PHYS 205 Chapter 19, Fall 2017

Chapter 19 Electric Potential Energy and The Electric Potential Outline Outline 1. Potential Energy 2. The Electric Potential Difference 3. The Electric Potential Difference Created by Point Charges 4. Equipotential Surfaces and their Relation to the Electric Field 5. Capacitors and Dielectrics 1- Potential Energy B A B A AB mgh mgh W GPE  GPE   GPE WAB   Work done by electric force on charge q from A to B: . . . W F d q E d     But PE = change in potential energy = -W   - B A PE V V V q       A B V Electric potential at A in Volts (V) V Electric potential at B in Volts (V) Change in potential energy in Joules (J) Charge of the particle in Coulombs (C) PE q      . - - B A J N m V V Ed Units Volts C C     We define: So that 2- The Electric Potential Difference Example 1 Work, Potential Energy, and Electric Potential The work done by the electric force as the test charge (+2.0x10-6C) moves from A to B is +5.0x10-5J. (a)Find the difference in EPE between these points. (b)Determine the potential difference between these points. B A WAB EPE  EPE  o AB o A o B A B q W q q V V     EPE EPE B A WAB EPE  EPE (a) 10 J 0.5 EPE EPE  5     AB A B W (b) 25 V 10 C 2.0 10 J 0.5 6 - 5        o AB A B q W V V Conceptual Example 2 The Accelerations of Positive and Negative Charges A positive test charge is released from A and accelerates towards B. Upon reaching B, the test charge continues to accelerate toward C. Assuming that only motion along the line is possible, what will a negative test charge do when released from rest at B? A positive charge accelerates from a region of higher electric potential toward a region of lower electric potential. A negative charge accelerates from a region of lower potential toward a region of higher potential. Conservation of Energy EPE kx mgh I mv E      2 2 1 2 2 1 2 2 1  Note: the eV electron Volt is defined as the energy that the electron / proton gains when accelerated through a potential difference of 1 Volt Example 4 The Conservation of Energy A particle has a mass of 1.8x10-5kg and a charge of +3.0x10-5C. It is released from point A and accelerates horizontally until it reaches point B. The only force acting on the particle is the electric force, and the electric potential at A is 25V greater than at C. (a) What is the speed of the particle at point B? (b) If the same particle had a negative charge and were released from point B, what would be its speed at A? A A B B EPE mv EPE mv    2 2 1 2 2 1 B A A B EPE EPE mv mv    2 2 1 2 2 1  B  A o A B V q V mv mv    2 2 1 2 2 1  B  A o B V q V mv   2 2 1 (a)        1.9 m s 10 kg 8.1 10 C 25 V 0.3 2 2 5 5         m V q V v B A o B        1.9 m s 10 kg 8.1 10 C 25 V 0.3 2 2 5 5            m V q V v B A o A (b) When the electric field is directed downward, point B is at a lower potential than point A A positive test charge that moves from A to B loses electric potential energy It will gain the same amount of kinetic energy as it loses potential energy     0 0 W PE W PE      Electric field Gravity Uniform E - - - 0 B A B A A B V V Ed V V E d V V E        Field between two parallel plates 3- The Electric Potential Difference Created by Point Charges VA is a scalar quantity. For several point charges qi , A e q V k r   algebraic sum TOT i i V V  The electric potential created at A by a point charge q is: Consider a charge q0 is brought from infinity to point A . . e final initial A k q ke q V V V V r r         0 0 . . . electrical potential energy of a pair of charges ek q q PE q V r      If the charges have the same sign, PE is positive Positive work must be done to force the two charges near one another The like charges would repel If the charges have opposite signs, PE is negative The Coulomb force does positive work . 0 e W F d    Example 5 The Potential of a Point Charge Using a zero reference potential at infinity, determine the amount by which a point charge of 4.0x10-8C alters the electric potential at a spot 1.2m away when the charge is (a) positive and (b) negative.    V 300 m 2.1 10 C 0.4 C .8 99 10 N m 8 2 2 9          r kq V (a) (b) V  300 V Example 6 The Total Electric Potential At locations A and B, find the total electric potential.       240 V 60 m .0 10 C 0.8 C 99 10 N m .8 20 m .0 10 C 0.8 C .8 99 10 N m 8 2 2 9 8 2 2 9              A V       0 V 40 m .0 10 C 0.8 C 99 10 N m .8 40 m .0 10 C 0.8 C .8 99 10 N m 8 2 2 9 8 2 2 9              VB Potentials and Charged Conductors No work is required to move a charge between two points that are at the same electric potential - .( - ) 0 B A W q V V   Conductors in Equilibrium E = 0 inside the conductor The electric field just outside the conductor is perpendicular to the surface The potential is a constant everywhere on the surface of the conductor VA = VB = cst The potential everywhere inside the conductor is constant and equal to its value at the surface Vinside = Vsurface = cst 4- Equipotential Surfaces and their relation to the Electric Field An equipotential surface is a surface on which all points have the same electric potential. Given some initial velocity, no work is required to move a charge at a constant speed on an equipotential surface. The electric field at every point on an equipotential surface is perpendicular to the surface rB rA . e B B k q V r  . e A A k q V r  Electric field lines Equipotential lines r1 r2   1 2 . . TOT k q k q V r r    s V E    Example 9 The Electric Field and Potential Are Related The plates of the capacitor are separated by a distance of 0.032 m, and the potential difference between them is VB-VA=-64V. Between the two equipotential surfaces shown in color, there is a potential difference of -3.0V. Find the spacing between the two colored surfaces. 10 V m 0.2 m 0.032 64 V 3        s V E 10 m 5.1 10 V m 0.2 V 0.3 3 3          E V s 5- Capacitors and Dielectrics Most common capacitors is the parallel plate capacitor. With A being the area of each plate and d being the separation between the plates. The capacitance, C, of a capacitor is defined as: Q C V  Units: Farad (F), 1 F = 1 C / V Where Q is the charge accumulated on each plate and V is the potential difference between the 2 plates. • The symbol for a capacitor is • For a parallel-plate capacitor whose plates are separated by air: o A C  d  2 12 2 is the permittivity of free space = 8.85 10 . 1 and Coulomb's constant 4 o e o C N m k      Combination of Capacitors In parallel: 1 2 1 2 1 2 ( ) ( ) but ( ) TOT TOT TOT TOT Q Q Q C V C V Q C V C C C            For n capacitors in parallel: 1 n eq i i C C   In series: 1 2 1 2 1 2 and 1 1 1 eq eq Q Q Q V V V V C C C C C C           1 1 1 n i eq i C C   For n capacitors in series: Capacitors with Dielectrics If a dielectric is inserted between the plates of a capacitor, the capacitance can increase markedly. Dielectric constant E  Eo d V E E o     A q E o o   V d A q o         d A C o  Parallel plate capacitor filled with a dielectric Capacitance of Parallel Plate Capacitor Conceptual Example 11 The Effect of a Dielectric When a Capacitor Has a Constant Charge An empty capacitor is connected to a battery and charged up. The capacitor is then disconnected from the battery, and a slab of dielectric material is inserted between the plates. Does the voltage across the plates increase, remain the same, or decrease? The dielectric becomes polarized. The presence of the positive charge on the dielectric effectively reduces some of the negative charge on the metal This allows more negative charge on the plates for a given applied voltage, so Q increases due to the additional flow of charges. So the capacitance increases when the dielectric is inserted between the plates. Energy Stored in a Capacitor At t=0, small charges dq are transferred from 1 plate to another. Work done by the battery = . . but dW V dq q q dq V dW C C     0 0 . where . Q Q TOT batt q dq W dW Q V C C      2 2 2 . 2 2 2 batt TOT CV q Q W C C     2 . energy stored in a capacitor 2 CVbatt  2 2 1 Energy CV    2 2 1 Energy Ed d o A         2 2 1 Volume Energy Energy density  o E  Volume Ad