Page 1 Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 0 1. INTRODUCTION A capacitor can store energy in the form of potential energy in an electric field. In this chapter we'll discuss the capacity of conductors to hold charge and energy. 1.1 Capacitance of an isolated conductor When a conductor is charged its potential increases. It is found that for an isolated conductor (conductor should be of finite dimension, so that potential of infinity can be assumed to be zero) potential of the conductor is proportional to charge given to it. q = charge on conductor V = potential of conductor q Isolated conductor q ? V ? q = CV Where C is proportionally constant called capacitance of the conductor. 1.2 Definition of capacitance : Capacitance of conductor is defined as charge required to increase the potential of conductor by one unit. 1.3 Important point about the capacitance of an isolated conductor : • It is a scalar quantity. • Unit of capacitance is farad in SI unis and its dimensional formula is M –1 L –2 I 2 T 4 • 1 Farad : 1 Farad is the capacitance of a conductor for which 1 coulomb charge increases potential by 1 volt. 1 Farad = Volt 1 Coulomb 1 1 ?F = 10 –6 F, 1nF = 10 –9 F or 1 pF = 10 –12 F • Capacitance of an isolated conductor depends on following factors : (a) Shape and size of the conductor : On increasing the size, capacitance increase. (b) On surrounding medium : With increase in dielectric constant K, capacitance increases. (c) Presence of other conductors: When a neutral conductor is placed near a charged conductor capacitance of conductors increases. • Capacitance of a conductor does not depend on (a) Charge on the conductor (b) Potential of the conductor (c) Potential energy of the conductor. 1.4 Capacitance of an isolated Spheical Conductor. Ex.1 Find out the capacitance of an isolated spherical conductor of radius R. Sol. Let there is charge Q on sphere. ? Potential R KQ V ? Hence by formula : Q = CV R CKQ Q ? C A P A C I T A N C E C A P A C I T A N C E Page 2 Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 0 1. INTRODUCTION A capacitor can store energy in the form of potential energy in an electric field. In this chapter we'll discuss the capacity of conductors to hold charge and energy. 1.1 Capacitance of an isolated conductor When a conductor is charged its potential increases. It is found that for an isolated conductor (conductor should be of finite dimension, so that potential of infinity can be assumed to be zero) potential of the conductor is proportional to charge given to it. q = charge on conductor V = potential of conductor q Isolated conductor q ? V ? q = CV Where C is proportionally constant called capacitance of the conductor. 1.2 Definition of capacitance : Capacitance of conductor is defined as charge required to increase the potential of conductor by one unit. 1.3 Important point about the capacitance of an isolated conductor : • It is a scalar quantity. • Unit of capacitance is farad in SI unis and its dimensional formula is M –1 L –2 I 2 T 4 • 1 Farad : 1 Farad is the capacitance of a conductor for which 1 coulomb charge increases potential by 1 volt. 1 Farad = Volt 1 Coulomb 1 1 ?F = 10 –6 F, 1nF = 10 –9 F or 1 pF = 10 –12 F • Capacitance of an isolated conductor depends on following factors : (a) Shape and size of the conductor : On increasing the size, capacitance increase. (b) On surrounding medium : With increase in dielectric constant K, capacitance increases. (c) Presence of other conductors: When a neutral conductor is placed near a charged conductor capacitance of conductors increases. • Capacitance of a conductor does not depend on (a) Charge on the conductor (b) Potential of the conductor (c) Potential energy of the conductor. 1.4 Capacitance of an isolated Spheical Conductor. Ex.1 Find out the capacitance of an isolated spherical conductor of radius R. Sol. Let there is charge Q on sphere. ? Potential R KQ V ? Hence by formula : Q = CV R CKQ Q ? C A P A C I T A N C E C A P A C I T A N C E Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 2 C = 4 ? ? 0 R (i) If the medium around the conductor is vacuum or air.: C vacuum = 4 ? ? 0 R R = Radius of spherical conductor. (may be solid or hollow) (ii) If the medium around the conductor is a dielectric of constant K from surface of sphere to infinity then C medium = 4 ? ? 0 KR (iii) vaccum / air medium C C = K = dielectric constant. 2. CAPACITOR : A capacitor or condenser consists of two coductors separated by an insulator or dielectric. (i) When uncharged conductor is brought near to a charged conductor, the charge on conductors remains same but its potential dcreases resulting in the increase of capacitance. (ii) In capacitor two conductors have equal but opposite charges. (iii) The conductors are called the plates of the capacitor. The name of the capacitor depends on the shape of the capacitor. (iv) Formulae related with capacitors: (a) Q = CV ? A B B B A A V – V Q V – V Q V Q C ? ? ? Q = Charge of positive plate of capacitor. V = Potential difference between positive and negative plates of capacitor C = Capacitance of capacitor. (v) The capacitor is represented as following : , (vi) Based on shape and arrangement of capacitor plates there are various types of capacitors: (a) Parallel plate capacitor (b) Spherical capacitor. (c) Cylindrical capacitor (v) Capacitance of a capacitor depends on (a) Area of plates. (b) Distance between the plates. (c) Dielectric medium between the plates. 2.1 Parallel Plate Capacitor Two metallic parallel plates of any shape but of same size and separated by small distance constitute parallel plate capacitor. Suppose the area of each plate is A and the separation between the two plates is d. Also assume that the space between the plates contains vacuum. We put a charge q on one plate and a charge –q on the other. This can be done either by connecting one plate with the positive terminal and the other with negative plate of a battery (as shown in figure a ) or by connecting one plate to the earth and by giving a charge +q to the other plate only. This charge will induce a charge – q on the earthed plate. The charges will appear on the facing surfaces. The charges density on each of these surfaces has a magnitude ? = q/A. Page 3 Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 0 1. INTRODUCTION A capacitor can store energy in the form of potential energy in an electric field. In this chapter we'll discuss the capacity of conductors to hold charge and energy. 1.1 Capacitance of an isolated conductor When a conductor is charged its potential increases. It is found that for an isolated conductor (conductor should be of finite dimension, so that potential of infinity can be assumed to be zero) potential of the conductor is proportional to charge given to it. q = charge on conductor V = potential of conductor q Isolated conductor q ? V ? q = CV Where C is proportionally constant called capacitance of the conductor. 1.2 Definition of capacitance : Capacitance of conductor is defined as charge required to increase the potential of conductor by one unit. 1.3 Important point about the capacitance of an isolated conductor : • It is a scalar quantity. • Unit of capacitance is farad in SI unis and its dimensional formula is M –1 L –2 I 2 T 4 • 1 Farad : 1 Farad is the capacitance of a conductor for which 1 coulomb charge increases potential by 1 volt. 1 Farad = Volt 1 Coulomb 1 1 ?F = 10 –6 F, 1nF = 10 –9 F or 1 pF = 10 –12 F • Capacitance of an isolated conductor depends on following factors : (a) Shape and size of the conductor : On increasing the size, capacitance increase. (b) On surrounding medium : With increase in dielectric constant K, capacitance increases. (c) Presence of other conductors: When a neutral conductor is placed near a charged conductor capacitance of conductors increases. • Capacitance of a conductor does not depend on (a) Charge on the conductor (b) Potential of the conductor (c) Potential energy of the conductor. 1.4 Capacitance of an isolated Spheical Conductor. Ex.1 Find out the capacitance of an isolated spherical conductor of radius R. Sol. Let there is charge Q on sphere. ? Potential R KQ V ? Hence by formula : Q = CV R CKQ Q ? C A P A C I T A N C E C A P A C I T A N C E Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 2 C = 4 ? ? 0 R (i) If the medium around the conductor is vacuum or air.: C vacuum = 4 ? ? 0 R R = Radius of spherical conductor. (may be solid or hollow) (ii) If the medium around the conductor is a dielectric of constant K from surface of sphere to infinity then C medium = 4 ? ? 0 KR (iii) vaccum / air medium C C = K = dielectric constant. 2. CAPACITOR : A capacitor or condenser consists of two coductors separated by an insulator or dielectric. (i) When uncharged conductor is brought near to a charged conductor, the charge on conductors remains same but its potential dcreases resulting in the increase of capacitance. (ii) In capacitor two conductors have equal but opposite charges. (iii) The conductors are called the plates of the capacitor. The name of the capacitor depends on the shape of the capacitor. (iv) Formulae related with capacitors: (a) Q = CV ? A B B B A A V – V Q V – V Q V Q C ? ? ? Q = Charge of positive plate of capacitor. V = Potential difference between positive and negative plates of capacitor C = Capacitance of capacitor. (v) The capacitor is represented as following : , (vi) Based on shape and arrangement of capacitor plates there are various types of capacitors: (a) Parallel plate capacitor (b) Spherical capacitor. (c) Cylindrical capacitor (v) Capacitance of a capacitor depends on (a) Area of plates. (b) Distance between the plates. (c) Dielectric medium between the plates. 2.1 Parallel Plate Capacitor Two metallic parallel plates of any shape but of same size and separated by small distance constitute parallel plate capacitor. Suppose the area of each plate is A and the separation between the two plates is d. Also assume that the space between the plates contains vacuum. We put a charge q on one plate and a charge –q on the other. This can be done either by connecting one plate with the positive terminal and the other with negative plate of a battery (as shown in figure a ) or by connecting one plate to the earth and by giving a charge +q to the other plate only. This charge will induce a charge – q on the earthed plate. The charges will appear on the facing surfaces. The charges density on each of these surfaces has a magnitude ? = q/A. Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 2 + + + + + + – – – – – – +q –q (a) or + + + + + + – – – – – – +q –q (b) If the plates are large as compard to the separation between them, then the electric field between the plates (at point B) is uniform and perpendicular to the plates except for a small region near the edge. The magnitude of this uniform field E may be calculated by using the fact that both positive and negative plates produce the electric field in the same direction (from positive plate towards negative plate) of magnitude ?/2 ? 0 and therefore, the net electric field between the plates will be, 0 0 0 2 2 E ? ? ? ? ? ? ? ? ? Outside the plates (at point A and C) the field due to positive sheet of charge and negative sheet of charge are in opposite directions. Therefore, net field at these points is zero. The potential difference between the plates is, ? 0 0 A qd d d . E V ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? The capacitance of the parallel plate capacitor is, d A V q C 0 ? ? ? or d A C 0 ? ? 2.2 Cylindrical Capacitor Cylindrical capacitor consists of two co-axial cylinders of radii a and b and length l. If a charge q is given to the inner cylinder, induced change –q will reach the inner surface of the outer cylinder. By symmetry, the electric field in region between the cylinders is radially outwards. By Gauss’s theorem, the electric field at a distance r from the axis of the cylinder is given by r q 2 1 E 0 l ? ? ? The potential difference between the cylinders is given by ? ? ? ? ? ? ? ? ? a b 0 a b r dr q 2 1 dr E V l ? ? ? ? ? ? ? ? ? ? ? b a In 2 q 0 l – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + + a b B l A – or, ? ? ? ? ? ? ? ? ? ? b a In 2 V q C 0 l 2.3 Spherical Capacitor A spherical capacitor consists of two concentric spheres of radii a and b as shown. The inner sphere is positively charged to potential V and outer sphere is at zero potential. The inner surface of the outer sphere has an equal negative charge. The potential difference between the spheres is b 4 Q – a 4 Q V 0 0 ? ? ? ? ? + – – – – – – – – – – + + + + + + + + + a b Hence, capacitance 0 4 ab Q C V (b a) ? ? ? ? Page 4 Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 0 1. INTRODUCTION A capacitor can store energy in the form of potential energy in an electric field. In this chapter we'll discuss the capacity of conductors to hold charge and energy. 1.1 Capacitance of an isolated conductor When a conductor is charged its potential increases. It is found that for an isolated conductor (conductor should be of finite dimension, so that potential of infinity can be assumed to be zero) potential of the conductor is proportional to charge given to it. q = charge on conductor V = potential of conductor q Isolated conductor q ? V ? q = CV Where C is proportionally constant called capacitance of the conductor. 1.2 Definition of capacitance : Capacitance of conductor is defined as charge required to increase the potential of conductor by one unit. 1.3 Important point about the capacitance of an isolated conductor : • It is a scalar quantity. • Unit of capacitance is farad in SI unis and its dimensional formula is M –1 L –2 I 2 T 4 • 1 Farad : 1 Farad is the capacitance of a conductor for which 1 coulomb charge increases potential by 1 volt. 1 Farad = Volt 1 Coulomb 1 1 ?F = 10 –6 F, 1nF = 10 –9 F or 1 pF = 10 –12 F • Capacitance of an isolated conductor depends on following factors : (a) Shape and size of the conductor : On increasing the size, capacitance increase. (b) On surrounding medium : With increase in dielectric constant K, capacitance increases. (c) Presence of other conductors: When a neutral conductor is placed near a charged conductor capacitance of conductors increases. • Capacitance of a conductor does not depend on (a) Charge on the conductor (b) Potential of the conductor (c) Potential energy of the conductor. 1.4 Capacitance of an isolated Spheical Conductor. Ex.1 Find out the capacitance of an isolated spherical conductor of radius R. Sol. Let there is charge Q on sphere. ? Potential R KQ V ? Hence by formula : Q = CV R CKQ Q ? C A P A C I T A N C E C A P A C I T A N C E Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 2 C = 4 ? ? 0 R (i) If the medium around the conductor is vacuum or air.: C vacuum = 4 ? ? 0 R R = Radius of spherical conductor. (may be solid or hollow) (ii) If the medium around the conductor is a dielectric of constant K from surface of sphere to infinity then C medium = 4 ? ? 0 KR (iii) vaccum / air medium C C = K = dielectric constant. 2. CAPACITOR : A capacitor or condenser consists of two coductors separated by an insulator or dielectric. (i) When uncharged conductor is brought near to a charged conductor, the charge on conductors remains same but its potential dcreases resulting in the increase of capacitance. (ii) In capacitor two conductors have equal but opposite charges. (iii) The conductors are called the plates of the capacitor. The name of the capacitor depends on the shape of the capacitor. (iv) Formulae related with capacitors: (a) Q = CV ? A B B B A A V – V Q V – V Q V Q C ? ? ? Q = Charge of positive plate of capacitor. V = Potential difference between positive and negative plates of capacitor C = Capacitance of capacitor. (v) The capacitor is represented as following : , (vi) Based on shape and arrangement of capacitor plates there are various types of capacitors: (a) Parallel plate capacitor (b) Spherical capacitor. (c) Cylindrical capacitor (v) Capacitance of a capacitor depends on (a) Area of plates. (b) Distance between the plates. (c) Dielectric medium between the plates. 2.1 Parallel Plate Capacitor Two metallic parallel plates of any shape but of same size and separated by small distance constitute parallel plate capacitor. Suppose the area of each plate is A and the separation between the two plates is d. Also assume that the space between the plates contains vacuum. We put a charge q on one plate and a charge –q on the other. This can be done either by connecting one plate with the positive terminal and the other with negative plate of a battery (as shown in figure a ) or by connecting one plate to the earth and by giving a charge +q to the other plate only. This charge will induce a charge – q on the earthed plate. The charges will appear on the facing surfaces. The charges density on each of these surfaces has a magnitude ? = q/A. Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 2 + + + + + + – – – – – – +q –q (a) or + + + + + + – – – – – – +q –q (b) If the plates are large as compard to the separation between them, then the electric field between the plates (at point B) is uniform and perpendicular to the plates except for a small region near the edge. The magnitude of this uniform field E may be calculated by using the fact that both positive and negative plates produce the electric field in the same direction (from positive plate towards negative plate) of magnitude ?/2 ? 0 and therefore, the net electric field between the plates will be, 0 0 0 2 2 E ? ? ? ? ? ? ? ? ? Outside the plates (at point A and C) the field due to positive sheet of charge and negative sheet of charge are in opposite directions. Therefore, net field at these points is zero. The potential difference between the plates is, ? 0 0 A qd d d . E V ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? The capacitance of the parallel plate capacitor is, d A V q C 0 ? ? ? or d A C 0 ? ? 2.2 Cylindrical Capacitor Cylindrical capacitor consists of two co-axial cylinders of radii a and b and length l. If a charge q is given to the inner cylinder, induced change –q will reach the inner surface of the outer cylinder. By symmetry, the electric field in region between the cylinders is radially outwards. By Gauss’s theorem, the electric field at a distance r from the axis of the cylinder is given by r q 2 1 E 0 l ? ? ? The potential difference between the cylinders is given by ? ? ? ? ? ? ? ? ? a b 0 a b r dr q 2 1 dr E V l ? ? ? ? ? ? ? ? ? ? ? b a In 2 q 0 l – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + + a b B l A – or, ? ? ? ? ? ? ? ? ? ? b a In 2 V q C 0 l 2.3 Spherical Capacitor A spherical capacitor consists of two concentric spheres of radii a and b as shown. The inner sphere is positively charged to potential V and outer sphere is at zero potential. The inner surface of the outer sphere has an equal negative charge. The potential difference between the spheres is b 4 Q – a 4 Q V 0 0 ? ? ? ? ? + – – – – – – – – – – + + + + + + + + + a b Hence, capacitance 0 4 ab Q C V (b a) ? ? ? ? Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 4 Ex.2 Find capacitance of the given system. 2Q Q d A=Area Sol. Arranging charges 0 0 0 0 A 2 Q 2 2 E ? ? ? ? ? ? ? ? ? ? ? ? Now, V = Ed = 0 A 2 Qd ? 2 Q 3 2 Q 3 2 Q 2 Q ? d A V 2 / Q C 0 ? ? ? ? 3. ENERGY STORED IN A CHARGED CAPACITOR –Q Q initially Finally middle state dq q –q Work has to be done in charging a conductor against the force of repulsion by the already existing charges on it. The work is stored as a potential energy in the electric field of the conductor. Suppose a conductor of capacity C is charged to a potential V 0 and let q 0 be the charge on the conductor at this instant. The potential of the conductor when (during charging) the charge on it was q (< q 0 ) is, C q V ? Now, work done in bringing a small charge dq at this potential is, dq C q Vdq dW ? ? ? ? ? ? ? ? ? total work done in charging it from 0 to q 0 is, C q 2 1 dq C q dW W 2 0 q 0 q 0 0 0 ? ? ? ? ? This work is stored as the potential energy, ? C q 2 1 U 2 0 ? Further by using q 0 = CV 0 we can write this expression also as, 0 0 2 0 V q 2 1 CV 2 1 U ? ? In general if a conductor of capacity C is charged to a potential V by giving it a charge q, then qV 2 1 C q 2 1 CV 2 1 U 2 2 ? ? ? Page 5 Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 0 1. INTRODUCTION A capacitor can store energy in the form of potential energy in an electric field. In this chapter we'll discuss the capacity of conductors to hold charge and energy. 1.1 Capacitance of an isolated conductor When a conductor is charged its potential increases. It is found that for an isolated conductor (conductor should be of finite dimension, so that potential of infinity can be assumed to be zero) potential of the conductor is proportional to charge given to it. q = charge on conductor V = potential of conductor q Isolated conductor q ? V ? q = CV Where C is proportionally constant called capacitance of the conductor. 1.2 Definition of capacitance : Capacitance of conductor is defined as charge required to increase the potential of conductor by one unit. 1.3 Important point about the capacitance of an isolated conductor : • It is a scalar quantity. • Unit of capacitance is farad in SI unis and its dimensional formula is M –1 L –2 I 2 T 4 • 1 Farad : 1 Farad is the capacitance of a conductor for which 1 coulomb charge increases potential by 1 volt. 1 Farad = Volt 1 Coulomb 1 1 ?F = 10 –6 F, 1nF = 10 –9 F or 1 pF = 10 –12 F • Capacitance of an isolated conductor depends on following factors : (a) Shape and size of the conductor : On increasing the size, capacitance increase. (b) On surrounding medium : With increase in dielectric constant K, capacitance increases. (c) Presence of other conductors: When a neutral conductor is placed near a charged conductor capacitance of conductors increases. • Capacitance of a conductor does not depend on (a) Charge on the conductor (b) Potential of the conductor (c) Potential energy of the conductor. 1.4 Capacitance of an isolated Spheical Conductor. Ex.1 Find out the capacitance of an isolated spherical conductor of radius R. Sol. Let there is charge Q on sphere. ? Potential R KQ V ? Hence by formula : Q = CV R CKQ Q ? C A P A C I T A N C E C A P A C I T A N C E Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 2 C = 4 ? ? 0 R (i) If the medium around the conductor is vacuum or air.: C vacuum = 4 ? ? 0 R R = Radius of spherical conductor. (may be solid or hollow) (ii) If the medium around the conductor is a dielectric of constant K from surface of sphere to infinity then C medium = 4 ? ? 0 KR (iii) vaccum / air medium C C = K = dielectric constant. 2. CAPACITOR : A capacitor or condenser consists of two coductors separated by an insulator or dielectric. (i) When uncharged conductor is brought near to a charged conductor, the charge on conductors remains same but its potential dcreases resulting in the increase of capacitance. (ii) In capacitor two conductors have equal but opposite charges. (iii) The conductors are called the plates of the capacitor. The name of the capacitor depends on the shape of the capacitor. (iv) Formulae related with capacitors: (a) Q = CV ? A B B B A A V – V Q V – V Q V Q C ? ? ? Q = Charge of positive plate of capacitor. V = Potential difference between positive and negative plates of capacitor C = Capacitance of capacitor. (v) The capacitor is represented as following : , (vi) Based on shape and arrangement of capacitor plates there are various types of capacitors: (a) Parallel plate capacitor (b) Spherical capacitor. (c) Cylindrical capacitor (v) Capacitance of a capacitor depends on (a) Area of plates. (b) Distance between the plates. (c) Dielectric medium between the plates. 2.1 Parallel Plate Capacitor Two metallic parallel plates of any shape but of same size and separated by small distance constitute parallel plate capacitor. Suppose the area of each plate is A and the separation between the two plates is d. Also assume that the space between the plates contains vacuum. We put a charge q on one plate and a charge –q on the other. This can be done either by connecting one plate with the positive terminal and the other with negative plate of a battery (as shown in figure a ) or by connecting one plate to the earth and by giving a charge +q to the other plate only. This charge will induce a charge – q on the earthed plate. The charges will appear on the facing surfaces. The charges density on each of these surfaces has a magnitude ? = q/A. Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 2 + + + + + + – – – – – – +q –q (a) or + + + + + + – – – – – – +q –q (b) If the plates are large as compard to the separation between them, then the electric field between the plates (at point B) is uniform and perpendicular to the plates except for a small region near the edge. The magnitude of this uniform field E may be calculated by using the fact that both positive and negative plates produce the electric field in the same direction (from positive plate towards negative plate) of magnitude ?/2 ? 0 and therefore, the net electric field between the plates will be, 0 0 0 2 2 E ? ? ? ? ? ? ? ? ? Outside the plates (at point A and C) the field due to positive sheet of charge and negative sheet of charge are in opposite directions. Therefore, net field at these points is zero. The potential difference between the plates is, ? 0 0 A qd d d . E V ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? The capacitance of the parallel plate capacitor is, d A V q C 0 ? ? ? or d A C 0 ? ? 2.2 Cylindrical Capacitor Cylindrical capacitor consists of two co-axial cylinders of radii a and b and length l. If a charge q is given to the inner cylinder, induced change –q will reach the inner surface of the outer cylinder. By symmetry, the electric field in region between the cylinders is radially outwards. By Gauss’s theorem, the electric field at a distance r from the axis of the cylinder is given by r q 2 1 E 0 l ? ? ? The potential difference between the cylinders is given by ? ? ? ? ? ? ? ? ? a b 0 a b r dr q 2 1 dr E V l ? ? ? ? ? ? ? ? ? ? ? b a In 2 q 0 l – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + + a b B l A – or, ? ? ? ? ? ? ? ? ? ? b a In 2 V q C 0 l 2.3 Spherical Capacitor A spherical capacitor consists of two concentric spheres of radii a and b as shown. The inner sphere is positively charged to potential V and outer sphere is at zero potential. The inner surface of the outer sphere has an equal negative charge. The potential difference between the spheres is b 4 Q – a 4 Q V 0 0 ? ? ? ? ? + – – – – – – – – – – + + + + + + + + + a b Hence, capacitance 0 4 ab Q C V (b a) ? ? ? ? Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 4 Ex.2 Find capacitance of the given system. 2Q Q d A=Area Sol. Arranging charges 0 0 0 0 A 2 Q 2 2 E ? ? ? ? ? ? ? ? ? ? ? ? Now, V = Ed = 0 A 2 Qd ? 2 Q 3 2 Q 3 2 Q 2 Q ? d A V 2 / Q C 0 ? ? ? ? 3. ENERGY STORED IN A CHARGED CAPACITOR –Q Q initially Finally middle state dq q –q Work has to be done in charging a conductor against the force of repulsion by the already existing charges on it. The work is stored as a potential energy in the electric field of the conductor. Suppose a conductor of capacity C is charged to a potential V 0 and let q 0 be the charge on the conductor at this instant. The potential of the conductor when (during charging) the charge on it was q (< q 0 ) is, C q V ? Now, work done in bringing a small charge dq at this potential is, dq C q Vdq dW ? ? ? ? ? ? ? ? ? total work done in charging it from 0 to q 0 is, C q 2 1 dq C q dW W 2 0 q 0 q 0 0 0 ? ? ? ? ? This work is stored as the potential energy, ? C q 2 1 U 2 0 ? Further by using q 0 = CV 0 we can write this expression also as, 0 0 2 0 V q 2 1 CV 2 1 U ? ? In general if a conductor of capacity C is charged to a potential V by giving it a charge q, then qV 2 1 C q 2 1 CV 2 1 U 2 2 ? ? ? Capacitance – Nirmaan TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 4 3.1 Energy Density of a Charged Capacitor This energy is localized on the charges or the plates but is distributed in the field. Since in case of a parallel plate capacitor, the electric field is only between the plates, i.e., in a volume (A × d), the energy density Ad V d A 2 1 d A CV 2 1 volume U U 2 0 2 E ? ? ? ? ? ? ? ? ? ? ? or ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? E d v E 2 1 d V 2 1 U 2 0 2 0 E ? 3.2 Calculation of Capacitance The method for the calculation of capacitance involves integration of the electric field between two conductors or the plates which are just equipotential surfaces to obtain the potential difference V ab . Thus, ? ? ? ? a b ab dr . E – V ? ? ? ? ? ? a b ab dr . E – q V q C 3.3 Heat Generated : (1) Work done by battery W = QV Q = charge flow in the battery V = EMF of battery (2) W = +Ve (When Battery discharging) W = –Ve (When Battery charging) (3) ? Q = CV (C = equivalent capacitance) so W = CV × V = CV 2 Now energy on the capacitor 2 CV 2 1 ? ? Energy dissipated in form of heat (due to resistance) H = Work done by battery – {final energy of capacitor - initial energy of capacitor} Ex.3 At any time S 1 switch is opened and S 2 is closed then find out heat generated in circuit. S 1 S 2 2V V Sol. V V –CV CV + – initially 2V –2CV 2CV + – finally Charge flow through battery = Q f – Q i = 2CV – CV = CVRead More

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