –7 –2 10  is developed on polythene. The mass transferred to polythene is ..... kg. (A) –19 11.38 10  (B) –19 5.69 10  (C) –19 2.25 10  (D) –19 9.63 10  2. The protonic charge in 100 gm of water is .......... c (A) 5 4.8 10  (B) 6 5.4 10  (C) 4 3.6 10  (D) 6 4.9 10  3. A copper sphere of mass 2 gm contains about 22 2 10  atoms. The charge on the nucleus of each atom is 29e. The fraction of electrons removed. (A) –10 2 10  (B) –12 1.19 10  (C) –11 1.25 10  (D) –11 2.16 10  4. The rate of alpha partical falls on neutral spheare is 1012 per second. The time in which sphere gets charged by 2µc is ......... sec. (A) 2.25 (B) 3.15 (C) 6.25 (D) 1.66 5. A charge Q is divided into two parts and then they are placed at a fixed distance. The force between the two charges is always maximum when the charges are ......... (A) Q Q , 3 3 (B) Q Q , 2 2 (C) Q 3Q , 4 4 (D) Q 4Q , 5 5 6. Two point charges repel each other with a force of 100 N. One of the charges is increased by 10% and other is reduced by 10%. The new force of repulsion at the same distance would be ........ N. (A) 121 (B) 100 (C) 99 (D) 89 7. Given that q1 + q2 = q if the between q1 and q2 is maximum, 1q q ............... (A) 1 (B) 0.75 (C) 0.25 (D) 0.5 8. Two small conducting sphere of equal radius have charges +1µc and – 2µc respectively and placed at a distance d from each other experience force F1. If they are brought in contact and separated to the same distance, they experience force F2. The ratio of F1 to F2 is .......... (A) –8 : 1 (B) 1 : 2 (C) 1 : 8 (D) –2 : 1 9. Three charges, each of value Q, are placed at the vertex of an equilateral triangle. A fourth charge q is placed at the centre of the triangle. If the charges remains stationery then, q = ............... (A) Q 2 (B) Q – 3 (C) Q – 2 (D) Q 3 Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 12 10. Two small charged spheres repal each other with a force –3 2 10 N  . The charge on one sphere is twice that of the other. When these two spheares displaced 10 cm further apart the force is –4 5 10 N  , then the charges on both the spheres are ........ (A) –19 –19 1.6 10 C, 3.2 10 C   (B) –19 –19 3.4 10 C,11.56 10 C   (C) –19 –19 33.33 10 C, 66.66 10 C   (D) –19 –19 2.1 10 C, 4.41 10 C   11. Three charges –q1, + q2 and –q3 are placed as shown in figure. The x component of the force on –q1 is proportional to ......... (A) 3 2 2 2 q q – sinθ b a (B) 3 2 2 2 q q – cosθ b a (C) 3 2 2 2 q q + sinθ b a (D) 3 2 2 2 q q + cosθ b a 12. Two equal negative charges –q are fixed at points (o, a) and (o, –a). A positive charge Q is released from rest at the point (2a, o) on the X - axis. The charge Q will .......... (A) move to the origin and remain at rest there (B) execute simple harmonic motion about the origin (C) move to infinity (D) execute oscillations but not simple harmonic motion 13. Four charges, each equal to –Q, are placed at the corners of a square and a charge +q is placed at its centre. If the system is in equilibrium, the value of q is ......... (A)   Q 1 2 2 4  (B)   Q – 1 2 2 4  (C)   Q – 1 2 2 2  (D)   Q 1 2 2 2  14. For the system shown in figure, if the resultant force on q is zero, then Q = ............... (A) –2 2Q (B) 2 2Q (C) 2 3Q (D) –3 2Q 1 –q1 +q2 X Y b a –q3  Q q FA FA a q (0, a) Q a Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 13 15. Two point positive charges q each are placed at (–a, o) and (a, o). A third positive charge qo is placed at (o, y). For which value of y the force at qo is maximum ......... (A) a (B) 2a (C) a 2 (D) a 3 16. Two identical charged spheres suspended from a common point by two massless strings of length l are initially a distance d (d << l ) apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the spheres approach each other with a velocity υ . Then function of distance x between them becomes ........... (A) α v x (B) –1 2 α v x (C) –1 α v x (D) 1 2 α v x 17. Three identical spheres each having a charge q and radius R, are kept in such a way that each touches the other two spheares. The magnitude of the electric force on any sphere due to other two is ........... (A) 2 0 1 5 2 4π 4 R R q        (B) 2 0 1 2 8π 3 R q        (C) 2 0 1 3 4π 4 R q        (D) 2 0 1 3 – 8π 2 R q        18. Two equal negative charges –q are fixed at points (o, a) and (o, –a) on the Y axis. A positive charge q is released from rest at the point x (x < < a) on the X-axis, then the frequency of motion is ......... (A) 2 3 0 q π ma  (B) 2 3 0 2q 4π ma  (C) 2 3 0 4q 2π ma  (D) 2 3 0 q 2π ma  19. Two identical balls having like charges and placed at a certain distance apart repel each other with a certain force. They are brought in contact and then moved apart to a distance equal to half their initial separation. The force of repulsion between them increases 4.5 times in comparison with the initial value. The ratio of the initial charges of the balls is ........ (A) 4 : 1 (B) 6 : 1 (C) 3 : 1 (D) 2 : 1 20. A point charge q is situated at a distance r from one end of a thin conducting rod of length L having a charge Q (uniformly distributed along its length). The magnitude of electric force between the two, is ............... (A) 2kqQ r(r + L) (B) kqQ r(r + L) (C) kqQ r(r – L) (D) kQ r(r + L) 21. Two point charges of +16µc and –9µc are placed 8 cm apart in air. ............... distance of a point from –9µc charge at which the resultant electric field is zero. (A) 24 cm (B) 9 cm (C) 16 cm (D) 35 cm Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 14 22. Point charges 4µc and 2µc are placed at the vertices P and Q of a right angle triangle PQR respectively. Q is the right angle, –2 PR = 2 10 m  and –2 QR =10 m . The magnitude and direction of the resultant electric field at c is ......... (A) 9 –1 0 4.28×10 NC , 45 (B) 8 –1 0 2.38×10 NC , 40.9 (C) 4 –1 0 1.73×10 NC , 34.7 (D) 10 –1 0 4.9×10 NC , 34.7 23. An inclined plane making an angle of 30o with the horizontal is placed in an uniform electric field E = 100 Vm–1. A particle of mass 1 kg and charge 0.01 c is allowed to slide down from rest from a height of 1m. If the coefficient of friction is 0.2 the time taken by the particle to reach the bottom is .......... sec. (A) 2.337 (B) 4.337 (C) 5 (D) 1.337 24. A small sphere whose mass is 0.1 gm carries a charge of –10 3 10 C  and is tieup to one end of a silk fibre 5 cm long. The other end of the fibre is attached to a large vertical conducting plate which has a surface charge of –6 –2 25 10 Cm  , on each side. When system is freely hanging the angle fibre makes with vertical is ............... (A) 41.80 (B) 450 (C) 40.80 (D) 45.80 25. A Semicircular rod is charged uniformly with a total charge Q coulomb. The electric field intensity at the centre of curvature is ....... (A) 2 2KQ πr (B) 2 3KQ πr (C) 2 KQ πr (D) 2 4KQ πr 26. The electron is projected from a distance d and with initial velocity 0 υ parallel to a uniformly charged flat conducting plate as shown in figure. It strikes the plate after travelling a distance l along the direction. The sur- face charge density of conducting plate is equal to (A) 0 0 2d mυ e  l (B) 2 0 0 d mυ e  l (C) 0 0 d mυ e  l (D) 2 0 0 2 2d mυ e  l 27. Two point masses m each carrying charge –q and +q are attached to the ends of a massless rigid non-conducting rod of length l. The arrangement is placed in a uniform electric field E such that the rod makes a small angle 50 with the field direction. The minimum time needed by the rod to align itself along the field is ........ (A) 2m t = π 3qE l (B) π m t = 2 2qE l (C) m t = qE l (D) m t = 2π E l v0 p X Y ----l----- Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com 15 28. Two uniformaly charged spherical conductors A and B having radius 1mm and 2mm are separated by a distance of 5 cm. If the spheres are connected by a conducting wire then in equilibrium condition, the ratio of the magnitude of the electric fields at the surfaces of spheres A and B is ......... (A) 4 : 1 (B) 1 : 2 (C) 2 : 1 (D) 1 : 4 29. Let 4 Q P(r) = r πR be the charge density distribution for a solid sphere of radius R and total charge Q. For a point ‘P’ inside the sphere at distance r1 from the centre of the sphere the magnitude of electric field is (A) 2 0 1 Q 4π r  (B) 2 1 4 0 Qr 4π R  (C) 2 1 4 0 Qr 3π R  (D) 0 30. Two point charges 1q 2µc  and 2 q 1µc  are placed at distance b = 1cm and a = 2 cm from the origin on the y and x axes as shown in figure. The electric field vector at point P (a, b) will subtend an angle  with the X - axis given by, (A) tan θ 4  (B) tanθ 1  (C) tanθ 3  (D) tan θ 2  31. A simple pendulum consists of a small sphere of mass m suspended by a thread of length l. The sphere carries a positive charge q. The pendulum is placed in a uniform electric field of strength E directed Vertically upwards. If the electrostatic force acting on the sphere is less than gravitational force the period of pendulum is (A) 1 2 qE m T = 2π g – l         (B) 1 2 T = 2π l g       (C) 1 2 T = 2π qE m l g +         (D) 1 2 ml T = 2 qE        32. Consider a system of three charges q/3, q/3 and –2a/3 placed at points A, B and C respectively as shown in the figure. It the radius of the circle is R and 0 CAB = 60  then the electric field at centre 0 is ........ (A) 2 0 q 8π R  (B) 2 2 0 q 54π R  (C) 2 0 q 6π R  (D) 0 O B C A 60 0 X Y q1 b a q2 P(a, b) Downloaded from www.studiestoday.com Downloaded from www.studiestoday.com