Electricity & Magnetism

A negatively charged particle -q is placed at the center of a uniformly charged ring, where the ring has a total positive charge Q. The particle, confined to move along the z axis, is displaced a small distance z along the axis (where z << R) and released. Show that the particle oscillates in simple harmonic motion with a frequency given by

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Ngac An Bang, Faculty of Physics, HUS GENERAL PHYSICS 2 Electricity & Magnetism 1  Text book: Fundamentals of Physics, David Halliday et al., 8th Edition. Physics for Scientists and Engineers, Raymond A. Serway and John W. Jewett, 6th Edition.  Instructor: Dr. Ngac An Bang Faculty of Physics, Hanoi University of Science ngacanbang@hus.edu.vn  Homework: will be assigned and may be collected.  Quizzes and Exams:  There will be at least two (02) 15-minute quizzes.  There will be a mid-term exam and a final exam.  Grading policy:  Homework and Quizzes: 20 %  Midterm exam: 20 %  Final exam: 60 % Ngac An Bang, Faculty of Physics, HUSPhysics 2 2 Electric Charge and Field Physics 2 Ngac An Bang, Faculty of Physics, HUS Lecture 1  Electric Charges  Coulomb’s Law  Electric Fields  Electric Field of a Continuous Charge Distribution  Motion of Charged Particles in a Uniform Electric Field 3 Electric Charge and Field Mother and daughter are both enjoying the effects of electrically charging their bodies. Each individual hair on their heads becomes charged and exerts a repulsive force on the other hairs, resulting in the “stand-up’’ hairdos that you see here. (Courtesy of Resonance Research Corporation) Ngac An Bang, Faculty of Physics, HUS 4 Electric Charge and Field Electric Charge Electric charge Some simple experiments demonstrate the existence of electric forces and charges Ngac An Bang, Faculty of Physics, HUS 5  There are two types of charge. Convention dictates sign of charge:  Positive charge  Negative charge  Like charges repel, and opposite charges attract. Electric Charge and Field Electric Charge Quantization of Charge  The smallest unit of “free” charge known in nature is the charge of an electron or proton, which has a magnitude of e = 1.602 x 10-19 C  Charge of any ordinary matter is quantized in integral multiples of the elementary charge e, Q = ± Ne. An electron carries one unit of negative charge, -e,  While a proton carries one unit of positive charge, +e.  Note that although quarks (u, d, c, s, t, b) have smaller charge in comparison to electron or proton, they are not free particles. Charge is quantised Ngac An Bang, Faculty of Physics, HUS 6 Electric Charge and Field Electric Charge Charge conservation • In a closed system, the total amount of charge is conserved since charge can neither be created nor destroyed. • A charge can, however, be transferred from one body to another. A universal conservation law • The β- reaction n → p + e + νe 0e = 1e -1e + 0e n(udd), p(uud) d → u + e + νe • Electron-positron annihilation e- + e+ → γ + γ • Pair production (γ-conversion) γ → e- + e+ Ngac An Bang, Faculty of Physics, HUS 7 Electric Charge and Field Electric Charge All materials acquire an electric charge  Neutral object: Total positive charge Q+= Total negative charge Q-.  Positively charged object: Q+ > Q-,  Negatively charged object: Q+ < Q-  In this part, we consider only two types of materials • Conductors: Electrical conductors are materials in which some of the electrons are free electrons that are not bound to atoms and can move relatively freely through the material; • Insulators: are materials in which electrons are bound to atoms and can not move freely through the material. Some basic concepts Ngac An Bang, Faculty of Physics, HUS 8 Electric Charge and Field Charge Manipulation Charge transfer by contact Charging Objects By Induction Ngac An Bang, Faculty of Physics, HUS 9 Electric Charge and Field Coulomb’s Law Coulomb’s Law Consider a system of two point charges, q1 and q2, separated by a distance r in vacuum.  The force F12 exerted by q1 on q2 is given by Coulomb's law  The force F21 exerted by q2 on q1 is given by  The Coulomb constant k in SI units has the value  The constant ε0 is known as the permittivity of free space and has the value r r r qqkr r qqkF  2 21 2 21 12   Ngac An Bang, Faculty of Physics, HUS 10 1221 FF   2 2 9 0 C Nm109875.8 4 1   k 2 2 12- 0 Nm C1028.854  Electric Charge and Field Coulomb’s Law Electric force Ngac An Bang, Faculty of Physics, HUS 11  The electric force between charges q1and q2 is (a) repulsive if charges have same signs (b) attractive if charges have opposite signs  The electric force is a radial force, thus, a conservative force.  More than one force, Superposition principle is applied. Electric Charge and Field Coulomb’s Law Example 1 The electron and proton of a hydrogen atom are separated (on the average) by a distance of approximately 5.3 x10-11 m. Find and compare the magnitudes of the electric force and the gravitational force between the two particles. • From Coulomb’s law, we find that the magnitude of the electric force is • Using Newton’s law of universal gravitation we find that the magnitude of the gravitational force is • The ratio of them is N102.8)m103.5( )C106.1( C Nm108975.8 8211 219 2 2 9 2 2     r ekFE Ngac An Bang, Faculty of Physics, HUS 12 N106.3)m103.5( )kg1067.1)(kg101.9( kg Nm1067.6 47211 2731 2 2 11 2       r mm GF peG 39102  G E F F  1. Does the ratio γ depend on the distance r between the electron and the proton?. 2. What is the fundamental difference between the two forces?. Questions Electric Charge and Field Coulomb’s Law Example 2 Find the force on the charge q3 assuming that q1 = -q2 = 6.0 μC, q3 = 3.0 μC, a = 2.0 x10-2 m  The total force F3 acting on the charge q3 is  The electric force F13 can be calculated as  The electric force F23 can be calculated as  Finally, 23133 FFF      ji a qqji a qqF r r qq r r r qqF     4 2 4 1 .sin.cos 24 1 ˆ 4 1 4 1 2 31 0 2 31 0 13 132 13 31 013 13 2 13 31 0 13     Ngac An Bang, Faculty of Physics, HUS 13  i a qq r r qq r r r qqF   231 0 232 23 32 023 23 2 23 32 0 23 4 1ˆ 4 1 4 1           ji a qqFFF  4 21 4 2 4 1 2 31 0 23133  Electric Charge and Field Coulomb’s Law Example 2 Find the force on the charge q3 assuming that q1 = -q2 = 6.0 μC, q3 = 3.0 μC, a = 2.0 x10-2 m  The total force F3 acting on the charge q3 is  The magnitude F3 is Angle ϕ can be calculated as          ji a qqFFF  4 21 4 2 4 1 2 31 0 23133  Ngac An Bang, Faculty of Physics, HUS 14 N0.3 4 21 4 2 4 1 2/122 2 31 0 3             a qqF  0 3 3 3.151 1 4 2 4/2 tan         x y F F Electric Charge and Field Electric Field Defining the electric field Ngac An Bang, Faculty of Physics, HUS  What is the mechanism by which one particle can exert a force on another across the empty space between particles?  Suppose a charge is suddenly moved. Does the force exerted on a second particle some distance r away change instantaneously? 15  A charge produces an electric field everywhere in space.  The force is exerted by the field at the position of the second charge.  The field propagates through space at the speed of light.  It’s a vector field. Electric Charge and Field Electric Field Defining the electric field Ngac An Bang, Faculty of Physics, HUS 16 The electric field vector at a point in space is defined as the electric force acting on a positive test charge q0 placed at that point divided by the test charge: E  F  0q FE   The SI unit of the electric field is N/C Electric Charge and Field Electric Field Electric field of a point charge An electric charge q produces an electric field everywhere.  If we put a positive test charge q0 at any point P a distance r away from the point charge q, the electrostatic force exerts on a test charge is  The electric field created by the charge q at point P is E  F  Ngac An Bang, Faculty of Physics, HUS 17 r r r qqF  2 0 04 1   E  r r r q q FE  2 00 4 1   Electric Charge and Field Electric Field Field lines 1.The electric field vector is tangent to the electric field line at each point 2.Field lines point away from positive charges and terminate on negative charge 3.Field lines never cross each other 4. The number of lines per unit area through a surface perpendicular to the lines is proportional to the magnitude of the electric field in a given region. Ngac An Bang, Faculty of Physics, HUS 18 Electric Charge and Field Electric Field Superposition principle At any point P, the total electric field due to a group of source charges equals the vector sum of the electric fields of all the charges.  If we place a positive test charge q0 near n point charges q1, q2, q3 , qn, then the net force F0 from n point charges acting on the test charge is  By definition, the electric field E at the position of the test charge is Ngac An Bang, Faculty of Physics, HUS 19    n i in FFFFFF 1 003020100 ...      n i i n i i n i i E q F q F q FE 11 0 0 0 1 0 0 0   Electric Charge and Field Electric Field Electric dipole An electric dipole is defined as a positive charge +q and a negative charge -q separated by a distance d. For the dipole shown in this figure, 1. Find the electric field E at P due to the dipole, where P is a distance y from the origin. 2. Find the electric field E at Q due to the dipole, where Q is a distance x from the origin. Ngac An Bang, Faculty of Physics, HUS 20 Electric Charge and Field Electric Field Electric dipole 1. Find the electric field E at P due to the dipole, where P is a distance y from the origin. Answer  The electric field E+ at P due to the charge +q  The electric field E - at P due to the charge –q  The electric field E at P due to the dipole    jsinicos24 1 4 1 220 2 0          yd q r r r qE Ngac An Bang, Faculty of Physics, HUS 21    jsinicos24 1 ' ' '4 1 220 2 0          yd q r r r qE       iyd d yd qE i yd qEEE   2/1 222 2 0 220 2 22 2 4 1 .cos2 2 4 1                       iyd qdE  2/3 220 2 4 1       Electric Charge and Field Electric Field Electric dipole moment  Definition of electric dipole moment:  The electric field E at P due to the dipole  In case of y >> d dqPe   Ngac An Bang, Faculty of Physics, HUS 22     2/32202/3220 24 1 2 4 1           yd Pi yd qdE e   3 04 1 y PE e    Electric Charge and Field Electric Field of a Continuous Charge Distribution Superposition principle Continuous charge distribution • Volume charge density • Surface charge density • Linear charge density Ngac An Bang, Faculty of Physics, HUS 23 Electric Charge and Field Superposition principle Continuous charge distribution • Charge distribution • Electric field at P due to Δq • Superposition   Vi i qQqQ d Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 24 r r r dqEd r r r qE i i i i i  2 0 2 0 4 1 4 1     EdEEE i i  Electric Charge and Field Electric Field of a Rod A non-conducting rod of length l with a uniform positive charge density λ and a total charge q is lying along the x-axis, as illustrated in figure. 1. Calculate the electric field at a point P(x0,0) located along the axis of the rod. 2. Calculate the electric field at a point Q(0,y0) located along its perpendicular bisector. Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 25 Electric Charge and Field Electric Field of a Rod A non-conducting rod of length l with a uniform positive charge density λ and a total charge q is lying along the x-axis, as illustrated in figure. 1. Calculate the electric field at a point P(x0,0) located along the axis of the rod. dxdq  i xx dqEd  2 00 )(4 1   Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 26 i lx qi lx lE i xx dxEdE l l   )4(4 1 )4(4 1 )(4 22 00 22 00 2/ 2/ 2 00           i x qE lx  2 00 0 4 1    Point charge Electric Charge and Field Electric Field of a Rod A non-conducting rod of length l with a uniform positive charge density λ and a total charge q is lying along the x-axis, as illustrated in figure. 2. Calculate the electric field at a point Q(0,y0) located along its perpendicular bisector.  jcosisin)(4 1 2200      xy dxEd Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 27  jcosisin)(4 1' 2200      xy dxEd   yyx EdEdEdEdE  )( j xy dxdEEdE y  .)( cos 4 j.cos 22 00    Electric Charge and Field Electric Field of a Rod j xy dxE  .)( cos 4 2200      )tan1( cos tan 22 0 2 0 2 2 0 0       yyx dydxyx Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 28 j y j y dyE  max 00 222 00 0 sin.2 4 1 cos)tan1( .cos 4 max max           j lyy lj ly l y E  44 1 4 2/ .2 4 1 22 000 22 000       j lyy qE  44 1 22 000    jy qE ly  2 00 0 4 1    Point charge Electric Charge and Field Electric Field of a Circular Arc Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 29 Electric Charge and Field Electric Field on the Axis of a Ring A non-conducting ring of radius R with a uniform charge density λ and a total charge Q is lying in the xy-plane, as shown in figure. Compute the electric field at a point P, located at a distance z from the center of the ring along its axis of symmetry.  Let’s consider a small length element dl on the ring. The amount of charge contained within this element is dq = λdl  The electric field dE created by the charge dq at point P is Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 30 r r r dqEd  2 04 1    EdEdEd z   2204 1 zR dqdE   Electric Charge and Field Electric Field on the Axis of a Ring  Using the symmetry argument illustrated in this figure, we see that the electric field at P must point in the z+ direction.  Upon integrating over the entire ring, we obtain     zz EdEdEdEdE  Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution 31 ndl zR zE n zR z zR dqE ndEEdE ring ring z    .)(4 1 )()(4 1 .cos. 2/322 0 2/12222 0           n zR QzE   2/322 0 )(4 1   Electric Charge and Field Electric Field on the Axis of a Ring  The electric field at point P a distance z from the center of the ring along its axis • At the center O: z = 0, E = 0 • In the limit z >> R :  Graphical representation n zR QzE   2/322 0 )(4 1   Ngac An Bang, Faculty of Physics, HUS -10 -8 -6 -4 -2 0 2 4 6 8 10 -0.4 -0.2 0.0 0.2 0.4 E(z )/E 0 z/R Electric Field of a Continuous Charge Distribution 32 n z Q zE   2 04 1)(   Point charge 2 0 0 4 1 R QE      2/320 1/ /)(  Rz RzEzE E(z )/E 0 Electric Charge and Field Electric Field due to a Charged Disk Ngac An Bang, Faculty of Physics, HUS Electric Field of a Continuous Charge Distribution A circular plastic disk of radius R that has a positive surface charge of uniform density σ on its upper surface is shown in the figure on the right. What is the electric field at point P, a distance z from the disk along its central axis? 33 k Rz zE ˆ1 2 220       Electric Charge and Field A point charge in an electric field A particle of charge q and mass m is placed in an electric field E, the electric force exerted on the charge is  If this is the only force exerted on the particle, it must be the net force and causes the particle to accelerate according to Newton’s second law  If the particle has a positive charge, its acceleration is in the direction of the electric field.  If the particle has a negative charge, its acceleration is in the direction opposite the electric field. EqF   amEqF    Ngac An Bang, Faculty of Physics, HUS Motion of Charged Particles in an Electric Field 34 Electric Charge and Field A point charge in a uniform electric field An electron enters the region of a uniform electric field as shown in the figure below, with initial velocity vi = 3.00 x106 m/s and E = 200 N/C. The horizontal length of the plates is l = 0.100 m. A. Find the acceleration of the electron while it is in the electric field. B. If the electron enters the field at time t = 0, find the time at which it leaves the field. C. If the vertical position of the electron as it enters the field is yi = 0, what is its vertical position when it leaves the field. Ngac An Bang, Faculty of Physics, HUS Motion of Charged Particles in an Electric Field 35 Electric Charge and Field A point charge in an electric field A negatively charged particle -q is placed at the center of a uniformly charged ring, where the ring has a total positive charge Q. The particle, confined to move along the z axis, is displaced a small distance z along the axis (where z << R) and released. Show that the particle oscillates in simple harmonic motion with a frequency given by Ngac An Bang, Faculty of Physics, HUS Motion of Charged Particles in an Electric Field 36 3 04 mR qQ    Electric Charge and Field Motion of Charged Particles in an Electric Field An electric dipole in an electric field • The total net force • The field exerts a torque on the dipole. The torque about the midpoint O of the dipole is • Using the definition of the electric dipole, We have the general expression for the torque 0)(   EqEqFFF  )(sin kdFFrFr     Ngac An Bang, Faculty of Physics, HUS 37 Epe   Ngac An Bang, Faculty of Physics, HUS That’s enough for today  Please try all the example problems given in your textbook. Few more problems will be given to you.  Feel free to contact me via email. Electric Charge and Field 38 See you all next week!

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