Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Chapter 30. Potential and Field To understand the production of electricity by solar cells or batteries, we must first address the connection between electric potential and electric field. Chapter Goal: To understand how the electric potential is connected to the electric field.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Potential and Field Work and potential energy are associated with assembling a charge distribution. + +Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Potential and Field The work per unit charge done by an electric force = the change in potential energy of a unit charge and the change in the electrostatic potential V measured in volts. F • ds = dWdV = dW /qE • ds = dV E d sCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Batteries • A chemical reaction entails an adjustment of electron concentration on different atoms. • A battery uses a chemical reaction (pair), intercepting an electron forcing it to pass through a circuit. • The battery terminals are in effect at different electric potentials • The battery potential is typically a few volts – the scale of atomic energy potential differences. http://www.av8n.com/physics/battery.htm http://www.batteryuniversity.com/index.htmCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. A. 1.0 V B. 2.0 V C. 5.0 V D. 6.0 V E. 7.0 V What total potential difference is created by these three batteries?Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. A. 1.0 V B. 2.0 V C. 5.0 V D. 6.0 V E. 7.0 V What total potential difference is created by these three batteries?Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Conductors in equilibrium • Move charge from one conductor to another. • The excess charge distributes on the surfaces in a unique way such that the two conductors are free of E field so volumes of constant potential. • The unique equilibrium surface charges densities are not uniform and in general tricky to calculate.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Electric power transmission • Imagine this picture is a cross section of two wires. • Do work to transport additional + charge from – to + at one end. The charge distributes along the length reproducing the picture but the total charge, the strength of the electric field, the potential difference are increased. • Get work at the other end by transporting + charge from + to -. • The other end might be 1000 miles away!Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Capacitance • In electrostatic equilibrium, the E field is perpendicular to the surface or each conductor, else the transverse component would cause current. • Doubling the surface charge distribution maintains the equilibrium conditions (E perpendicular to surfaces) but doubles |E| everywhere and the potential difference V. • In general V=Q/C where C [Coulomb/Volt = Farad] is called the capacitance and depends only on the geometry. V =1CQCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Energy stored in a capacitor To transfer a charge dq from – to + given an existing potential difference V(q)=q/C requires differential work dW = Vdq = (q/C)dq Summing the differential work as q ranges from 0 to Q gives the total work to establish the +Q and –Q charge separation and the total energy stored U. W =qC0Q∫dq =Q22C U =Q22C=12QΔV =12C ΔV( )2Aside: One can show U is a volume integral of an energy density u~E2.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Parallel plate capacitor The electric field and surface charge densities are uniform. The potential difference is V=Ed -Q +Q d + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - E V + −V − = − −qEd( )/qΔV = Ed =ηd /εo= QdεoA⎛ ⎝ ⎜ ⎞ ⎠ ⎟ C =εoA / dNote the capacity is proportional to area A and inversely proportional to separation.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Spherical capacitor 12 Charge Q moved from outer to inner sphere Gauss’ law says E=kQ/r2 Potential difference ΔV = E • dsab∫Along path shown ΔV =kQr2ab∫= −kQ1rab= kQ1a−1b⎛ ⎝ ⎜ ⎞ ⎠ ⎟Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Example capacitorCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Capacitor combinationsCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Combinations of Capacitors If capacitors C1, C2, C3, … are in parallel, their equivalent capacitance is (note C is proportional to area) If capacitors C1, C2, C3, … are in series, their equivalent capacitance isCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Equivalent Capacitors 16 C =ε0Ad Two capacitors connected in parallel C1 C2 Ceq 15 V 10 V 15 V 10 V 15 V 10 V Add Areas: Ceq = C1+C2 Add Charge: Qeq = Q1+Q2 = Veq • Same voltage: V1 = V2Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Energy Stored in a CapacitorCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 30.9 Storing energy in a capacitor QUESTIONS:Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 30.9 Storing energy in a capacitorCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Energy in the Electric Field The energy density of an electric field, such as the one inside a capacitor, is The energy density has units J/m3.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Effect of insulators in the gapCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Dielectric physics Insulator is polarized by the field between the conducting plates forming in effect two narrow gap (large C) capacitors in series. Put another way, for fixed charge on the conductors the polarization produces an opposing E field reducing the voltage between the conductors by some
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