Title 2. ELECTROSTATIC POTENTIAL AND CAPACITANCE.pdf ✅

TOPIC-1 Electric Potential Concepts Covered:  Electric potential  Potential difference  Equipotential surfaces  Electrical potential energy of system of two point charges and of electric dipole. Revision Notes Electric potential  Electric potential: Amount of work done by an external force in moving a unit positive charge from infinity to a point in an electrostatic field without producing an acceleration.  It is written as V = W q where, V= Electric potential of that point W = work done in moving charge q through the field, q = charge being moved through the field.  The SI units of electric potential are J C , Volt, Nm C . Potential difference  Electric potential difference: Amount of work done in moving a unit charge from one point to another in an electric field. Electric potential difference D V = Work Charge Charge = = ∆PE W q Work done between two points A and B, WAB = – VAB × q where, VAB = VB – VA is potential difference between A and B.  In an electric field, the work done by electric field to move a test charge q by a distance dl is dW. dW = q E dl → → . ∆V V V V W q q E dl q E dl AB B A AB A B A B = = − = − = − = − → → ∫ → → ∫ . . [Board 2020] Electric potential due to point charge  The electric potential by point charge q, at a distance r from the charge is V q r E = 1 4πε0 · where, e0 is absolute electrical permittivity of free space.  Electric potential is a scalar quantity.  Dimension of Electric potential is [M L2 T–3A–1]. Dipole and system of charges l Electric dipole consists of two equal but opposite electric charges which are separated by a certain distance. l The net potential due to a dipole at any point on its equatorial line is always zero. So, work done in moving a charge on an equatorial line is zero.  Electric potential due to dipole at a point at distance r and making an angle q with the dipole moment p is V p r = 1 4 0 2 πε θ . cos = >> → 1 4 0 2 πε . . ( ) p r r r a   Potential at a point due to system of charges is the sum of potentials due to individual charges. Learning Objectives After going through this chapter, the students will be able to:  Define electric potential and potential difference.  Understand the electrical potential energy of an electric dipole in an electrostatic field.  Calculate the capacitance of a parallel plate capacitor with and without a dielectric medium between the plates.  Differentiate between conductors and insulators based on the behavior of free and bound charges.  Design configurations of capacitors in series and parallel to achieve desired capacitance.  Evaluate the effectiveness of dielectric materials in altering the capacitance of a capacitor. 2 CHAPTER ELECTROSTATIC POTENTIAL AND CAPACITANCE List of Topics Topic-1: Electric potential Topic-2 : Capacitance
CBSE Question Bank Chapterwise & Topicwise, PHYSICS, Class-XII  In a system of charges q1, q2, q3, ...qn having positive vectors r1  , r2  , r3  , ..... rn  relative to point P, the potential at point P due to total charge configuration is algebraic sum of potentials due to individual charges. KEY-TERMS Electric potential: The amount of work needed to move a unit charge from a reference point to a specific point against an electric field. Dipole: A pair of equal and oppositely charged or magnetized poles separated by a certain distance. Equipotential surfaces  Equipotential surface is a surface in space on which all points have same potential. It requires no work to move the charge on such surface.  Electric field is always perpendicular to the equipotential surface.  Spacing among equipotential surfaces allows to locate regions of strong and weak electric field.  Equipotential surfaces never intersect each other. If they intersect then the intersecting point of two equipotential surfaces results in two values of electric potential at that point, which is impossible.  Potential energy of a system of two charges, U q q r = 1 4 0 1 2 πε 12  Potential energy of a system of three charges, U q q r q q r q q r =       + +       1 4 0 1 2 12 1 3 13 2 3 πε 23  Potential energy due to single charge in an external field: Potential energy of a charge q at a distance r in an external field, U q = V r→ ( ) Here, V r( ) → is the external potential at a distance r.  Potential energy due to two charges in an external field, U q V r q V r q q r = + + → → 1 1 2 2 0 1 2 12 1 4 ( ) ( ) . πε [Board 2020]  Potential energy of a dipole in an external field: When a dipole of charge q1 = + q and q2 = – q having separation '2a' is placed in an external field ( E → ). U(q) = –pEcos q [Board 2020] Here, p (Dipole moment) = 2aq and q is the angle between electric field and dipole. Example-1 Two point charges, q1 = 10 × 10–8 C, q2 = –2 × 10–8 C are separated by a distance of 60 cm in air. (i) Find at what distance from the 1st charge, q1 would the electric potential be zero. (ii) Also calculate the electrostatic potential energy of the system. (All India 2008) Solution (i) Given : q1 = 10 × 10–8C, q2 = –2 × 10–8C AB = 60 cm = 0.60 = 0.6m Let AP = x Let AP = x then PB = 0.6 – x q1 q2 A B Potential P due ot charge q1 = Kq AP 1 Potential P due to charge q2 = Kq BP 2 Q Potential at P = 0 ⇒ Kq AP Kq BP 1 2 + = 0 ⇒ q AP 1 = −q PB 2 \ 10 10 8 × − x = − − × − − ( ) . 2 10 0 6 8 x ⇒ 2x + 10x = 6 ⇒ 12x = 6 ⇒ x = 1 2 = 0.5m ∴ Distance from first charge = 0.5 m = 50 cm. (ii) Electrostatic energy of the system is En = Kq q r 1 2
Electrostatic Potential and Capacitance = − × × × × × − − − 9 10 10 2 10 60 10 9 7 8 2 = − × × − − 18 10 60 10 6 2 = −3 10 × 10–4 = – 3 × 10–5 joule \ U or En = – 3 × 10–5 joule MNEMONICS Concept: Characteristics of equipotential surface Mnemonics: Exclusive peace and No war; Noble India is super power. Interpretations: Exclusive peace: Electric field is perpendicular to the surface No war: No Work is done on moving a charge on the surface Noble India: Never Intersects Super Power: Same potential everywhere on the surface KEY FORMULAE Electric Potential, V W q = , measured in volt; 1 volt = 1 Joule / 1 coulomb. Electric potential difference or “voltage” ( ) ∆ ∆ V V V U q W q f i = = − = . Electric potential due to a point charge q at a distance r away: V q r = 1 4πε0 · Finding V from E: Vf – Vi = – E d r i f → → ∫ . Potential energy of two point charges in absence of external electric field: U q q r =       1 4 0 1 2 πε 12 Potential energy of two point charges in presence of external electric field: q V r q r q q r 1 1 2 2 1 2 0 1 4 ( ) + + V( ) πε 2 SUBJECTIVE TYPE QUESTIONS Very Short Answer Type Questions (1 mark each) 1. Which physical quantity has SI unit NmC–1 ? U [Delhi Set-1, 2020 MODIFIED] Ans. Electric Potential. 1 2. A charged particle (+q) moves in a uniform electric field E → in the direction opposite to E → . What will be the effect on its electrostatic potential energy during its motion ? A [Outside Delhi Set-1, 2020] 3. A point charge +Q is placed at point O as shown in the figure. Is the potential difference VA – VB positive, negative or zero? A [Set-1, 2016] Ans. Positive. [Marking Scheme, 2016] 1 4. A charge 'q' is moved from a point A above a dipole of dipole moment 'p' to a point B below the dipole in equatorial plane without acceleration. Find the work done in the process. A [Outside Delhi Set-1, 2016] 5. On moving a charge 20 C by 2 cm, 2 J work is done. What is the potential difference between the points ? A [OEB] 6. If E1 and E2 be the electric field strength of a short dipole on its axial line and on its equatorial line respectively, then what is the relation between E1 and E2 ? U [OEB] Ans. E1 = 2E2 1 [Explanation: Electric field at a point on equatorial line of a dipole is Kp/r 3 and that on axial line is 2Kp/r 3 .] Short Answer Type Questions-I (2 marks each) 1. Deduce an expression for the potential energy of a system of two points charges q1 and q2 located at positions r1  and r2  respectively in an external field E → . A & E [SQP 2020-21] Ans. V1 = Electric potential at the point having position vector r1  V2 = Electric potential at the point having position vector r2  q1V1 = Work done in bringing q1 from infinity to r1 against the external field q2V2 = Work done in bringing q2 from infinity to r2 against the external field 1 1 4 0 1 2 πε 12 × q q r = Work done on q2 against the force exerted by q1 These questions are for practice and their solutions are available at the end of the chapter.