PHY2049: Chapter 241What You Already KnowÎCoulomb’s lawÎElectric fieldsÎGauss’ lawÎElectric fields for several configurations Point Line Plane (nonconducting) Sheet (conducting) Ring (along axis) Disk (along axis) Sphere Cylinder Dipole (along || and ⊥ axes)PHY2049: Chapter 242Chapter 24: Electric Potential ÎElectric Potential EnergyÎElectric PotentialÎEquipotential SurfacesÎPotential of Point ChargeÎPotential of Charge DistributionÎCalculating the Field from the PotentialÎPotential Energy from a System of ChargesÎPotential of Isolated Charged ConductorsPHY2049: Chapter 243Reading Quiz: Chapter 24ÎAn equipotential surface is: a) a surface where the electric field is constant b) always parallel to the electric field c) a surface where the potential is zero d) always perpendicular to the electric field e) a surface where the electric field is zeroPHY2049: Chapter 244Reading Quiz: Chapter 24ÎThe volt is a unit of: a) potential energy b) electric field c) potential d) forcePHY2049: Chapter 245Reading Quiz: Chapter 24ÎElectric potential is: a) a scalar quantity b) a vector quantity c) can be either scalar or vectorPHY2049: Chapter 246Electric Work and Potential EnergyÎPoint charges q1, q2: Work moving charge q2from A → BÎPath independence: “conservative force” Define potential energy of two point charge()12212 12 122ˆˆBBABAABABAABkq qW F ds r ds r ds drrkq q kq q kq qWdrrrr=⋅= ⋅ ⋅===−∫∫∫• Depends only on endpoints• Path independent•Like gravitation12BA ABkq qUU W Ur−≡− ⇒=PHY2049: Chapter 247Electric Force is ConservativeÎHolds in all electrostatic situations (not just point charge) Proof: integrate over any charge distributionÎWork done by electric field moving charge q from i to f Calculate from difference of potential energiesChargesifelecfiifWUUU=−Δ = −Work:qPHY2049: Chapter 248Problem: Electric Potential Energy ÎTwo identical +12 mC point charges are initially spaced 5 cm from each other. If they are released at the same instant from rest, how fast will they be moving when they are very far from each other? Assume m1= m2= 1.0 g. ii f fKUKU+= +()2221202 0ffiikq kqmv vdmd+= +⇒=()()()()29539 10 0.0121.6 10 m/s10 0.05fv−×==×PHY2049: Chapter 249Gravitational & Electric Potential Energy GravityGABhWmgUU==−EABdWqEUU==−Electricd+++++++-------ABABPoint B at lower potential energy than point A (q>0)hPHY2049: Chapter 2410Electric Potential Î Potential = PE per unit chargeÎ Potential difference: general E fieldÎ Potential difference: constant EÎ Potential higher at + charges and “falls” to lower value at − charges +q: Moves from higher to lower V −q: Moves from lower to higher V/VUq=abVV Ed−=bbaaVV Eds−=− ⋅∫ba+++++++++++-------------------+QEdPHY2049: Chapter 2411Units for V and EÎUnits of potential: “volt” V = U/q ⎯→ Volt = Joule / CoulombÎUnits of electric field F = Eq ⎯→ E = F/q → Newton / Coulomb V = Ed ⎯→ E = V/d → Volt / MeterPHY2049: Chapter 2412Example of Potential of Point ChargeÎPoint charge q(using V= 0 at r = ∞)ÎExample: Potential at surface of proton (r= 10-15m)()()919615910 1.6101.44 10 1.44MV10kqVr−−××== = × =kqVr=PHY2049: Chapter 2413Energy Units: Electron VoltsÎ1 eV = energy of charge e accelerated through 1 VoltÎLet q = 4e and V = 2000 V()19191eV 1.6 10 C 1V1.6 10 J−−=×=×i()19 154 2000 8000eV 8keV8000 1.6 10 1.28 10 JKK−−=× = ==×× =×PHY2049: Chapter 2414ConcepTest: Electric Energy ÎA proton and an electron are each accelerated across a region of constant E field. Which has larger acceleration? (a) proton (b) electron (c) both have equal acceleration (d) neither one acceleratesF = Eea = F/m = Ee/mme mpElectron is much lighter than protonPHY2049: Chapter 2415ConcepTest: Electric EnergyÎWhich has the biggest increase in KE? (a) proton (b) electron (c) both have the same increase in KE (d) KE = 0 for bothK = Fd = EedVe> VpPHY2049: Chapter 2416Equipotential SurfacesÎEquipotentials: Contours of constant potential No work to move charge along contour: W = -qΔV = 0ÎE ⊥ equipotential surface If E⎥⎥≠ 0, would need work to move charge along surface See http://www.falstad.com/emstatic/PHY2049: Chapter 2417Equipotential: Constant E FieldExample: CapacitorConstant EPHY2049: Chapter 2418Equipotential: Point ChargeEquipotentialsPHY2049: Chapter 2419Equipotential: DipolePHY2049: Chapter 2420Topographic Map:Equal Altitude ContoursContour: Line of constant gravitational potentialPHY2049: Chapter 2421Calculating E From Electric Potential VÎElectric field in terms of potentialxyzxyzxyzdU F ds F dx F dy F dzdV E ds E dx E dy E dzVVVEEExyz=− ⋅ =− − −=− ⋅ =− − −∂∂∂=− =− =−∂∂∂Divide by qPHY2049: Chapter 2422Example: Electric Field of Point ChargeÎGet E by differentiating potential22ˆ,,kq x y z kqErrrrrr⎛⎞=≡⎜⎟⎝⎠()()1/2 3/2 3222 222, etc.xyzkq kqx kqxExrxyz xyzEE∂=− = =∂++ ++()1/2222rxyz=++Coulomb’s
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