Electrostatic Potential and Capacitance iit jee

Hand-Out Chapter - 2Physics: Electrostatic Potential and Capacitance Electrostatic Potential and Capacitance APPROACH TO CHAPTER: ELECTROSTATIC POTENTIAL AND CAPACITANCE The numerical problems based on electric potential are relatively easy. Numerical problems based on capacitors, especially capacitor combinations, need cautious approach. Derivations are simple and learning the same can help you perform better in the examinations. (i) Units and Symbols for numericals Quantity (ii) Symbol SI Unit Electric potential/Electric potential difference V Volt (V) Capacitance C Farad (F) Dielectric constant K No unit Polarisation P Cm –2 Key Formulae Electric Potential & Potential difference (i ) General for mulae: Potential, V  (a) W q Potential difference, V  (b) Based on the idea that work per unit charge gives potential. WB – WA q Helps define the unit – Volt. Helps to think of the variation (ii) Electric potential due to a point charge (in vacuum): V (iii) 1 q 4πε 0 r (i) with distance (ii) if the charge is surrounded Electric potential at a point due to a system of charges by a medium. n V i 1 ( iv ) of electric potential 1 qi 4πε 0 ri Electric potential due to a dipole: (i) along the axial line, V  (ii) equatorial line, V = 0 Physics/Class XII Remember: It’s r2, not r, in 1 p 40 r 2 the denominator. 1 © 2012 Vidyamandir Classes Pvt. Ltd. In parallel combination.. Series combination C C1 C 2 individual capacitance. Electrostatic potential ener gy Potential energy of a system of charges: (a) 1 q1q 2 A two – charge system (in vacuum) U  40 r (b) (ii) Remember: (i) An n–charge system (in vacuum) U  1 4 0 n n  i 1 j 1 j i potential energy relates to the work done on the system that gets stored. © 2012 Vidyamandir Classes Pvt.. . the effective Effective capacitance: capacitance is less than the least (i) 1 1 1    . the effective capacitance is more than the largest (ii) Parallel combination C = C1 + C2 + … Physics/Class XII 2 individual capacitance. q1q j rij Potential energy of a dipole in an electric field  U  –p. Helps to think of how a di- C electric medium affect the 0 A (with vacuum between the plates) d capacitance of a system.  KA C 0 (with a medium of dielectric constant K between the plates) d (c) In series combination.2 Physics: Electrostatic Potential and Capacitance (v) Relation between electric potential and Helps to find out field strength electric field intensity: when E– potential difference between two points and the dV dr distance between them are given. Ltd.E CAPACITANCE Capacitance is the charge required per unit rise in q C V (a) General form (b) Parallel plate capacitor electric potential.Hand-Out Chapter . Hand-Out Chapter ..2 Physics: Electrostatic Potential and Capacitance (d) B.. q  A Apply (iii) Effective capacitance: (a) Parallel Combination C q V Combination Same Different Approach Parallel Voltage Charge Find the total charge Series q = q1 + q2 + q3 + . (i) Helps you to express energy 1 Energy stored by a capacitor: U  CV 2 2 stored in terms of q & V as well as q & C. Charge Voltage Find the total voltage CV = C1V + C2V + C3V + .. . 2 W   pE sin  d  pE  cos  2 – cos 1  Find the work done in twisting the 1 dipole against the torque  W  –pE cos   p. DERIVATIONS Potential energy of a dipole in an electric field dW   d  pE sin  d Start with the torque experienced by a dipole in an electric field.. Ltd.E (ii) Capacitance of a parallel plate capacitor E  0 Get expressions for the charge q and the potential difference (V) between V  Ed the plates. C = C1 + C2 + … (b) Series combination Physics/Class XII 3 © 2012 Vidyamandir Classes Pvt. q q q q = + + +. . Equipotential surfaces are : 1. Physics/Class XII 4 © 2012 Vidyamandir Classes Pvt.. Parallel planes for a uniform electric field Van de Graaff’s generator Use: To generate high electric potential of the order of millions of volts... Principle: Charge given to a hollow conductor gets transferred to the outer surface.. C C1 C 2 (iv) Energy stored by a capacitor: Start with q = CV q = CV Then differentiate dq = C dV Write the work done to move charge dq against dW = V dq a pd of V 1 U  CV 2 2 Substitute for dq Integrate it to get the total work done to build up charge from 0 to q Work done is stored as potential energy C..Hand-Out Chapter . C C1 C 2 C3 1 1 1 = + + .2 Physics: Electrostatic Potential and Capacitance V = V1 + V2 + V3 + . Concentric spheres for an isolated point charge 2. Properties/ Definitions / Descriptions Equipotential surface: Any surface which has the same electrostatic potential at every point is called an equipotential surface. Ltd.. Physics/Class XII 5 © 2012 Vidyamandir Classes Pvt. Ltd. .2 Physics: Electrostatic Potential and Capacitance Diagr am: Write down the use of the device and the principle based on which it functions Parts: Hollow sphere Conveyer belt Two combs Function of each part: Bottom (first) comb: To spray charge on to the conveyer belt. Conveyer belt: Carries charge to the top (second) comb. Top (second) comb: Carries charge to the hollow sphere.Hand-Out Chapter . Hollow sphere gets charged to a high potential. 1.: (i) E = 2. [Ans. [Ans.5 × 1010 3.: 8 ×10–2 J] 3 A 5 Fcapacitor is charged by a 100 V supply. What will be the capacitance if the distance between the plates be reduced by half and the space between them is filled with a substance of dielectric constant k = 6.: 0. (3 Marks) [Ans. when placed with its axis making an angle of 60° with a uniform electric field experiences a torque of 4 3 Nm.: Ui = 4. 1.35 mJ] 2005 A parallel plate capacitor with air between the plates has a capacitance of 8 pF. Aε 0 K1  K 2  ] 2d A 4 Fcapacitor is charged by a 200 V supply. if the dipole has charges of  8 nC. 2.: 5V] (ii) the initial and final energies. ‘V’ across the plates of two capacitors A and B versus increase of charge. Find the net capacitance of t he capacitor. 4. Ltd.55 × 10–9 C] (2 Marks) The given graph shows the variation of charge q versus potential difference V for two capacitors. C1 Corresponds to B] Physics/Class XII 6 © 2012 Vidyamandir Classes Pvt. 5.5 mJ. N .: 4. The supply is then disconnected and the charged capacitor is connected to another uncharged 2 F capacitor.037 J] Deduce an expression for the electric potential due to an electric dipole at any point on its axis.: 3. (ii) –1 J] C [Ans. (2 Marks) [Ans. 3. The supply is then disconnected and the charged capacitor is connected to another uncharged 3 Fcapacitor. [Ans. 2006 Define the term ‘dielectric constant’ of a medium in terms of capacitance of a capacitor. An electric dipole of length 4 cm. How much electrostatic energy of the first capacitor is lost in the process of attaining the steady situation. [Ans.: 0.: 96 pF] Two dielect ric slabs of dielectric const ants K 1 and K 2 are filled in bet ween the two plates. 2004 The graph shows the variation of voltage. How will you account for the difference in energy. The two capacitors C1 and C2 have same plate separation but the plate area of C2 is double than that of C1.Hand-Out Chapter . Uf = 0. Compute the magnitude of this charge.2 Physics: Electrostatic Potential and Capacitance 1. Mention one contrasting feature of electric potential of a dipole at a point as compared to that due to a single charge. Calculate the (i) magnitude of the electric field. (2 Marks) [Ans.5 J] A 10 F capacitor is charged by a 30 V dc supply and then connected across an uncharged 50 Fcapacitor. o f the parallel plate capacit or as shown in t he figure. 2. How much electrostatic energy of the first capacitor is lost in the process of attaining the steady situation. Give reason for your answer. 4. (2 Marks) 2. Find the total energy stored in the capacitors in the given network. Calculate: (3 Marks) (i) the final potential difference across the combination. [Ans. Which of the two capacitors has higher capacitance. ‘Q’ stored on them. [Ans. each of area A. .: C2 Corresponds to A. (1 Mark) The electric field and electric potential at any point due to a point charge kept in air is 20 NC –1 and 10 JC–1 respectively. Which of the lines in the graph correspond to C1 and C2 and why. (ii) potential energy of the dipole. Cx = 5 F Cy = 20 F] (ii) Calculate the potential difference between the plates of X and Y. [Ans. The separation between the pates is now reduced by half and the space between them is filled with a medium of dielectric constant 5. [Ans. 3. (i) Calculate capacitance of each capacitor if equivalent capacitance of the combination is 4 miro Farad.25 . 2008 Derive an expression for the potential energy of an electric dipole of the dipole moment in an electric field. (i) 156. (b) Obtain the equivalent capacitance of the network given below. 2. Show that this energy can be expressed in terms of electric field as 1  0 E 2 Ad where A 2 1. Draw the field lines when the charge density of the sphere. connected by battery 12 volt battery in series. Two capacitors of capacitance 6 F and 12 Fare connected in series with a battery. X and Y. 4] Physics/Class XII 7 © 2012 Vidyamandir Classes Pvt. A dielectric slab of thickness d and dielectric constant K is now placed between the plates. Calculate the (i) charge on the sphere (ii)total electric flux passing through the sphere [Ans. Compute the total battery voltage. The dielectric strength of the gas surrounding the electrode is 5 × 107 Vm–1. 1. Justify your answer in each case.: C = 80 F] 2007 A parallel plate capacitor. (ii)  2 × 1013 (V – m)] (a) Derive an expression for the energy stored in a parallel plate capacitor C. [Ans. Ltd. 12 V] (iii) What is the ratio of electrostatic energy stored in X and Y. Draw a schematic sketch and write briefly its working. (i) positive.2 Physics: Electrostatic Potential and Capacitance 4. if any. is the area of each plate and d is the separation between the plates.: V = 3 Volt] A parallel plate capacitor with air between the plates has a capacitance of 8 pF. Calculate the value of capacitance of the capacitor in the second case. determine the charge and voltage across C4. A Van De Graaff type generator is capable of building up potential difference of 15 × 106 V. 4. 0.5 m in diameter has a surface charge density of 100 C/m2. q = 150 nC. (a) 4. V = 300 V] Explain the principle on which Van De Graaff generator operates. [Ans. How will the energy stored in a fully charged capacitor change when the separation between the plates is doubled and a dielectric medium of dielectric constant 4 is introduced between the plates. For a supply of 300 V. The battery used to charge it is then disconnected. is charged to a potential difference V. derive an expression for the electric field intensity at any point outside a uniformly charged thin spherical shell of radius R and surface charge density C /m. (b) A uniformly charged conducting sphere of 2. X has air between the plates while Y contains a dielectric of medium εr = 4. What change. [Ans. 5. Using Gauss’ law. charged to a potential difference V. 1. . 2. [Ans. Derive the expression for the energy stored in a parallel plate capacitor of capacitance C with air as medium between its plates having charges Q and –Q. will take place in (i) charge on the plates (ii) electric field intensity between the plates (iii) capacitance of the capacitor. (ii) negative. [Ans. each with plate area A and separation d. The voltage across the 6 Fcapacitor is 2 V.3m] (5 Marks) 2009 Two parallel plate capacitor. What is the minimum radius of the spherical shell required. have the same area of plates and same separation between them.Hand-Out Chapter . C = 500 pF. 4. a dipole placed in a uniform electric field is in (i) stable. Figure shows two identical capacitors C1 and C2 each of 1. Or (a) Define electric flux. if battery remains connected and dielectric slab of thickness d and constant K is now placed between the plates what is the change if any will take place in :(i) charge on plates. If the distance between the plates is doubled. After sometime ‘S’ is left open and dielectric slabs of dielectric constant K = 2 are inserted to fill completely the space between the plates of the two capacitors. and sepration ‘d’ is charges to a potential difference V. Is the potential difference VA  VB positive.Hand-Out Chapter . state with reason how the following change: (i) electric field between the plates (ii) capacitance. connected to a battery of 2 V. 5 F] [Ans. A point charge Q is placed at point O as shown in the figure. and (iii) energy stored in the capacitor. [Ans. Is there any restriction on the upper limit of the high voltage set up in this machine? Explain. does it mean that all the free electrons of the metal are moving in the same direction? Explain the principle of a device that can build up high voltages of the order of a few million volts. (a) Depict the equi-potential surfaces for a system of two identical positive point charges placed a distance ‘d’ apart.c.5 F capacitance. The equivalent capacitance of the combination between A and B in the given figure is 4 micro Farad.: 48. (c) How is the field directed if (i) the sheet is positively charged.2 Physics: Electrostatic Potential and Capacitance 2. A parallel-plate capacitor is charged to a potential difference V by a d. [Ans. V ] 5 5 2010 1. 2011 1. (b) Using Gauss’s law. negative or zero. (ii) electric field intensity (iii) capacitance of capacitor Justify your answer in each case. When electrons drift in a metal from lower to higher potential. source.I. (ii) unstable equilibrium. units. 2.5 12 V. Write its S. The capacitor is then disconnected from the source. . if Q is (i) positive (ii) negative. (b) Deduce the expression for the potential energy of a system of two point charges q1 and q2 brought from infinity to    the points r1 and r2 respectively in the presence of external electric field E . 3. Initially switch ‘S’ is closed. In which orientation. prove that the electric field at a point due to a uniformly charged infinite plane sheet is independent of the distance from it. 48 C] (iii) What will be the potential drop across each capacitor. 2. 2012 1. (i) Calculate the capacitance of the capacitor C. (ii) Calculate charge on each capacitor if a 12 volt battery is connected across terminals A and B. Ltd. (ii) negatively charged? Physics/Class XII 8 © 2012 Vidyamandir Classes Pvt. Draw a schematic diagram and explain the working of this device. 2. A parallel plate capacitor each with plate area A. How will the (i) charge and (ii) potential difference between the plates of the capacitors be affected after the slabs are inserted.