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A sphere of radius A has a charge Q uniformly spread throughout its volume. Find the difference in the electric potential, in other words, the voltage difference, between the center and a point 2A from the center. r2 Vr2 Vr1 E dr 0 r1 since E 0 inside the conductor. For any two points r1 and r2 inside the conductor Vr1 Vr2 The conductor’s surface is an equipotential. Equipotential Surfaces An equipotential surface is a surface on which the electric potential V is the same at every point. Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work on such a charge. So, electric field must be perpendicular to the surface at every point so that the electric force qE is always perpendicular to the displacement of a charge moving on the surface. Field lines and equipotential surfaces are always mutually perpendicular. Method of images: What is a force on the point charge near a conducting plate? Equipotential surface - -- The force acting on the positive charge is exactly the same as it would be with the negative image charge instead of the plate. a The point charge feels a force towards the plate with a magnitude: 1 2 q F 40 (2a) 2 Method of images: A point charge near a conducting plane. E ? Equipotential surface - -- P r a E 1 aq 40 (a 2 r 2 )3 2 1 aq E 40 (a 2 r 2 )3 2 2 aq E 40 (a 2 r 2 ) 3 2 Equilibrium in electrostatic field: Earnshaw’s theorem There are NO points of stable equilibrium in any electrostatic field! How to prove it? Gauss’s Law will help! Imaginary surface surrounding P P If the equilibrium is to be a stable one, we require that if we move the charge away from P in any direction, there should be a restoring force directed opposite to the displacement. The electric field at all nearby points must be pointing inward – toward the point P. But that is in violation of Gauss’ law if there is no charge at P. Thomson’s atom 1899 If charges cannot be held stably, there cannot be matter made up of static point charges (electrons and protons) governed only by the laws of electrostatics. Such a static configuration would collapse! Capacitors Consider two large metal plates which are parallel to each other and separated by a distance small compared with their width. y Area A L The field between plates is V E 0 [V (top) V (bottom)] E y dy dy L 0 0 0 0 L L A QL [V (top) V (bottom)] L L 0 0 A 0 A QL V A 0 The capacitance is: A 0 Q Q C QL V L A 0 Cylindrical Capacitor Spherical Capacitor A 0 Q Q C V QL L A 0 Capacitors in series: 1 1 1 1 ... Ctot C1 C2 C3 Capacitors in parallel: Ctot C1 C2 C3 ... 1 1 2 2 W CV Q 2 2C [C ] farad Quiz 1) If a 4-F capacitor and an 8-F capacitor are connected in parallel, which has the larger potential difference across it? Which has the larger charge? 2) A B Two capacitors are connected in series as shown. If they were initially uncharged, what will be the charge inside the dotted box after connecting points A and B to a battery of voltage V? 3) If the wire connecting the capacitors is bent so that capacitors look like A B how do you now call the arrangement? a C3 C1 C2 b C5 C4 C1=C5=8.4 F and C2=C3=C4=4.2 F The applied potential is Vab=220 V. a)What is the equivalent capacitance of the network between points a and b? b) Calculate the charge on each capacitor and the potential difference across each capacitor. E d r 0 in electrosta tics E E dr 0 B C E2 E1 D A E dr 0 D E1dr E2dr E1 ( B A) E2 ( D C ) 0 B A C E1 E2 Most capacitors have a non-conducting material, or dielectric, between their conducting plates. When we insert an uncharged sheet of dielectric between the plates, experiments show that the potential difference decreases to a smaller value V. Q C0 without dielectric V0 Q C with dielectric V C C0 When the space between plates is completely filled by the dielectric, the ratio C K C0 is called dielectric constant. Smaller, Denser, Cheaper Moore’s Law (1965): every 2 years the number of transistors on a chip is doubled Have a great day!