Sharjah Indian School, Sharjah Boys Wing

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1 Notes on ELECTROSTATICS Sharjah Indian School, Sharjah Boys Wing Electrostatics deals with the study of forces, fields and potentials arising from static charges. Static Electricity When an object is rubbed with another suitable object, electrons are transferred from one to the other. The body which gains electrons becomes negative whereas, the body which loses electrons becomes positively charged. {e.g. ebonite (-) and fur (+), glass rod (+) and silk (- ) etc.}. The exchange of electrons takes place by the expense of the energy supplied to the electrons in doing work against friction. Hence such electricity is called frictional electricity. As the charges remain stationary, it is also called static electricity. CONDUCTORS & INSULATORS (Read page number 5) PROPERTIES OF ELECTRIC CHARGES a) Attractive property:- Like charges repel and unlike charges attract. b) Additivity:- The total charge of a system is the algebraic sum of the individual charges. i.e., Q = Q 1 + Q 2 + Q c) Conservation of charge:-the total charge of the isolated system is always conserved.it shows the creation of negative charges on a body is always accompanied by that of equal amount of positive charges on another surface. d) Quantization:- The charge possessed by an object is always an integral multiple of the fundamental charge (i.e., charge of an electron e = 1.6 x C). It follows that Q = ±ne. At the macroscopic level, one deals with charges that are enormous compared to the magnitude of charge e. Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored. e) Electrostatic induction:- It is the phenomenon of redistribution of charges in a neutral body, whenever there is charged body near it. This phenomenon is responsible for the attraction of neutral objects towards charged ones. (Such as, attracting small bits of paper towards a charged ebonite rod). It further indicates that attraction is not the sure test of charges. dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school sc Page 1

2 GOLD LEAF ELECTROSCOPE It is a device used to test the presence of charges on a body (Shown beside). An uncharged electroscope is used to detect if a body is charged or not. Further, a chargedelectroscope is useful to detect the nature of charges on the body. The degree of divergence of the gold leaves is directly proportional to the magnitude of charges contained in the given body. (For further reading refer page number 4) CHARGING OF A BODY A body can be charged by the process of (i) conduction or (ii) induction. Further, two objects can also be charged simultaneously by induction. (Refer page 6 for method of charging ) COULOMB S LAW or INVERSE SQUARE LAW It statesthat the force of attraction or repulsion between two point charges is directly proportional to the product of the magnitude of the two charges and inversely proportional to the square of distance between them. If two point charges q 1 and q 2 are separated by a distance r then the magnitude of the force (F) is given by In free space or air, the value of k in S.I units is found to be 9 x N m 2 C -2. The above Law can be used to define the S.I unit of charge as follows: 1 Coulomb (C) is the charge that when placed at a distance of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude N on each other. The constant k = 1/4, where represents the physical quantity called the permittivity of the given medium. In vacuum, it is given as 0 = C 2 N -1 m -2.. Thus = 9 x N m 2 C -2. Relative permittivity ( r ) :- It is the ratio of the permittivity of a medium to that of free space. r = / 0 It is also called dielectric constant (K) It has no units. Hence the force between two charges in a medium can be written as, F (1) and that in vacuum is given by (2) Page 2 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept.

3 Divide (2) by (1). Thus the relative permittivity of a medium can also be defined as the ratio of the force between two point charges in free space to that between the same two charges separated by the same distance in the given medium. Also, F =. It shows that the presence of a medium reduces the force between two charges. COULOMB S LAW IN VECTOR FORM Let the position vectors of charges q 1 and q 2 ber 1 and r 2 respectively. Then the force can be represented as, Where is the unit vector acting along the direction of the force. PRINCIPLE OF SUPERPOSITION OF CHARGES It states that the resultant force acting on a point charge due to different charges in its vicinity is the vector sum of the forces exerted by each charge on it, each force is being calculated using Coulomb s Law. F = F 1 + F 2 + F It signifies that the force between two charges is independent of the presence of other charges. CONTINUOUS DISTRIBUTION OF CHARGES Linear Density ( ) :- It is the charge per unit length of a charged conductor. i.e., = q/l Unit :- C m -1 Surface charge density ( ):- It is the charge per unit area of a surface. ie., = q/a Unit :- C m -2. Volume charge density ( ):- It is the charge per unit volume of a body. ie., = q/v Unit :- C m -3. dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school Page 3

4 PRINCIPLE OF SUPERPOSITION OF CHARGES It states that the resultant force acting on a charge due to different charges is the vector sum of the individual forces acting on it due to the different charges, each force is being calculated using Coulomb s Law. i.e, F = F 1 + F 2 + F 3 +. It signifies that the force between two charges is independent of the presence of the other charges. ELECTRIC FIELD Electric field intensity(e) at a point due to a charge is the force per unit charge experienced by a test charge (q) at the given point. It can thus be expressed as. This is true for negligibly small value of q. However, the electric field at a point doesnot depend on the test charge. The force between a source charge Q and test charge q is given by, F = Therefore, the electric field at the point is E = Unit:- N/C It is a vector quantity. Significance:- If the value electric field at a point is known, the value of force acting on any given charge placed at that point can be calculated. LINES OF FORCE (Electric field lines) An elctric field line is the direction in which force is experienced by a positive charge in an electric field. It is the curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point. Figure shows the field lines around some simple charge configurations. The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward. General properties of electric field lines: (i)field lines start from positive charges and end at negative charges. If there is a single charge, they may start or end at infinity. Page 4 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school

5 (ii)in a charge-free region, electric field lines can be taken to be continuous curves without any breaks. (iii) Two field lines can never cross each other. (If they didthe field at the point of intersection will not haveunique direction, which is absurd, as electric field is a vector quantity) (iv) Electrostatic field lines do not form any closed loops. This follows from the conservative nature of electric field. Distinguish between charge and mass Mass 1. It is not quantised. 2. The force between them is always attractive. 3. It may increase with speed. 4. It is always positive Charge 1. It is quantised. 2. The force between them can be attractive or repulsive. 3. It is independent of speed. 4. It can be positive or negative. Mention the similarities and dissimilarities between electrostatic and gravitational force. Similarities Dissimilarities Both obey inverse square law. Both are coservative forces. Both are central forces. Grav. Force does not depend on the medium whereas electrostatic force depends on the medium in which the charges are kept. Gr. Force always attractive, but ele. force can be attractive or repulsive. ELECTRIC DIPOLE An electric dipole is a pair of equal and opposite point charges q and q,separated by a distance 2a. The product of one of the charges and the distance between them is called the diople moment. i.e.,dipole moment p = q x 2a. It is a vector quantity directed from q to +q. The total charge of the electric dipole is obviously zero. ` Page 5 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school

6 Case(i) When the point is on the dipole axis The Electric field of an electric dipole Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in figure. Then and Where p is the unit vector acting in the direction of the dipole moment. Thus the total field can be found using the Principle of superposition as, i.e., E = where p = q x 2a Case(ii) When the point is on the equatorial plane (Perpendicular bisector) So, for a short dipole, we have, Let P be the point at a distance r on the perpendicular bisector of an electric dipole of dipole moment p as shown below: The magnitudes of the electric fields due to the twocharges +q and q are given by E = They are found to be equal and are in the directions shown. The components, normal to the dipoleaxis cancel away. components along the dipole axisadd up. Hence the total electric field is given by The It can be further written as E = where p = q x 2a i.e., E = The the direction of electric field is antiparallel to the dipole moment. Also for a short dipole, the field at a point on its axial line is twice that at a point at the same distance on its equatorial line. Page 6 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school

7 DIPOLE IN A UNIFORM EXTERNAL FIELD Consider a permanent dipole (i.e., the dipole that exists irrespective of the field E, not that is induced by E.)of moment p in a uniformexternal field E, as shown in the figure.the two forces are equal and opposite. Thus net force F = qe qe = 0 So, the net force onthe dipole is zero, since E is uniform. Therefore, it does not experience any translatory motion. However, the forces are equal and opposite, at different points, it results in a torqueon the dipole. Magnitude of torque = force x perpendicular distance = q E 2 a sin i.e, = pe sin, where p = q x 2a The direction of torque is normal to the plane of the paper, coming out of it. In vector form, = p X E [ Note:- If the field is non-uniform, there will be a net force inaddition to the torque. The direction of net force depends on the orientation of p with respect to E.] ELECTRIC FLUX ( ) It is the number of electric field lines passing through a surface normal to it. Further, electric field strength can be defined by the number of field lines passing per unit area of the surface normal to it. i.e., E = / S. or Electric flux ( ) = E S So, the unit is N C -1 m 2. If the surface is inclined, the flux can be found by defining area vector ( S), the vector whose magnitude is the same as the area of the surface, whereas, the direction, outward normal to the surface as shown. Thus electric flux can be expressed as Electric flux ( ) = E S cos In vector form, = E SHence electric flux is a scalar quantity. GAUSS S LAW It states that the surface integral of electric field around a charge enclosed by a surface (i.e., the total electric flux through the surface) is 1/ 0 times the net charge enclosed by the surface. i.e., Thus, whenever you find that the net electricflux through a closed surface is zero, we conclude that the total chargecontained in the closed surface is zero. dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school Page 7

8 Significance of the Gauss s Law: (i) Gauss s law is true for any closed surface, no matter what its shape or size. (ii) The term q net on the right side of Gauss s lawincludes thesum of all charges enclosed by the surface. The charges may be locatedanywhere inside the surface. (iii) The surface that we choose for the application of Gauss s law is calledthe Gaussian surface. (iv) Gauss s law is based on the inverse square dependence ondistance contained in the Coulomb s law. Any violation of Gauss slaw will indicate departure from the inverse square law. APPLICATIONS OF GAUSS S LAW (i) Electric field due to an infinitely long charged wire. Consider an infinitely long thin straight wirewith uniform linear charge density. Let Ebe the intensity of electric field at a point distance r from the wire. Imagine a Gaussian surface around a given length l of the wire. Since the field is everywhere radial, fluxthrough the two ends of the cylindricalgaussian surface is zero. Flux through the Gaussian surface = flux through the curved cylindrical part of the surface = E 2 rl According to Gauss s Law, Hence E 2 rl = Thus E = But = Thus the graph between E and r can be shown as: Where is the unit vector normal to the wire. (ii) Electric field due to an infinitely charged sheet. r Let be the uniform surface charge density of an infinite plane sheet. By symmetry, the electric field will not depend on y and z coordinates and its direction at every point must be parallel to the x-direction. dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school Page 8

9 Assume the Gaussian surface to be a rectangular parallelepiped of cross sectional area A, as shown. Thus, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux. So, By Gauss s law, i.e., 2EA = So, E = or Electric field E = ( = q/a) In vector form, E =, where is the unit vector normal to the plane and going away from it. [Special cases (Refer note book)] (iii) Electric field due to a charged spherical shell. Let be the surface charge density of a charged spherical shell of radius R. Consider a point P outside theshell with radius vector r. To calculate E at P, we take thegaussian surface to be a sphere of radius r and with centreo, passing through P. Case (a):- Inside (r R) By Gauss s law, But the net charge enclosed by the surface is zero. i.e., E.4 r 2 = 0 or E = 0 It follows that no electric field exists inside a thin charged shell. This phenomenon is called electrostatic shielding. Case (b):- Outside the shell (r R) Here the Gussian surface encloses the total Charge on the shell. By Gauss s law, i.e., E. 4 r 2 = Thus E = It shows that, for a point outside the shell, it acts as a point charge kept at its centre. Further, = So, E = E = Page 9 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school

10 Sharjah Indian School, Sharjah Boys Wing ELECTROSTATIC POTENTIAL Electrostatic potential at a point is defined as the work done per unit positive charge on a test charge to bring it from infinity to the given point. Let W be the work done to bring a test charge q from infinity to a point in an electric field. Then, by definition, Electric potential, V = Hence the unit of electric potential is J/C or Volt It is a scalar quantity. Expression for potential at a point due to a point charge. Let r be the distance to a point from a source charge Q. If a test charge q is kept at a distance x from Q, the force acting on it is, r F = +Q Thus the work done to displace it through dx is given by W = - The negative sign indicates the work is done against the force of repulsion. i.e., W = - So, the work done to bring it from infinity to the given point is W = - i.e., W = -kqq[ ] this gives W = Hence by definition, electric potential, V = Expression for potential at a point due to a system of charges. The electrostatic potential at a point due to a system of charges is the algebraic sum of the potentials due to individual charges at the given point. i.e., V = V 1 + V 2 + V V n So, V = - kqq Expression for potential at a point due to a short electric dipole. Consider a short electric dipole of dipole moment p, making an angle with the line joining the center to a point P as shown in the figure. The potential at the point P is the algebraic sum of potentials due to q and -q. i.e., V = V 1 + V 2 = V = kq ( kq [ (1) (r 2 -r 1 ) Since the dipole is short, r1 r 2 = r. Thus Further, cos = (from figure) Thus equation (1) becomes, V = But p = q. 2a. Thus we get, V = Page 10 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept.

11 (Note: (i) the above equation follows that at the perpendicular bisector of the dipole, i.e. when = 90 o, V= 0. However electric field is not zero (ii) Electric potential at a point on the axial line is V = as = 0) Potential difference (p.d) Potential difference between two points is defined as the work done per unit positive charge to move a test charge from one point to another. i.e., Potential difference (V) = Thus, the work required to move a charge q between two points possessing a p.d V is given by, W = qv This indicates that when a charge q is released from rest it gains kinetic energy given by, ½ mv 2 = qv In the case of electron, ½ mv 2 = ev This leads to the definition of a practical unit of energy electron Volt (ev) 1 ev is defined as the energy gained by an electron under unit potential difference. Thus 1 ev = 1.6 x J. Potential energy of a system of charges The potential at a distance r 12 is given by V = Hence the energy on q 2 is U = (using W = qv). This gives the potential energy of system of two charges as, U = Further, the potential energy of the system containing many charges (as shown beside) can be found by Note:- It shows that the potential energy of a systemis the characteristic of the present state of configuration, and not the way the state is achieved. EQUIPOTENTIAL SURFACES The surfaces throughout which potential remains the same are called equipotential surfaces. Thus the potential difference between any two points on an equipotential surface is zero (V= 0). So the work done to move any charge from one point to another on an equipotential surface is also zero. (W=qV). Also, the electric field lines are always perpendicular to the corresponding equipotential surfaces (proof given later). e.g. the equipotential surface corresponding to an electric field along x axis is the yz plane. Some examples of equipotential surfaces are shown in the next page. Page 11 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept.

12 (i) due to a point charge (ii) due to a uniform electric field (iii) due to a dipole (iv) due to two identical positive charges (v) due to an electric field with increasing magnitude. Show that electrostatic field is conservative. Consider two points A and B having position vectors r1 and r2 as shown. Assume that the charge Q is kept at the origin. The force on a unit positive charge at a distance r is given by, E = So the work done to move it from A to B is A B W AB =. By definition, this gives the potential difference. i.e. = (1) Similarly, the work done to move it from B to A is W BA = i.e. = (2) Thus the total work done to move it in the closed loop is given by adding equation (1) and (2). It gives The line integral of electric field over a closed loop is zero. i.e. nonet workis done to move the charge in a closed loop in an electric field. It shows that the electric field is conservative. Relation between electric field and potential difference Consider a uniform electric field E in which two point A and B are separated by a distance dx. The force acting on a charge q in the field is F = qe The work done to move the charge from B to A against the field dw = - Fdx = - qe dx (1).A dx. B Let dv be the potential difference between A and B. Then dw = q dv (2) Equating (1) and (2) q dv = - qe dx E =. i.e., electric field is the negative gradient of potential with distance. Page 12 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept. Q

13 Conclusions:- (i) for a uniform electric field between two charged plates separated by a distance d, it can be written as, (Thus the unit of electric field can be V/m) (ii) If the electric field at a point is zero, = 0. Then the potential difference V = a constant, not necessarily to be zero. e.g. inside a charged shell. (iii) Electric field is in the direction in which the potential decreases steepest. POTENTIAL ENERGY OF A SYSTEM OF TWO CHARGES IN THE PRESENCE OF AN EXTERNAL ELECTRIC FIELD Consider a uniform electric field in which two charges q 1 and q 2 kept at the points with position vectors r 1 and r 2 are kept. The work required to bring the charges to the respective positions are q 1 V(r 1 ) and q 2 V(r 2 ) respectively. The potential energy between the two charges is given by U = Hence the total potential energy the system is given by) U = q 1 V(r 1 ) + q 2 V(r 2 ) POTENTIAL ENERGY OF AN ELECTRIC DIPOLE IN A UNIFORM ELECTRIC FIELD When an electric dipole is placed in a uniform electric field at an angle, with the field, it experiences a torque given by = pe sin The work required to rotate the dipole from 1 to 2 is, W = = W = - pe [cos Thus work done is given by W = pe (cos 1 cos 2 ) (1) However, the potential energy of the dipole in the field at a given orientation is the amount of work done to turn it from the position of maximum torque to the given position. Hence by substituting the value of 1 = 90 o and 2 = in the above equation we get, Potential energy, U = - pecos In vector form, U = p. E Special Note:- At equilibrium, = 0. This happens when = 0 or = 180. At = 0, potential energy U = -pe. Now the equilibrium is said to be STABLE. At = 180, potential energy U = +pe. Now the equilibrium is said to be UNSTABLE. ELECTROSTATICS OF CONDUCTORS (Read pages 68 & 69) DIELECTRICS Dielectrics are non-conducting substances. In contrast to conductors they have no (or negligible number of) charge carriers. In a dielectric, the free movement of charges is not possible. Page 13 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept.

14 Polar and Non-Polar Dielectrics If the centres of the negative and positive charges of molecules of a substance do not coincide, such substances possess a net dipole moment in the normal state. Those substances are called polar substances. E.g. HCl, H 2 O If the centres of the negative and positive charges of molecules of a substance coincide, such substances possess no net dipole moment in the normal state. Those are called non-polar substances. E.g. CO 2, H 2. DIELECTRIC POLARISATION When a non-polar dielectric is kept in a uniform electric field, the field will induce dipoles in the molecules, separating the centres of negative and positive charges.the induced dipole moment is in the direction of the field and is proportional to the field strength. Such substances are called linear isotropic dielectrics. When a polar dielectric is kept in the external field, all the dipoles are made to align in the direction of the field. In the absence of the field, due to thermal agitations, these dipoles are aligned randomly. The behavior of polar and non-polar substances in the presence and absence of an external electric field is depicted in the figure given beside. This phenomenon of making the dipoles aligned in the direction of the external electric field is called dielectric polarization. The dipole moment per unit volume of a dielectric Is called polarization (P). Thus for a linear isotropic dielectric, Polarization (P) E. or P = e E. Where e is a constant of the medium called electric susceptibility( e ). Hence it is defined as the Polarization of the dielectric under unit electric field. dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept. Page 14

15 Sharjah Indian School, Sharjah Boys Wing CAPACITORS AND CAPACITANCE Capacitors are one of the main building blocks of electronic circuits. They are used for the temporary storage of electric charges and thereby electrostatic energy. A capacitor or a condenser is an arrangement of two conductors separated by an insulating medium. Capacitors are widely used in oscillatory circuits, radio tuner, filter circuits etc. CLASSIFICATION:- According to the shape the conductors used in capacitors, they are classified into parallel plate capacitors, spherical capacitors, cylindrical capacitors etc. According to the nature of the dielectric medium used, they are classified as paper condenser, air capacitor, electrolytic capacitor etc. PRINCIPLE OF CAPACITORS A B Consider a metallic plate A connected to a positive potential V. + - A second uncharged plate is held near it so that negative charges are + - induced on B at the near end of A. Thus the farther end becomes positive. + - On earthing the plate B, the potential difference between the plates is increased considerably. This strong electric field between the plates helps in the storage of electrostatic energy temporarily within the electric field. CAPACITANCE The charge stored in a capacitor is directly proportional to the potential difference between the plates. i.e. Q V or Q = CV Where C is constant of the given capacitor, called its capacitance. Symbol of a capacitor (i) Fixed capacitor (ii) Variable capacitor PARALLEL PLATE CAPACITOR A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance (See the figure). Let the medium between the plates to be vacuum. Let A be the area of each plate and d the separation between them. The two plates have charges Q and Q. Since the area is quite large, the result of infinitely Charged sheet can be used. Page 15 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept.

16 Thus Electric field outside the plates = 0 Electric field between the plates = / 0. But = Q / A. Thus Electric field, E = (1) Also, we have E = where V is the potential difference between the plates. So eqn.(1) becomes, = i.e., Capacitance, C = The above equation shows that the capacitance of the capacitor depends only on its geometrical factors. ENERGY STORED IN A CAPACITOR Consider a capacitor of capacitance C connected to a battery, providing a potential difference V. The work required to store a charge dq is given by dw = V.dQ (as W= qv) Hence the total work needed to charge the capacitor fully to Q is given by, W = = (V= Q/C) i.e. W = This further gives the total energy stored in the capacitor as U = Substituting Q = CV, we get, U = ½ CV 2 Graphically, The graph between the charge stored in a capacitor and the p.d applied can be plotted as shown beside. Q It further shows that the slope of the graph gives Capacitance (C) A and the area enclosed by the graph gives the energy stored in the capacitor. Thus energy, U = area of the triangle O B V U = ½.V.Q i.e. U = ½ CV 2 where Q = CV Energy density (u):- It is the energy possessed per unit volume of a charged capacitor. Let U be the energy stored in a capacitor of plates separation d and area A. Then, the energy density, u = = But C = Hence u = = Thus, u = 2 / 0 ( = Q/A) Or u = ½ 0 E 2. (E = / 0 ) Page 16 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept.

17 COMBINATION OF CAPACITORS 1. SERIES. Let three different capacitors of capacitances C 1, C 2 and C 3 be connected in series as shown in the figure. When connected to a battery of voltage V, + C C C 3 - the two plates at the ends acquire charges from the respective terminals of the battery. The inner plates are induced with opposite charges as shown. V 1 V 2 V 3 Thus the series combination can be regarded as a + - single capacitor with larger separation of plates. V So the total capacitance of the combination becomes less. (C ) Hence, (i) The charge (Q) remains the same throughout. (ii) The total potential difference supplied by the battery is distributed among the three capacitors. So V = V 1 + V 2 +V 3 (iii) If C s is the effective capacitance of the combination, the above equation can be written as, This gives The reciprocal of the effective capacitance of series combination is the sum of the reciprocals of the individual capacitances. Note:- (i) For two capacitors it can be written as Cs = (ii) The effective capacitance is always less than the least of the individual values. 2. PARALLEL. Let three different capacitors of capacitances be connected in parallel as shown in the figure. When connected to a battery of voltage V, each capacitor ` acquires charges directly from the battery. Thus the combination can be + C 1 - regarded as a single capacitor with larger area of plates.+ Q 1 - plates are induced with opposite charges as shown. + C 2 - So, the total capacitance of the combination becomes more. (C ) + Q 2 - Hence, (i) The p.d remains the same for all the capacitors. + C 3 - (ii) The total charge absorbed from the battery is distributed+ Q 3 - among the capacitors. So Q = Q 1 + Q 2 +Q 3 (iii) If C p is the effective capacitance of the combination, the above equation can be written as, C p V= C 1 V + C 2 V + C 3 V This gives C p = C 1 + C 2 + C 3. i.e. the effective capacitance in parallel combination is the sum of the individual capacitances. It also suggests that the effective capacitance in parallel combination is always greater than the greatest of the individual values of capacitances. (For notes on effect on capacitance by the insertion of dielectric slab and conducting slab, refer note book) dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept. Page 17

18 Van de Graaff Generator It is a device used to accelerate charged particles to a high velocity, by creating high voltages of the order of a few million volts. Principle: Consider a large spherical conducting shell of radius R, having charge Q inside which a small sphere of radius r and charge q is kept at the centre as shown. The potential at A is given by V A = [kq/r + kq/r] A The potential at B is given by, B V B = [kq/r + kq/r] It shows that the smaller sphere is always at a higher potentialthanthe larger one, irrespective of the charges given to them. This potential difference V B V A, so produced can be used to accelerate charged particles, as ½ mv 2 = qv Construction and Working: It consists of a large spherical shell Supported on an insulating column. A long narrow endless insulating belt, capable of moving with the help of a motor, is wound around two pulleys P 1 and P 2 as shown. The charges sprayed on to the belt from the source from the metal brush, are transferred to another conducting brush connected to the large shell These charges further spread out uniformly on the outer surface. This builds up a very high voltage inside the shell and can used to accelerate charged particles to a high speed. Limitations: (i) Not suitable to accelerated neutral particles. (ii) The dielectric strength of the medium limits the amount of K.E that can be gained by particle. [Notes: (i) Corona discharge:- the phenomenon of discharge of electricity from sharp, pointed ends due high surface charge density ( = q/a] (ii) Dielectric strength:- It is the maximum electric field amaterial can tolerate without breakdown, i.e., without starting toconduct electricity through partial ionization. Page 18 dept. of physics, shrajahindian school, Sharjah boys wing dept.of physics. sharjahindian school dept. of physics, shrajahindian school, Sharjah boys wing Prepared by M.Rajeev Dept. of Physics Sharjah Indian School Boys Wing