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Electrical Potential Young and Freedman Chapter 23 Work It takes work to lift a mass in a GRAVITATIONAL FIELD The work done increases the Gravitational Potential Energy Work q q Q It takes work to lift a charge in an electric field This increases the charges Electrical Potential Energy Review: Work, Energy, Potential Energy, Conservation of Energy v r Unit are Newtons · Meters = Joules !W = F " !x e nc sta Di e rc Fo e on kd or W Wa #b = ! b a v r b F " dl = ! F cos $dl a Forces like Gravity and the Electric Force are Conservative Forces [i.e. No Friction]. Work done by conservative forces can be described as a change in Potential Energy Wa #b = (U a " U b ) = "(U b " U a ) = " !U ΔU can be positive or negative ΔU positive – Work is “done” in moving from point a to point b ΔU negative – Energy is “released” in moving from point a to point b To avoid confusion it is often helpful to think about a ball rolling up or down a hill ΔU positive – Work is “done” in rolling the ball UP the hill – higher up means higher EP ΔU negative – Energy is “released” when the ball rolls down - lower down means lower EP Electrostatic Potential Energy in a Uniform Electric Field + b yb ya r E a + q y r E = ! Eyˆ constant - r Move a positive charge in a vertical direction dl = yˆ dl r b v b b W = $ F # dl = $ " qEyˆ # yˆ dl = $ " qEdl = " qE ( yb " ya ) = " qE!y a a a W = " !U Looks like U gravity = mg!h !U a "b = qE!y b + c Now move from b → c + r E a - W =! y c v r F " dl b r r But F is perpendicular to dl So no work is done and there is no change in U !U a "c = qE!y Now, instead, go directly from a → c and compute ΔU c b Δy a + φ L + r E - W =! c a y v r c F " dl = ! F cos(# )dl a c W = " qE cos(# ) ! dl a W = " qE cos(! ) L Path length is given by: y = L cos(! ) # L = "y cos(! ) Therefore: W = " qE!y Work done on a charge in an electric Field is INDEPENDENT OF PATH Since Wa #b = (U a " U b ) = "(U b " U a ) = " !U is independent of path, all that matters are the values of the potential energy at the start and at the end Work and Potential Energy – Keeping the signs straight Wa #b = ! b a r v b F " dl = ! F cos $dl Work done by a Force a + + + + - - - - Work and Potential Energy Wa #b = ! b a r v b F " dl = ! F cos $dl Work done by a Force a The sign is critically important r r F = qE r F" ME" Direction of Motion The Work in the equations above refers to the Work DONE BY THE ELECTRIC FIELD If I slowly move the charge,* I will always be applying a force which is equal and opposite to the Electrostatic Force. The work DONE BY ME will be equal and opposite to the work done by the E field. * no change in kinetic energy The path of integration does NOT matter for a “conservative” force such as the Electrostatic Force Wa #b = ! b a r v b F " dl = ! F cos $dl a Example: Electric potential energy of two point charges Fr = b b a a Wa )b = ( Fr dr = ( qq0 qq0 dr = 2 4*+ 0 r 4*+ 0 ( b a qq0 4!" 0 r 2 qq0 & 1 1 # 1 $$ ' !! dr = 2 4*+ 0 % ra rb " 0r The path of integration does NOT matter for “conservative” forces such as the electrostatic force r v b Wa #b = ! F " dl = ! F cos $dl a a qq0 & 1 1 # $$ ' !! Wa (b = U a ' U b = '+U = 4)* 0 % ra rb " b NO ABSOLUTE ZERO for potential energy • It is the CHANGE in potential energy that is important • We can pick any distance to be the “baseline” for potential energy • For convenience we pick the zero of potential energy to refer to the Energy of the system when the charges are VERY far away from one another “infinity” m h M We pick ZERO for potential energy to be “Infinite Separation” qq0 ' *U = 4() 0 qq0 ' *U = 4() 0 &1 1# $$ ' !! % ra rb " & 1# $$ 0 ' !! rb " % Let ra = ∞ Define U (r ) = qq0 1 4!" 0 r The Potential Energy: qq0 1 U (r ) = 4!" 0 r is related to the amount of work that must be done to bring two charges together from far away. I must do POSITIVE work to bring two like charges together I must do NEGATIVE work to bring two like charges together Electrostatic potential energy for several point charges What is the electrostatic potential energy of q0 due to q1, q2, and q3 ? Electrostatic Potential Energy is a SCALAR and adds algebraically This is different from the Electric Field; a VECTOR that adds vectorially Problem: A system of point charges q1 $ q2 q3 ' *e $ e e ' U= & + )= & + ) 4 "# 0 % a 2a ( 4 "# 0 % a 2a ( *e2 $ 3 ' U= & ) 4 "# 0 % 2a ( Check sign - Sign is negative, which is correct for opposite charges. The system is at a lower potential energy than if q1 was at infinity ! Problem: A system of point charges U q1 $ q2 q3 ' *e $ e e ' U =q $ q& +q ') = e &$ *e + )e ' 24 "# 0 %1 a 2 ( 4 "# 0 % a 2a ( 2a = & + )= & + )=0 2 4 "#*e $ 3 'a ( 0 % a U= & ) 4 "# 0 % 2a ( 4 "# 0 % a a( Check sign - Sign is negative, which is correct for opposite charges. The system is at a lower potential energy than if q1 was at infinity ! Electrostatic potential The Electrostatic Potential Energy of a point charge q0 and a collection of Charges qi : Sample Problem #3 q1 q2 q3 What is the TOTAL potential energy of this charge distribution? Start by removing ALL charges to ∞. This corresponds to the “zero” of Potential Energy Step 1: All charges at ∞ U=0 Step 2: Bring Charge q3in from ∞ . This requires no work because there are no other charges present. U=0 Step 3: Bring charge q2 in from infinity q1 1 U= 4'( 0 q3 q2 & q 2 q3 # $$ !! % r23 " Step 4: Bring charge q1 in from infinity q1 q2 1 U= 4'( 0 q3 & q2 q3 q1q2 q1q3 # $$ !! + + r12 r13 " % r23 The electrostatic potential energy of collection of point charges 1 U= 4"# 0 ! i< j qi q j rij Note: BE CAREFUL NOT TO DOUBLE COUNT Electrostatic Force Electrostatic Field (Chapter 21) r 1 qi F (r ) = q ! r̂ 2 i i 4"# 0 ri v 1 qi ( ) E r =! r̂ 2 i i 4"# 0 ri We defined the concept of the Electric Field as a generalization of the Electric Force. The Electric Field assigns a VECTOR to each point in Space Electrostatic Potential Energy of a charge q0 and a collection of charges qi q0 U= 4() 0 ' q1 q2 q3 $ q0 %% + + + ..."" = & r1 r2 r3 # 4() 0 qi !i r i If the charges qi remain fixed and I move q0, I can express Potential Energy as a Function of the position of q0 q0 U (r ) = 4"# 0 qi !i r i Electrostatic Potential 1 q U= 4 "# 0 r U 1 qi V= = $ q0 4 "# 0 i ri Potential of a set of POINT CHARGES Potential energy due to a POINT CHARGE ! IMPORTANT: Like Work!-Electrostatic Potential is a “relative” quantity. r r Wa"b = U a #U b = $ a F % dl = b divide by q0 ! Va # Vb = ! b a r r $ a q0 E % d l b r r b E " dl = ! E cos $dl a UNITS : Volt = Joule/Coulomb ΔV = 1.5 volts Electrostatic Potential Energy Electrostatic Potential q0 U (r ) = 4"# 0 qi !i r i 1 V (r ) = 4"# 0 qi !i r i We defined the concept of the Electric Potential as a generalization of the Electric Potential Energy. The Electric Potential assigns a Scalar to each point in Space Finding potential by integration Find the potential at a distance r from a point charge using integration Va # Vb = ! b a r r b E " dl = ! E cos $dl a reduces to V "0 = $ # r Edr = q =" 4%&0 r q V= 4 %& 0 r $ # r q dr 2 4 %& 0 r # ' q * = 0 " )" , ( 4 %& 0 r + r Example 23.9: Potential of oppositely charged parallel plates Find the potential at any height y between two oppositely charged parallel plates Va # Vb = ! b a V (y ) = r r b E " dl = ! E cos $dl a U (y ) q0 Ey = = Ey q0 q0 We have chosen V=0 at point b Example 23.8: A charged conducting sphere A solid conducting sphere of radius R has a total charge q. Find the potential both outside and inside the sphere. E is 0 everywhere within the sphere, else charge would move to make it zero. Hence no work is done on a test charge moving around within the sphere and the potential is the same at every point within the sphere. The value of the potential is defined by the surface of the sphere. NOTE: A conducting sphere is always an equipotential surface Equipotential Surfaces and Electric Field Lines Equipotential Surfaces are always perpendicular to Electric Field Lines Chapter 23 Summary Chapter 23 Summary cont. End of Chapter 23 You are responsible for the material covered in T&F Sections 23.1-23.4 You are expected to: • Understand the following terms: Work, Potential Energy, Electrostatic Potential, Potential Difference, Equipotential Surface • Understand the relationship between work and potential energy. In particular, be able to distinguish between work done “by the field” and work done by the force moving a charge in a field. • Be able to calculate the electrostatic potential energy of simple charge geometries using either the sum rule for discrete geometries or the integral rule for continuous geometries • CLEARLY DISTINGUISH BETWEEN POTENTIAL ENERGY AND ELECTROSTATIC POTENTIAL involved. • Be able to calculate the electrostatic potential at a point in space due to simple charge geometries using either the sum rule for discrete geometries or the integral rule for continuous geometries • Understand the convention of setting the zero of potential energy at “infinity” Recommended Y&F Exercises chapter 23: • 6,9,14,21,23,35,40