Physics 220 Lecture 1: Electric Charge - PDF

Welcome… …to Physics 220.  Physics Learning Center Attending the PLC is voluntary open learning environment but very encouraged When: to follow faculty and peer learning Where: to follow assistants provide problem...

Welcome… …to Physics 220.  Physics Learning Center Attending the PLC is voluntary open learning environment but very encouraged When: to follow faculty and peer learning Where: to follow assistants provide problem solving and homework help Lecture 1 agenda: Electric Charge. Review of some things you hopefully learned in high school. Coulomb’s Law (electrical force between charged particles). You must be able to calculate the electrical forces between one or more charged particles. The electric field. You must be able to calculate the force on a charged particle in an electric field. Electric field due to point charges. You must be able to calculate electric field of one or more point charges. Motion of a charged particle in a uniform electric field. You must be able to solve for the trajectory of a charged particle in a uniform electric field.  Electric Charge What is charge? basic property of matter (just like, say, mass) humans cannot directly sense charge but some animals can we can observe charge indirectly via its effects on matter customary symbol: q or Q, unit: [q] = C (Coulomb) Two kinds of charge: + - like charges repel unlike charges attract Law of conservation of charge: net amount of charge does not change in any process (Not on your starting equation sheet, but a fact that you can use any time.) Charges are quantized (come in units of e= 1.6x10-19 C). Elementary particles that make up atoms: Protons + charge +e = +1.6x10-19 C Neutrons uncharged Electrons - charge –e = –1.6x10-19 C Helium atom Lecture 1 agenda: Electric Charge. Review of some things you hopefully learned in high school. Coulomb’s Law (electrical force between charged particles). You must be able to calculate the electrical forces between one or more charged particles. The electric field. You must be able to calculate the force on a charged particle in an electric field. Electric field due to point charges. You must be able to calculate electric field of one or more point charges. Motion of a charged particle in a uniform electric field. You must be able to solve for the trajectory of a charged particle in a uniform electric field.  Coulomb’s Law Force between two point charges q1 and q2: force is vector, directed along connecting line  magnitude: q1q 2 F k 2 12 r12 r12 is the distance between the charges 2 9 Nm 1 C2 k 9 10 2  with  0 8.85 10  12 2. C 4 0 Nm  a note on starting equations q1q 2 F k 2 is an official starting equation 12 r12 in homework and exam solutions, official starting equations can be used without deriving them all other steps of a solution need to be derived you may formulate the official starting equation in different variables, QA QB for example the equation F k is “legal” E 2 D *“Starting” does not mean that a starting equation has to be the first thing that appears on your paper. It might be several lines before you use a starting equation. Coulomb’s Law is strictly valid for point charges only. It is a good approximation for small uniformly charged objects. r12 + - If more than two charges are involved, the net force is the vector sum of all forces (superposition). For objects with complex shapes, you must add up all the forces acting on each separate charge (calculus!!). + - + - + - Example: a positive charge Q1 = +Q is located a distance d along the y-axis from the origin. A second positive charge Q2 = +Q is located at the origin and a negative charge Q3 = -2Q is located on the x-axis a distance 2d away from Q1. Calculate the net electrostatic force on Q1 due to the other two charges. Example: a positive charge Q1 = +Q is located a distance d along the y-axis from the origin. A second positive charge Q2 = +Q is located at the origin and a negative charge Q3 = -2Q is located on the x-axis a distance 2d away from Q1. Calculate the net electrostatic force on Q1 due to the other two charges. y Q1=+Q d 2d Q2=+Q Q3=-2Q x Calculate the net electrostatic force on Q1 due to the other two charges.  y F F2  F3  q1q2 ˆ +Q +Q  ˆ Q2 ˆ F2 k 2 j k 2 j k 2 j F2 r12 d d  Q1=+Q F3 F3xˆi  F3yˆj  F3  d 2d F3 F3 cos ˆi  F3 sin ˆj  Q2=+Q x  q1q3 q1q3 Q3=-2Q F3 k 2 cos i  k 2 sin ˆj ˆ q1q2 r13 r13 F12 =k 2 r12  +Q -2Q  3 ˆ +Q -2Q  1 ˆ 1 3 F3 k i  k j 2d 2d 2 2 sin   cos  2 2 2 2 Note: F2 and F3 are not drawn to scale (F3 is “too long”). Calculate the net electrostatic force on Q1 due to the other two charges. y  +Q -2Q  3 ˆ +Q -2Q  1 ˆ F3 k i  k j 2d 2d 2 2 2 2 F2  2Q2 3 ˆ 2Q2 1 ˆ F3 k 2 i  k 2 j Q1=+Q 4d 2 4d 2  F3  3 kQ2 ˆ 1 kQ2 ˆ d F3  i  j 2d 2 2 4 d 4 d  Q2=+Q x Q3=-2Q  Q2 ˆ 3 kQ2 ˆ 1 kQ 2 ˆ F F2  F3 k 2 j+ i  j q1q2 d 4 d2 4 d2 F12 =k 2 r12 1 3  3 kQ2 ˆ 3 kQ2 ˆ sin   cos  F i  j 2 2 4 d 2 4 d2 Note: F2 and F3 are not drawn to scale (F3 is “too long”). Comments: Once you have become an expert at problems like this, you can combine and perhaps even skip some steps. Skipping steps on work to be graded is not recommended! You may express your answer in unit vector notation, as on the previous slide. 3 kQ2 3 kQ2 Or you may write Fx  2  Fy  2  4 d 4 d You may also express your answer as a magnitude and direction. All three of the above ways of writing F completely specify the vector.  y F2 F Q1=+Q  F3 d 2d  Q2=+Q Q3=-2Q x If Q1 were free to move, what direction would its initial acceleration be? How would I calculate the acceleration? Would the acceleration remain constant as Q1 moved? Could I use the equations of kinematics (remember them from Physics 110?) to describe the motion of Q1? Lecture 1 agenda: Electric Charge. Just a reminder of some things you learned back in grade school. Coulomb’s Law (electrical force between charged particles). You must be able to calculate the electrical forces between one or more charged particles. The electric field. You must be able to calculate the force on a charged particle in an electric field. Electric field due to point charges. You must be able to calculate electric field of one or more point charges. Motion of a charged particle in a uniform electric field. You must be able to solve for the trajectory of a charged particle in a uniform electric field.  Coulomb’s Law: it’s just part of a bigger picture Coulomb's Law: 1 q1q 2 r12 F = , + - 12 4πε 0 r12 2 Q1 Q2 Charged particles produce forces over great distances. How does a charged particle "know" another one is “there?” Introduce concept of electric field new way of thinking about the Coulomb force  The Electric Field a charged particle creates a F12 “field” in all space. + other charged particles F13 sense the field and experience a force in F31 response + like - Distinguish source charges charges F21 unlike repel charges and test charges attract A charged particle modifies the properties of the space around it. Definition of electric field: one or more source charges  E define the electric field via force they exert on a test charge q0:   F0 E= The subscript “0” reminds you the force is on the “test charge.” I won’t require the q0 subscripts when you use this equation for boardwork or on exams.  F = qE This is your second starting equation. It is a vector equation that tells you magnitude and direction of the force!     F0  N The units of electric field are  E  =   =    q0  C newtons/coulomb. In another chapter, you will learn that the units of electric field can also be expressed as volts/meter: N V  E = = C m The electric field can exist independent of whether there is a charged particle around to “feel” it. Remember: the electric field direction is + the direction a + charge would feel a force. A + charge would be repelled by another + charge. Therefore the direction of the electric field is away from positive (and towards negative). http://regentsprep.org/Regents/physics/phys03/afieldint/default.htm  Gravitational Fields The idea of a field is not new to you. You experienced fields (gravitational) in Physics 110.  m1m 2 FG =G 2 , attractive r12   FG Units of g g(r) = are actually m N/kg!  g(r) is the local gravitational field. On earth, it is about 9.8 N/kg, directed towards the center of the earth. A particle with mass modifies the properties of the space around it. Lecture 1 agenda: Electric Charge. Review of some things you hopefully learned in high school. Coulomb’s Law (electrical force between charged particles). You must be able to calculate the electrical forces between one or more charged particles. The electric field. You must be able to calculate the force on a charged particle in an electric field. Electric field due to point charges. You must be able to calculate electric field of one or more point charges. Motion of a charged particle in a uniform electric field. You must be able to solve for the trajectory of a charged particle in a uniform electric field.  The Electric Field Due to a Point Charge Coulomb's law says q1q 2 F =k 2 , 12 r12 treat q1 as source charge and q2 as test charge, divide by q2, the electric  field due to point charge q1 is q1 q |E q1 |=k 2 or, E=k 2 r12 generally r This is your third starting equation. If we definer̂ as a unit vector from the source point to the field point… source point r̂ + field point …then the vector equation for the electric field of a point charge becomes: You may start with either  q equation for the electric field E=k 2 rˆ (this one or the one on the r previous slide). But only use this one if you REALLY know what you are doing! Example: calculate the magnitude of the electric field at the electron’s distance away from the proton in a hydrogen atom (5.3x10-11 m). +e -e EP + - D kq k(+e) 9 109 1.6 10 19  N EP   2  r2 D 5.3 10   11 2 C 11 N E P 5.110 C For comparison, air begins to break down and conduct electricity at about 30 kV/cm, or 3x106 V/m.  A Dipole A combination of two electric charges with equal magnitude and opposite sign, separated by a fixed distance, is called a dipole. +q + - -q d The distance between the charges is d. Dipoles are “everywhere” in nature. This is an electric dipole. Later in the course we’ll study magnetic dipoles.  The Electric Field of a Dipole Example: calculate the electric field at point P, which lies on the perpendicular bisector a distance L from a dipole of charge q. P L +q + - -q d Example: calculate the electric field at point P, which lies on the perpendicular bisector a distance L from a dipole of charge q. y  E+ E = E  E P   E y = 0 (symmetry ) E x = 2E ,x (symmetry E- ) r L r E x = +2E  cos    +q + - -q x d Example: calculate the electric field at point P, which lies on the perpendicular bisector a distance L from a dipole of charge q. y E x = +2E  cos  E+  P d/2 d  E x = +2E  = +E  r r E- r L r k q d kqd Ex =+ 2 = 3 r r r  d/2 d/2   qd ˆ +q + - -q x E = 3 i 4 0 r d “Charge on dipole” is positive by convention, so no absolute value signs needed around q.  P E qd E 4 o r 3 L Caution! The above equation for E applies only to points along the +q + - -q perpendicular bisector of the d dipole. It is not a starting equation. (r is not a system parameter, but let’s not worry about that right now) Lecture 1 agenda: Electric Charge. Review of some things you hopefully learned in high school. Coulomb’s Law (electrical force between charged particles). You must be able to calculate the electrical forces between one or more charged particles. The electric field. You must be able to calculate the force on a charged particle in an electric field. Electric field due to point charges. You must be able to calculate electric field of one or more point charges. Motion of a charged particle in a uniform electric field. You must be able to solve for the trajectory of a charged particle in a uniform electric field.  Motion of a Charged Particle in a Uniform Electric Field charged particle in electric field experiences force, and if it is free to move, an acceleration. - - - - - - - - - - - - - if the only force is due to the electric field - E   F  F ma qE. + + + + + + + + + + + + + E is constant a constant, use the equations of kinematics* (Physics 110?). *If you get called to the board, you can use the Physics 1135 starting equations. They are posted. Example: an electron moving with velocity v0 in the positive x direction enters a region of uniform electric field that makes a right angle with the electron’s initial velocity. Express the position and velocity of the electron as a function y of time. - - - - - - - - - - - - - x -e v0 E + + + + + + + + + + + + + v fx vix  a x t 1 2 x fi x  vix t  a x t  What would be different for a proton? 2 Make sure you understand what a uniform electric field is. Express the position and velocity of the electron as a function of time.   y ˆ FE = qE = -e  E = -eEj = ma  eE ˆ a=- j x m -e v0 E FE a eE ax = 0 ay = - m Let’s work the rest of the problem one component at a time. Express the position and velocity of the electron as a function of time. eE y ax = 0 ay = - m 0 0 1 2 x = x i + vix t + a x t x 2 -e v0 E FE a x = v0 t 0 0 1 2 y = yi + viy t + a y t 2 Position: 1 eE 2 1 2 1 eE 2 x = v0 t y=- t y = a yt = - t 2 m 2 2 m Express the position and velocity of the electron as a function of time. eE y ax = 0 ay = - m 0 v x = vix + a x t x -e v0 E FE a v x = v0 0 v y = viy + a y t Velocity : eE eE v x = v0 vy = - t v y = a y t= - t m m What is the shape of the electron’s path? 1 eE 2 y x = v0 t y=- t 2 m x x t= -e v0 v0 E FE a 2 1 eE  x   1 eE  2 y=-   = - 2  x 2 m  v0   2 mv 0  The trajectory of the electron is a parabola, concave down. Just like the trajectory of a ball thrown horizontally in the gravitational field of the Earth.  Concluding Remarks Homework Hints (may not apply every semester) There are two kinds of electric field problems in today’s lecture:  1. Given an electric field, calculate the force F = qE on a charged particle. q 2. Given one or more charged particles, E=k r2 calculate the electric field they produce. Make sure you understand which kind of problem you are working on!  Homework Hints (may not apply every semester) Symmetry is your friend. Use it when appropriate. Don’t use it when not appropriate.  GmM FG,pair  2 , attractive r The above equation is on the Physics 110 Starting Equation Sheet, which is posted in the recitation classrooms. You are free to use Physics 110 starting equations at any time.  Homework Hints (may not apply every semester) Your starting equations so far are:  q1q 2  F0 q F k 2 E= E=k 2 12 r12 q0 r (plus Physics 110 starting equations). Remove the absolute value signs ONLY IF you know that all charges are positive.   F0 F0 NEVER do this: E =  q0 =  (why?) q0 E