Electricity and Magnetism 18 Chapter ELECTRIC ENERGY AND CAPACITANCE 2.1 Potential Difference and Electric Potential 1) Electric potential energy, electric potential difference and electric potential Consider the system constituted by the charges Q and q in Fig 2.1 The electric field E due to the charge Q E= Q 4πε o r [Vm-1], [NC-1] (2.1) The electrostatic force F on the charge q F = qE = qQ 4πε o r [N] (2.2) When the charge q is released from point A, it moves away from Q, along the electric field line The loss in potential energy of the system as the charge q moves from A to B is the work done by the electric force r r uu qQdr dW = F.dr = Fdr = 4πε o r rB ⇒ W = qQdr ∫ 4πε r rA o = qQ  1   -  4πε o  rA rB  [J] (2.3) The work done by the electrostatic force is path independent It depends only on the initial point A and the final point B and is the same for all paths between A and B Fig 2.1 The change in the potential energy of the system ∆U = UB - UA = -W (2.4) where UA and UB are the potential energy of the system when the charge q is at A and B, respectively When q moves from A to B, ∆U < : the electric force F does work W > and the system loses energy If we set the potential energy of the system UB = at infinity, i.e rB = ∞ (the reference point of zero potential at infinity) then it follows from (2.3) and (2.4) that Electricity and Magnetism UA = qQ 4πε o rA 19 [J] (2.5) The electric potential (the potential energy per unit charge) at point A is defined as VA = UA Q = q 4πε o rA [J/C], [V] (2.6) The electric potential difference between A and B is the difference in potential energy per unit charge ∆V = W Q 1 1 ∆U = =  -  = VB - VA q 4πε o  rA rB  q [J/C], [V] (2.7) 2) Equipotential surfaces The points on an equipotential line all have the same electric potential Equipotential lines are always perpendicular to the electric field In three dimensions, the lines form equipotential surfaces Movement along an equipotential line (or an equipotential surface) requires no work because such movement is always perpendicular to the electric field For a point charge, the equipotential lines are circles centered on the charge (Fig 2.2.a) The dashed lines illustrate the scaling of voltage at equal increments The equipotential lines get further apart with increasing r (a) (b) (c) Fig 2.2 : Dashed lines : equipotential lines Solid lines : electric field lines The electrical potential of a dipole shows mirror symmetry about the center of the dipole (Fig 2.2.b) They are everywhere perpendicular to the electric field lines For parallel conducting plates like those in a capacitor, the electric field lines are perpendicular to the plates and the equipotential lines are parallel to the plates (Fig 2.2.c) Electricity and Magnetism 2.2 20 Potential Difference in a Uniform Electric Field The potential difference between two points A and B in a uniform electric field (Fig 2.3) V = Ed [V] (2.8) Fig 2.3 2.3 Electric Potential and Potential Energy Due to Point Charges 1) The electric potential due to a single point charge at a distance r from that point charge V= q 4πεo r (2.9) The electric potential due to a collection of point charges V= 4πεo n ∑ rii q (2.10) i =1 2) The electric potential of a dipole at a distance r from the dipole can be found by superposing the electric potential of two point charges (Fig 2.4) V= q (r− − r+ ) q q = 4πε o r+ 4πε o r− 4πε o r+ r− Fig 2.4 (2.11) Fig 2.5 Electricity and Magnetism 21 If r >> d then r- - r+ ≈ dcos(θ) and r-r+ ≈ r2 (2.11) can be approximated by V= p cos(θ) (2.12) 4πεo r where p = qd is the dipole moment The electric dipole moment for a pair of opposite charges of magnitude q is defined as the magnitude of the charge times the distance between them and the defined direction is toward the positive charge It is a useful concept in atoms and molecules where the effects of charge separation are measurable, but the distances between the charges are too small to be easily measurable It is also a useful concept in dielectrics and other applications in solid and liquid materials 2.4 Electric potential due to continuous charge distributions The electric potential due to a continuous charge distribution V= 4πεo dq ∫r (2.13) 2.5 Electric potential due to a charged conductor Since the electric field E = for all points inside an isolated conductor, an excess charge placed on an isolated conductor lies entirely on its surface All points on the conductor have the same potential (even if the conductor has an internal cavity and even if that cavity contains a net charge 2.6 Capacitance A capacitor consists of two isolated conductors (the plates) with charges +q and –q Initially when the battery is not connected, the two plates are neutral When the battery is connected, electrons will flow until the potential difference between plate A and positive terminal of the battery is zero, and the potential difference between plate B and the negative terminal of the battery is zero The capacitance C [F] is defined as q = CV (2.14) where V is the potential difference between the plates The direction of V relates to q as given in Fig.2.7 Fig 2.6 Fig 2.7 1) A parallel-plate capacitor (Fig 2.8) Gauss’ law ⇒ q = εoAE ⇒ E = q (A : the area of the plate) εo A Electricity and Magnetism 22 By definition V= ∫ r r qd Ed s = Ed = εo A (the integral is taken in the direction of the electric field E, see also Fig 2.7) ⇒ q ε A = o V d C= [F] (2.15) Fig 2.8 : A parallel-plate capacitor Fig 2.9 : A cylindrical capacitor 2) A cylindrical capacitor (Fig 2.9) Gauss’ law ⇒ q = εo(2πrL)E ⇒ E = q 2πεo rL (2πrL : the area of the curved part of the Gaussian surface, L the length of the cylinder) Let a and b to be the radius of the inner cylinder and the outer cylinder, respectively By definition b V= ∫a r r Ed s = ∫ b q b qdr ln( ) = 2πεo L a a 2πε o rL (a < b : radii) (the integral is taken in the direction of the electric field E, see also Fig 2.7) ⇒ 2πεo L q = b V ln( ) a C= (2.16) 3) A spherical capacitor The capacitor consists of a solid conducting sphere of radius a surrounded by a spherical shell of inner radius b These are the plates of the capacitor The solid sphere has a +Q on its top surface, which induces a charge of -Q on the inner surface of the outer shell This in turn induces +Q charge on the outer surface of the outer shell Gauss’ law ⇒ q = εo(4πr2)E ⇒ E = q 4πεo r (4πr2 : the area of the sphere) By definition b V= ∫a r r Ed s = b qdr ∫ a 4πεo r = q 1 ( − ) 4πεo a b (a < b : radii) (the integral is taken in the direction of the electric field E, see also Fig 2.7) ⇒ C= q 4πεoab = V b−a 4) An isolated sphere b → ∞ : (2.17) ⇒ C = 4πεoa (2.17) (2.18) Electricity and Magnetism 23 2.7 Combinations of Capacitors 1) Capacitors in Parallel Ceq = ΣCi (2.19) 2) Capacitors in Series = Ceq 2.8 ∑ Ci (2.20) Energy Stored in a Charged Capacitor The work W required to bring the total capacitor charge up to q q = CV dW = Vdq = q dq C ⇒ W = q2 CV = 2C this work W is stored as potential energy U in the capacitor U= CV q = 2C [J] (2.21) The energy density u is the potential energy per unit volume In case of a parallel-plate capacitor ε E2 u= o (2.22) 2.9 Capacitors with Dielectrics 1) Dielectric Non polar dielectric: the center of positive charges coincide with the one of negative charges ⇒ the molecules are neutral Polar dielectric : the center of positive charges doesn’t coincide with the one of negative charges ⇒ each molecule is a dipole A) polar dielectric B) non polar dielectric Fig 2.10 : In absence of an external electric field Electricity and Magnetism 24 Under the effect of an external electric field, the molecules of non polar dielectric become dipoles The electric dipoles tend to line up with the external electric field A) vacuum B) dielectric Fig 2.11 : With the same charge, the electric field in case A is stronger than case B 2) The electric field produced by charge inside a dielectric The electric field of a point charge inside a dielectric E= q 4πk o ε o r (2.23) Gauss’ law with a dielectric koεoΦ = q (2.24) ko : dielectric constant Material Air (1 atm) Polystyrene Paper Transformer oil Pyrex Porcelain Dielectric constant ko 1.00054 2.6 3.5 4.5 4.7 6.5 Dielectric strength (kV/mm) 24 16 14 Problems Electric potential 2.1) Find the electric potential inside and outside a sphere of radius R and of constant volume charge density ρ 2.2) Find the electric potential inside and outside a spherical conducting shell of radius R with total charge +Q 2.3) A total charge of +Q is uniformly distributed along the length of a rod of length L (Fig P2.1) Determine the electric potential at point P, a distance a from one end of the rod as shown Fig P2.1 Electricity and Magnetism 25 2.4) Determine an expression for the potential difference between two points A and B in Fig P2.2 Suppose that the electric field is constant 2.5) Determine an expression for the potential difference between two points A and C in Fig P2.2 Suppose that the electric field is constant Fig P2.2 Fig P2.3 2.6) The thin plastic rod shown in Fig P2.3 has length L = 12cm and a nonuniform linear charge density λ = αx where α = 28.9pC/m2 With V = at infinity, find the electric potential at P1 and P2 Where d1 = 8cm, d2 = 3cm 2.7) Three particles, charge q1 = 10µC, q2 = -20µC, q3 = 30µC, are positioned at the vertices of an isosceles triangle as shown in Fig P2.2 If a = 10cm and b = 6cm, how much work must an external agent to exchange the position of a) q1 and q3 b) q1 and q2 Fig P2.4 2.8) A non conducting sphere has radius R = 2cm and uniformly distributed charge q = 3.5fC Take the electric potential at the sphere center to be Vo = What is V at radial distance a) r = 1.5 cm b) r = R Capacitance 2.9) A metal plate of thickness a is inserted in-between the plates which are separated by a distance d (Fig P2.5) Find the capacitance of the system Electricity and Magnetism Fig P2.5 26 Fig P2.6 2.10) What happens if the outer surface of the capacitor is connected to Earth ? (Fig P2.6) Answer: The electrons from Earth neutralize the outer surface only The inner surface still maintains a total charge of -Q, which means that the electric field within the capacitor is unaffected Thus, the potential difference remains the same, and therefore, there is no loss of energy from the capacitor 2.11) Consider a parallel plate capacitor with rectangular plates and a sheet of metal of thickness a The dimensions of the capacitors are given in the Fig P2.7 Let x be the length of the metal plate that is inserted between the capacitor plates Let +Q and -Q be the charges on the plates of the capacitor a) Find the capacitance Co and the energy Uo stored in the capacitor before the metal sheet is inserted b) Find the capacitance C and the energy U stored in the capacitor after insertion of the metal sheet as function of x c) Find the force on the metal sheet ? Which direction does it tend to move the metal sheet ? Fig P2.7 Fig P2.8 2.12) Two long conducting wires of length L and radius a lie parallel a distance s apart (Fig P2.8) The upper wire carries charge Q and the lower charge -Q Since L >> s, we may assume the wires are effectively infinitely long for purpose of finding the electric fields and potentials a) Find the electric field E in the plane in between the wires b) Find the potential ϕ in the plane between the wires Find V, the potential different between the wires c) Find the capacitance C of the two wire system d) Find the total electrical energy stored in the system 2.13) Find the capacitance C of a cylindrical capacitor of length L and radii a and b (Fig P2.9) Electricity and Magnetism 27 top view side view Fig P2.9 2.14) The parallel plate capacitor in Fig P2.10 has plate area A = 100 cm2 and plate separation d = cm A potential difference Vo = 50 V is applied between the plates The battery is then disconnected A dielectric slab of thickness b = 0.8 cm and dielectric constant k = is placed between the plates after the battery was removed a) Before the dielectric slab is inserted, find the capacitance and the charge on the plate b) After the slab has been introduced, find - the electric field in the gaps between the plates and the dielectric slab - the electric field in the dielectric slab - the capacitance between the plates - the potential difference between the plates Fig P2.10 Electricity and Magnetism 28 Homeworks H2.1 A total charge of +Q [fC] is uniformly distributed along the length of a rod of length L [mm] (Fig H2.1) Determine the electric potential at point P, a distance a [mm] from one end of the rod as shown Fig H2.1 Fig H2.2 n Q L a 1 21 24 2 22 25 3 23 26 4 24 27 5 25 28 6 26 29 7 27 24 8 28 25 9 29 26 10 10 21 27 11 11 22 28 12 12 23 29 13 13 24 25 14 14 25 26 15 15 26 27 16 16 27 28 n Q L a 17 22 21 18 23 22 19 24 23 20 25 24 21 26 25 22 27 26 23 28 27 24 29 28 25 22 29 26 10 23 21 27 11 24 22 28 12 25 23 29 13 26 24 30 14 27 25 31 15 28 26 32 16 29 27 n Q L a 33 23 22 34 24 23 35 25 24 36 26 25 37 27 26 38 28 27 39 29 28 40 22 29 41 23 22 42 10 24 23 43 11 25 24 44 12 26 25 45 13 27 26 46 14 28 27 47 15 29 28 48 16 21 29 n Q L a 49 24 25 50 25 26 51 26 27 52 27 28 53 28 29 54 29 22 55 24 23 56 25 24 57 26 25 58 10 27 26 59 11 28 27 60 12 29 28 61 13 25 29 62 14 26 25 63 15 27 26 64 16 28 27 H2.2 Consider a parallel plate capacitor with rectangular plates and a sheet of metal of thickness a The dimensions of the capacitors are given in [mm] (Fig H2.2) Let x be the length of the metal plate that is inserted between the capacitor plates Let +Q and -Q be the charges [fC] on the plates of the capacitor a) Find the capacitance Co and the energy Uo stored in the capacitor before the metal sheet is inserted b) Find the capacitance C and the energy U stored in the capacitor after insertion of the metal sheet as function of x c) Find the force on the metal sheet ? Which direction does it tend to move the metal sheet ? n Q a d w L 1 1.1 1.2 20 31 2 1.2 1.3 21 32 3 1.3 1.4 22 33 4 1.4 1.5 23 34 5 1.5 1.6 24 35 6 1.1 1.7 25 36 7 1.2 1.2 26 37 8 1.3 1.3 27 38 9 1.4 1.4 28 39 10 10 1.5 1.5 29 40 11 11 1.1 1.6 30 41 12 12 1.2 1.7 31 42 13 13 1.3 1.2 32 43 14 14 1.4 1.3 33 44 15 15 1.5 1.4 34 45 16 16 1.1 1.5 35 46 Electricity and Magnetism 29 n Q a d w L 17 1.5 1.2 31 20 18 1.6 1.3 32 21 19 1.7 1.4 33 22 20 1.8 1.5 34 23 21 1.9 1.6 35 24 22 1.5 1.7 36 25 23 1.6 1.2 37 26 24 1.7 1.3 38 27 25 1.8 1.4 39 28 26 10 1.9 1.5 40 29 27 11 1.5 1.6 41 30 28 12 1.6 1.7 42 31 29 13 1.7 1.2 43 32 30 14 1.8 1.3 44 33 31 15 1.9 1.4 45 34 32 16 1.5 1.5 46 35 n Q a d w L 33 1.3 1.2 22 31 34 1.4 1.3 23 32 35 1.5 1.4 24 33 36 1.6 1.5 25 34 37 1.7 1.6 26 35 38 1.8 1.7 27 36 39 1.3 1.2 28 37 40 1.4 1.3 29 38 41 1.5 1.4 30 39 42 10 1.6 1.5 31 40 43 11 1.7 1.6 22 41 44 12 1.8 1.7 23 42 45 13 1.3 1.2 24 43 46 14 1.4 1.3 25 44 47 15 1.5 1.4 26 45 48 16 1.6 1.5 27 46 n Q a d w L 49 1.7 1.2 26 20 50 1.8 1.3 27 21 51 1.3 1.4 28 22 52 1.4 1.5 29 23 53 1.5 1.6 30 24 54 1.6 1.7 31 25 55 1.7 1.2 22 26 56 1.8 1.3 23 27 57 1.3 1.4 24 28 58 10 1.4 1.5 25 29 59 11 1.5 1.6 26 20 60 12 1.6 1.7 27 21 61 13 1.7 1.2 26 22 62 14 1.8 1.3 27 23 63 15 1.3 1.4 28 24 64 16 1.4 1.5 29 25 ... 22 21 18 23 22 19 24 23 20 25 24 21 26 25 22 27 26 23 28 27 24 29 28 25 22 29 26 10 23 21 27 11 24 22 28 12 25 23 29 13 26 24 30 14 27 25 31 15 28 26 32 16 29 27 n Q L a 33 23 22 34 24 23 35 25 ... 25 24 36 26 25 37 27 26 38 28 27 39 29 28 40 22 29 41 23 22 42 10 24 23 43 11 25 24 44 12 26 25 45 13 27 26 46 14 28 27 47 15 29 28 48 16 21 29 n Q L a 49 24 25 50 25 26 51 26 27 52 27 28 53 28 ... H2.1 Fig H2 .2 n Q L a 1 21 24 2 22 25 3 23 26 4 24 27 5 25 28 6 26 29 7 27 24 8 28 25 9 29 26 10 10 21 27 11 11 22 28 12 12 23 29 13 13 24 25 14 14 25 26 15 15 26 27 16 16 27 28 n Q L a 17 22