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1.8. Relative Motion A B Z Y X C Galilean coordinate transformation S’ motorial reference system S static reference system z r r0 r ' S r z' S' r0 P r' O' x' O y x Relative motion, in nature, is the transformation of reference frame. y' dr dr0 dr ' v v0 v ' dt dt dt vabsol ut e vconvect ed vr el at i ve z S r z' P S' r0 r' O' x' O x y y' aabsolute aconvected arelative z S r S' r0 r' O' x' Extreme case: O y x relative to S , S system does uniform motion aconvected 0 z' P aabsolute arelative a a' y' a a' The acceleration of the particle measured by an observer in one frame of reference is the same as that measured by any other observer moving with constant velocity relative to the first frame. Galilean Experiment Exercise A Boat Crossing a River A boat heading due north crosses a wide river with a speed of 10.0 km/h relative to the water. The water in the river has a uniform speed of 5.00 km/h due east relative to the Earth. Determine the velocity of the boat relative to an observer standing on either bank. is vBE v RE the velocity of the boat relative to Earth . is the velocity of the river relative to Earth. v is the velocity of the boat relative to Earth. BE vBE vBR vRE vBE v vRE 2 BR 2 11.2 (km / h) Look at the examples given in P65 to 67! 2017/5/3 9 Charles Augustin Coulomb (1736-1806) invented an instrument called the torsion balance - which later showed great utility for measuring very small electrical forces. 2017/5/3 10 Coulomb stated that the electrical force between two electrified objects is proportional to the inverse square of the distance between them and to the product P of their electrical masses. q1q2 f 2 r This achievement marked the point where electricity became thoroughly open to mathematical analysis. 2017/5/3 11 Orsted (1777-1851) was remembered as the man responsible for discovering the first experimental evidence for the relationship between electricity and magnetism. I the Magnetic Effect of Electrical Currents 2017/5/3 12 Just over a month after the discovery, Jean-Baptise Biot (1774-1862) and Felix Savart (1791-1841) were the first to provide a precise analysis of the effect. Biot and Savart announced the Biot-Savart Law which can be used to calculate the magnetic field for a segment of current carrying wire. Idl r dB 3 r 2017/5/3 13 Andre-Marie Ampere (1775-1836) was the first person to develop a technique for measuring electricity. he built an instrument which later refined being known as the galvanometer. Ampere’s theories became fundamental for 19th century developments in electricity and magnetism. Ampere’s law Ampere’s circulation theorem 2017/5/3 14 Michael Faraday (1791-1867) made one of the most important discoveries in the sciences of electricity and magnetism; namely, electromagnetic induction. James Clerk Maxwell (1831-1879) . His most important contribution was the extension and mathematical formulation of previous works on electricity and magnetism by Faraday, Ampere ad others into a linked set of partial differential equations. These equations, now collectively known as Maxwell’s Equations. 2017/5/3 15 Maxwell ’ s equations predicted that the phenomenon of light was electromagnetic in nature. Maxwell ’ s equations summarized a series of experimental laws in the field of electromagnetics. 2017/5/3 16 In 1888 the German physicist Hertz (H. Hertz, 18571894) firstly discovered the electromagnetic wave predicted by the Maxwell equations, thus the theoretical system of the classical electromagnetism was finally established. 2017/5/3 17 18 2017/5/3 Outline of Chapter 2 Electrostatic Field Two Important Physical Quantities Two Fundamental Theorems 2017/5/3 The field distributed round a static charge is called the electrostatic field. Force E Work U Field with sources conservative field 19 Gauss’s Law Circulation Theorem Main Contents Coulomb’s Law,Superposition Principle of Coulomb Force Definition and computation of E Electric Flux, Gauss’s Law Circulation Theorem of electrostatic field, Electric Potential The relationship of E and potential gradient Motion of charged particles in electric field 2017/5/3 20 Requirements Master the Calculation of Field Intensity Find the Field Intensity with Gauss Law Find Electric Potential 2017/5/3 21 §2.1 Point Charge, Coulomb’s Law 1. Properties of Electric Charges A charged body without size and shape, i.e., a charged geometrical point. (1) Two Kinds of Electric Charge Positive Charge, Negative Charge Like charges repel each other, and unlike charges attract each other. Where “like” means two charges with the same sign. 2017/5/3 22 When the girl touches a Van de Graaff generator, she received an excess of positive charge, causing her hair to stand on end. 2017/5/3 23 (2) Quantization of Electric Charge Electric Quantity e 1.6 10 19 C (3) Conservational Low of Charge The net charge in an isolated system is always conserved. Or : total charge is constant in any process. Conserved quantities: energy, momentum, angular momentum, charge…… 2017/5/3 24 (4) Relativistic Invariability of Electric Quantity The electric quantity is independent of the velocity and acceleration. This characteristics is not identical with mass. Q +++ Q 2017/5/3 25 2. Coulomb’s Law in the Vacuum French physicist. His major contributions to science were in the areas of electrostatics and magnetism. Coulomb’s Law gives the force between two charged particles at rest. Charles Coulomb (1736-1806) 2017/5/3 26 Coulomb Law in the Vacuum The direction of interaction force between two small charged balls at rest is along the line connected them, the magnitude F of interaction force is direct proportional to the product of charges q1 and q2 , and is opposite proportional to the square of distance r separated them, like charges repel each other, and unlike charges attract each other. 2017/5/3 27 Attention “vacuum” means that, except the charges considered suppose there is no any others. This law applies exactly only to particles. 2017/5/3 28 Mathematics Expression f 12q1 Vector form, q1q q12q2 f12 K r12 33 4 0rr r12 q2 r r21is a vector directed from the charge q2 toward the q1. Permittivity of vacuum, 0 8.85 10 12 C 2 / Nm 2 2017/5/3 29 Coulomb constant, k 1 40 The interaction force between point charges at rest is usually called electrostatic force or Coulomb force. When dealing with Coulomb law, remember that force is a vector quantity and must be treated accordingly. Discussion r 0 2017/5/3 F ? 30 Example The Hydrogen Atom Find the ratio of Coulomb force and the universal gravitation between electron and nucleus in a hydrogen atom. Solution The electron and proton of a hydrogen atom are separated (on average) by a distance of approximately 5.3 10 11 m,and the masses are M 1840me me 9.11 10 31 kg From Coulomb’s law, we find that the magnitude of the attractive electric force is F 2017/5/3 e2 40 r 2 8.2 108 N 31 Using Newton’s law of universal gravitation, we find that the magnitude of the gravitational force is Mme f G 2 3.6 10 47 N r The ratio f 40 GMme e 2 4.4 10 40 F Thus the gravitational force between charged atomic particles is negligible compared with the electric force. 2017/5/3 32 3. Superposition Principle of Electrostatic Force The resultant force on any one particle equals the vector sum of the individual forces due to all other particles. The resultant force on particle 1 due to particles 2 and 3 is given by the vector sum f1 f 12 f 1i i 2017/5/3 f1 33 q1 q2 + + f13 - q3 4. Find Electrostatic Force (1). Force Due to the System of Point Charges. The resultant force on particle 0 due to the other particles is given by the vector sum N q0 qi ri F 3 i 1 4 0 ri 2017/5/3 34 Quick Quiz There is a point charge q at every tip of the isosceles right triangle as shown in figure, AC=BC=a . Find Coulomb force on the point charge at tip C. Y +A a + B 2017/5/3 35 a + C X Key: F 2017/5/3 q 2 40 a 2 (i j ) 36 (2). Force Due to Continuous Charge Distributions dq Strategy Q 1. Divide the charge distribution into small elements, each of which contains a small amount of charge dq. r q0 + P 2. Model the element as a point charge, and find the dF at the point P due to one of these elements. 3. Evaluate the total electrostatic force at P due to the charge distribution by performing a vector sum of the contributions of all the charge elements. (i.e., by applying the superposition principle.) 2017/5/3 37 Charge Density If a total charge Q is uniformly distributed along a line of length l ,the linear charge density is defined by Q = l If Q is uniformly distributed on a surface of area A, the surface charge density is defined by Q A If Q is uniformly distributed throughout a volume V, the volume charge density is defined by Q V 2017/5/3 38 How to Represent dq lin e a r c h a rg e d e n s ity , dq dl s u rfa c e c h a rg e d e n s ity , d q d S v o lu m e c h a rg e d e n s id ity , d q d V 2017/5/3 39 Element Analysis Method dq Q dl dq ds dV Q q0dq r dF 4 0 r 3 r q0 + P q0dq r 3 4 r 0 F The integral covers the whole system of charged body. Symmetry: with both distributions of point charges and continuous charge distributions, take advantage of any symmetry in the system to simplify your calculations. 2017/5/3 40 Example As shown in figure, Electric quantities Q distribute uniformly on a fine bar of length L .Find the electrostatic force acting on the point charge q0 . Q,L b + q0 2017/5/3 q0 Q 1 1 f ( ) 4πε0 L b b L 41 Sum-up – Physical model Point charge – Coulomb law – Calculate the electrostatic force 2017/5/3 42