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Short Version : 20. Electric Charge, Force, & Fields 20.1. Electric Charge 2 kinds of charges: + & . Total charge = algebraic sum of all charges. Like charges repel. Opposite charges attract. Q qi d 3 x r i All electrons have charge e. e 1.60 1019 C = elementary charge All protons have charge +e. 1st measured by Millikan on oil drops. Theory (standard model) : basic unit of charge (carried by quark) = 1/3 e. Quark confinement no free quark can be observed. Smallest observable charge is e. Conservation of charge: total charge in a closed region is always the same. Coulomb’s law (force between 2 point charges) : F12 k q1 q2 rˆ 2 r [q] = Coulomb = C k c 2107 N m 2 / C 2 c 2.99792458 108 k 9.0 109 N m2 / C 2 r12 4 , 3 m 4 ˆi 3 ˆj m r12 rˆ 16 9 m 5 m 4ˆ 3ˆ 1 i j m 4 , 3 m 5 5 5 Conceptual Example 20.1. Gravity & Electric Force The electric force is far stronger than the gravitational force, yet gravity is much more obvious in everyday life. Why? Only 1 kind of gravitational “charge” forces from different parts of a source tend to reinforce. 2 kinds of electric charges forces from different parts of a neutral source tend to cancel out. Making the Connection Compare the magnitudes of the electric & gravitational forces between an electron & a proton. Fg G mem p k e2 FE 2 r r2 9 10 N m / C 1.6 10 C FE ke Fg G me m p 6.67 1011 N m2 / kg 2 9.11 1031 kg 1.67 1027 kg 9 2 2.27 1039 2 2 19 2 Point Charges & the Superposition Principle Extension of Coulomb’s law (point charges) to charge distributions. Superposition principle: Fnet Fnet Fi i F23 F13 Task: Find net force on q3 . Independent of each other Example 20.2. Raindrops Charged raindrops are responsible for thunderstorms. Two drops with equal charge q are on the x-axis at x = a. Find the electric force on a 3rd drop with charge Q at any point on the y-axis. y F F1 F2 F1 F2 Q r k qQ 2 2 0 , sin r r y q x1 = a q x2 = a r a2 y2 k qQ k qQ cos , sin cos , sin 2 2 r r x 2 k qQ yˆ j 3 r 2 k qQ y a 2 y 2 3/2 ˆj sin y r 20.3. The Electric Field Electric field E at r = Electric force on unit point charge at r. E 1 F q F = electric force on point charge q. E = F/q g = F/m [E]=N/C =V/m V = Volt Implicit assumption: q doesn’t disturb E. Rigorous definition: E lim Gravitational field Electric field q0 1 F q Force approach: Charges interact at a distance (difficult to manage when many charges are present). Fails when charge distributions are not known. Field approach: Charge interacts only with field at its position. No need to know how field is generated. Given E: FqE The Field of a Point Charge E 1 F qtest Field at r from point charge q : 1 k q qtest E qtest r 2 Field vectors for a negative point charge. kq ˆr 2 rˆ r 20.4. Fields of Charge Distributions Superposition principle E Ei (Discrete sources) i i k qi rˆi 2 ri (Point charges) The Electric Dipole Electric dipole = Two point charges of equal magnitude but opposite charges separated by a small distance.. Examples: Polar molecules. Heart muscle during contraction Electrocardiograph (EKG) Radio & TV antennas. H2O Example 20.5. Modeling a Molecule A molecule is modeled as a positive charge q at x = a, and a negative charge q at x = a. Evaluate the electric field on the y-axis. Find an approximate expression valid at large distances (y >> a). y Ey k q y k q y 2 0 2 r r r r Ex k q a k q a 2k qa r2 r r2 r r3 E2 E Q=1 E1 r r y q x1 = a q x2 = a x a 2k qa 2 y 2k qa y 3 2 3/2 (y >> a) Dipole ( q with separation d ): E qd r3 for r >> d = 2a Typical of neutral, non-spherical, charge distributions ( d ~ size ). Dipole moment : p = q d. On perpendicular bisector: On dipole axis: d = vector from q to +q y k p k E 3 ˆi 3 p y y E2 E 2k pˆ E i 3 x Q=1 E1 r r y (Prob 習題 51) q x1 = d/2 p q x2 = d/2 x Continuous Charge Distributions All charge distributions are ultimately discrete ( mostly protons & electrons ). Continuum approximation: Good for macroscopic bodies. Volume charge density [ C/m3 ] Surface charge density [ C/m2 ] Line charge density [ C/m ] E d E k dq rˆ 2 r Example 20.6. Charged Ring A ring of radius a carries a uniformly distributed charge Q. Find E at any point on the axis of the ring. By symmetry, E has only axial (x-) component. Ex dEx Ring E a kQx 2 x 2 3/2 ˆi Ring kx a2 x 2 3/2 Ring k dq x r2 r dq a kQx 2 x 2 3/2 On axis of uniformly charged ring Example 20.7. Power Line A long electric power line running along the x-axis carries a uniform charge density [C/m]. Find E on the y-axis, assuming the wire to be infinitely long. dEy y By symmetry, E has only y- component. dE dE Ey dEy Line P r r y dq Line k dx y k dq y r 2 r Line r 2 r k y x dq x y dx 2 x 2 3/2 k y y 2 x 2 2 y x 1 1 k y 2 2 2 k y y y 2k E ρˆ Perpendicular to an infinite wire 20.5. Matter in Electric Fields Point Charges in Electric Fields Newton’s 2nd law a q E m (point charge in field E) Trajectory determined by charge-to-mass ratio q/m. Constant E constant a. E.g., CRT, inkjet printer, …. Uniform field between charged plates (capacitors). Example 20.8. Electrostatic Analyzer Two curved metal plates establish a field of strength E = E0 ( b/r ), where E0 & b are constants. E points toward the center of curvature, & r is the distance to the center. Find speed v with which a proton entering vertically from below will leave the device moving horizontally. Too fast, hits outer wall For a uniform circular motion: Too slow, hits inner wall v2 b m e E0 r r v e E0 b m Dipoles in Electric Fields Uniform E: F q E q E 0 Total force: Torque about center of dipole: τ d d pˆ q E pˆ q E d q pˆ E 2 2 τ pE Work done by E to rotate dipole : W f i qE sin qE sin Potential energy of dipole in E (i = /2) p q d pˆ = dipole moment f W F tˆ r d i d d p E 2 f i t // tangent sin d p E cos f cos i U W p E cos f p E ( U = 0 for p E ) Non-uniform field: Total force: F q E q E 0 Example: dipole-dipole interaction | F | > | F+ | Force on end of B is stronger; hence net force is toward A c.f. Van der Waals interaction, long range part. Application: Microwave Cooking & Liquid Crystals Microwave oven: GHz EM field vibrates (dipolar) H2O molecules in food heats up. Liquid Crystal Display (LCD) dipolar molecules aligned but positions irregular Exploded view of a TN (Twisted Nematic) liquid crystal cell showing the states in an OFF state (left), and an ON state with voltage applied (right) Conductors, Insulators, & Dielectrics Bulk matter consists of point charges: e & p. Conductors: charges free to move ( electric currents ), e.g., e (metal), ion ( electrolytes ), e+ion (plasma). Insulators: charges are bounded. Dielectrics: insulators with intrinsic / induced dipoles. internal field from dipoles Induced dipole Alignment of intrinsic dipoles.