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Chapter 19 Electric Charge and Electric Field ▲ When a comb has been passed through hair, it can attract paper scraps. Why does this happen? ▲ Why atoms and molecules can be hold together to form liquids and solids? ▲ What really happens in an electric circuit? ▲ How do electric motors and generators works? ▲ And what is light? 2 Electromagnetism J. C. Maxwell 1831-1879 ☆Electromagnetic Field ☆Gauss’s law ☆Ampère’s law ☆Faraday’s law ☆Maxwell’s equations Then God said: E 0 / 0 B E t B 0 B 0 J 0 0 E t and there was light. 3 Electric charge 1) Two types: positive & negative Unlike charges attract; like charges repel. 2) Quantized: elementary charge e 1.6 1019 C 3) Law of conservation of electric charge: The net amount of electric charge produced in any process is zero. ▲ Insulators, conductors, semiconductors 4 Coulomb’s law (1) Electric forces between two point charges: Q1Q2 1 Q1Q2 F k 2 r r 2 r 4 0 r where r is unit vector Q1 r F Q2 r in SI units: k 8.988 109 9.0 109 N m2 / C 2 0 1 4 k 8.85 1012 C 2 / N m 2 0 : permittivity of free space 5 Coulomb’s law (2) Q1Q2 1 Q1Q2 F k 2 r r 2 r 4 0 r 1) It describes force at rest, or electrostatics 2) It is valid for two point charges 3) Principle of superposition: If several charges are present, the net force on any one of them will be the vector sum of forces due to each of the others. 6 Electric field Charges interact each other by electric field Charges create electric field interact on Q1 Q2 Field is a special form of matter How to describe? by using a test charge Fb Electric field (strength): . b E F / q or E lim F / q q 0 Fc .c .a Fa Q 7 Properties of electric field E F /q F qE 1) The field doesn't depend on the test charge q 2) It is a vector field, magnitude & direction 1 Q r 3) For a point charge E 2 4 0 r 4) For more charges, total field E Ei i 8 Electric equilibrium Example1:Two charges, -Q and -3Q, are a distance l apart. How can we place a third charge nearby to reach an equilibrium? Q x Solution: Position? Q Q 1 3Q 0 x 2 2 4 0 x 4 0 (l x) 1 1 l 3Q x 3 1 l 0.366Q 2 What is the Charge? Q1 1 3Q 63 3 0 Q1 Q 0.402Q 4 0 x 2 4 0 l 2 2 1 9 Continuous charge distribution The electric field can be calculated by integral ① Divide it into infinitesimal charges dQ 1 dQ r ② Contribution from dQ: dE 2 4 0 r ③ Consider all the components dEx , dE y , dEz ④ Finish the integration: Ex dEx , E y dE y , Ez dEz 10 A line of charge Example2: Charge is distributed uniformly along a line. Find the electric field at any given point P around it. (Charge per unit length is λ) Solution: ① x-y axes dE y dEy ② dQ dx dx ③ dE 4 0 r 2 ④ dE dE cos x dE y dE sin P dEx a r o x dx dq x 11 ⑤ Ex dx 2 cos 4 0 r cos θdθ 4 0 a 2 dE dEx dE cos y dEy dE y dE sin dx dE 4 0 r 2 P dEx 1 (sin θ2 sin θ1 ) 4 0 a dx Ey sin 2 4 0 r sin θdθ 4 0 a 2 a r 1 o x dx dq 2 x r=a/sin , x=- a·cot , dx=ad /sin2 1 (cos θ1 cos θ2 ) 4 0 a E Ex i E y j 12 y Ex (sin θ2 sin θ1 ) 4 0 a Ey (cos θ1 cos θ2 ) 4 0 a Discussion: P a 2 1 o x 1) If it is very long or infinite, 1 =0, 2 = Ex 0, Ey 2 0 a → useful result 2) For surface distribution: 13 A plane of charge Question: Charge is distributed uniformly on an infinite plane. Find the electric field at any given point P around it. (Charge per unit area is σ) E Ey dx 2 0 r 2 0 2 0 d 2 2 = ·dx cos 1 dE E dx y dE P. r a o x dx x 14 Uniform charged ring Example3: A thin ring of radius R holds a total charge Q distributed uniformly. Find the electric field at point P on the axis, x from its center. dQ Solution: dE 2 4 0 r dQ E Ex cos 2 Ring 4 0 r y dQ R o r P x x dE Q cos Q x 2 4 0 r 4 0 ( x 2 R 2 )3/ 2 15 Q cos Q x E 2 4 0 r 4 0 ( x 2 R 2 )3/ 2 R o P x E Discussion: . 1) x=0 or x >> R, E=? o 2) At what position along the axis, E= Emax ? 3) If there is a small gap in the circle, E o = ? 4) If there is only a semi-circle, E o = ? 2 0 R 16 Uniform charged disk Example5: Charge is distributed uniformly over a thin disk of radius R. Determine E at point P on the axis, x from its center. (Charge per unit area is σ) Solution: dQ 2 rdr xdQ dE 4 0 ( x 2 r 2 )3 2 E R o R o P r x x dr x 2 rdr [1 2 2 3 2 2 0 4 0 ( x r ) x x2 R2 ] 17 E [1 2 0 R x x R 2 2 ] o P x x Discussion: When R →∞: Parallel-plate capacitor E 2 0 + infinite plane E=0 E 0 - E=0 18 Field lines Visualize the electric field → electric field lines 19 Properties of field lines 1) The field point in the direction tangent to the field line at any point. 2) The magnitude of E is proportional to the number of lines crossing unit area ⊥ the lines. 3) Field lines start on + charges, end on – charges. 4) Field lines never cross each other; and there are no closed field lines. 20 Electric dipoles (1) Combination of two equal charges of opposite sign is called an electric dipole. (Q Q) Dipole moment: p Ql Ea 2E cos Q l2 2 4 0 ( y ) 4 2 l 4 0 ( y )3/2 4 2 l p Ea E E l y 4 2 E p 4 0 y 3 (y .a E 2 E l) Q y l Q 21 Electric dipoles (2) Eb Q l 4 0 ( x ) 2 2 2p 4 0 x 3 o Q Eb E E (x Q l 4 0 ( x ) 2 2 l) E b . x Q E l. Ea E 2 px l2 2 4 0 ( x ) 4 2 p 4 0 y 3 (y l) Discussion: In which cases: E r 3 / r 2 / r 1 / r 0 ? 22 Dipole in external field E Q 1) The total force: F F F 0 2) Torque on the dipole: l F 1 1 F l sin θ F l sin θ 2 2 o. θ F p Q Uniform field QE l sin θ pE sin θ or p E Discussion: what is the torque on the dipole when 0, / 2, ? 23 Vibrating charge Thinking: Negative charge -Q is distributed on a ring uniformly. A positive charge q is placed from the center of ring a small distance x. Show that it will undergo SHM when released, and what is T ? Q x E 4 0 ( x 2 R 2 )3/ 2 x o q, m R x -Q 24