EGR-326 ELECTROMAGNETIC FIELDS COULOMB’S LAW SEPTEMBER 4, 2023 DORDT UNIVERSITY FALL 2023 DR. WYENBERG SPHERICAL CYLINDRICAL CARTESIAN ππ ππ ππ SPHERICAL CYLINDRICAL CARTESIAN ππ ππ2 + π§π§ 2 π₯π₯ 2 + π¦π¦ 2 + π§π§ 2 ππ ππ ππ ππ sin ππ π§π§ ππ cos ππ ππ tan−1 ππ ππ π§π§ ππ ππ ππ π₯π₯ ππ sin ππ cos ππ ππ cos ππ π§π§ ππ cos ππ π§π§ π¦π¦ ππ sin ππ sin ππ π§π§ ππ sin ππ π₯π₯ 2 + π¦π¦ 2 tan π§π§ π¦π¦ −1 tan π₯π₯ −1 π₯π₯ 2 + π¦π¦ 2 tan−1 π§π§ π₯π₯ π¦π¦ π§π§ π¦π¦ π₯π₯ SPHERICAL SPHERICAL ππΜ ππΜ CARTESIAN CYLINDRICAL πποΏ½ πποΏ½ πποΏ½ ππΜ πποΏ½ πποΏ½ sin ππ ππΜ + cos ππ πποΏ½ CYLINDRICAL CARTESIAN πππποΏ½ + π§π§π§π§Μ π₯π₯ π₯π₯οΏ½ + π¦π¦π¦π¦οΏ½ + π§π§π§π§Μ ππ2 + π§π§ 2 π§π§πποΏ½ − πππ§π§Μ ππ2 + π§π§ 2 πποΏ½ πποΏ½ πποΏ½ πποΏ½ π₯π₯οΏ½ sin ππ cos ππ ππΜ + cos ππ cos ππ πποΏ½ − sin ππ πποΏ½ cos ππ πποΏ½ − sin ππ πποΏ½ π§π§Μ cos ππ ππΜ − sin ππ πποΏ½ π§π§Μ π§π§Μ π¦π¦οΏ½ cos ππ ππΜ − sin ππ πποΏ½ sin ππ sin ππ ππΜ + cos ππ sin ππ πποΏ½ + cos ππ πποΏ½ π§π§Μ sin ππ πποΏ½ + cos ππ πποΏ½ π₯π₯ 2 + π¦π¦ 2 + π§π§ 2 π₯π₯ 2 π§π§ π₯π₯ π₯π₯οΏ½ + π¦π¦π¦π¦οΏ½ + π¦π¦ 2 + π§π§ 2 π₯π₯ 2 + π¦π¦ 2 − π₯π₯ π¦π¦οΏ½ − π¦π¦π₯π₯οΏ½ π₯π₯ 2 + π¦π¦ 2 π₯π₯ π₯π₯οΏ½ + π¦π¦π¦π¦οΏ½ π₯π₯ 2 + π¦π¦ 2 π₯π₯ π¦π¦οΏ½ − π¦π¦π₯π₯οΏ½ π₯π₯ 2 + π¦π¦ 2 π§π§Μ π₯π₯οΏ½ π¦π¦οΏ½ π§π§Μ π₯π₯ 2 π₯π₯ 2 + π¦π¦ 2 + π¦π¦ 2 + π§π§ 2 π§π§Μ PROBLEM 1.27 The dipole field is described as: Where A is constant, and ππ > 0. π΄π΄ π¬π¬ = 3 2 cos ππ ππππ + sin ππ ππππ ππ a) Identify the surface on which the field is entirely perpendicular to the π₯π₯π₯π₯ plane and express the field on that surface in cylindrical coordinates. b) Identify the coordinate axis on which the field is entirely perpendicular to the π₯π₯π₯π₯ plane and express the field there in cylindrical coordinates. c) Specify the surface on which the field is entirely parallel to the π₯π₯π₯π₯ plane. CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS An electric charge ππ2 separated from a charge ππ1 by a distance ππ21 will experience a force: πΉπΉβ21 = ππ1 ππ2 Μ 2 ππ21 4 ππ ππ0 ππ21 (Here πΉπΉβ21 reads “the force on charge 2 by charge 1” and ππ21 Μ reads “the unit vector pointing in the direction of charge 2 with respect to charge 1) CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS πΉπΉβ21 = ππ1 ππ2 Μ 2 ππ21 4 ππ ππ0 ππ21 Graphically, this is perhaps a little clearer. In the case that both charges are the same sign: ππ1 ππβ21 ππ2 πΉπΉβ21 CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS πΉπΉβ21 = ππ1 ππ2 Μ 2 ππ21 4 ππ ππ0 ππ21 If both charges are opposite sign, note that πΉπΉβ21 points in the opposite direction of ππ21 Μ : ππ1 ππβ21 πΉπΉβ21 ππ2 But don’t worry; we don’t have to keep track of all this; it’s encoded in the vector formula above. ππ? CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS If the second charge is absent, we can still visualize what force would be caused by charge 1 at each location in space. It would have a magnitude: ππβ?1 ππβ?1 ππ? ππ1 ππβ?1 ππ? ππ1 ππ? πΉπΉ?1 = 2 4 ππ ππ0 ππ?1 ππβ?1 ππ? ππ? CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS Without knowing what charge we will be measuring the force on, we can find the force per unit charge by dividing by the unknown charge: ππβ?1 ππβ?1 ππ? ππ1 ππβ?1 πΉπΉ?1 ππ1 = 2 ππ? 4 ππ ππ0 ππ?1 ππβ?1 ππ? CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS πΉπΉ?1 ππ1 = 2 ππ? 4 ππ ππ0 ππ?1 And then remember that this force field exists everywhere in space; no matter where we place the second charge, it will feel a force from the first charge ππ1 CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS πΈπΈ?1 = πΉπΉ?1 ππ1 = 2 ππ? 4 ππ ππ0 ππ?1 This collection of forces is a force field – specifically, we call it the Electric Field CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS πΉπΉβ?1 πΈπΈ?1 = ππ? The Electric Field has a magnitude and a direction at every point in space. CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS πΉπΉβ?1 + πΉπΉβ?2 + πΉπΉβ?3 + β― πΈπΈ? = ππ? More generally, the Electric Field is the net force on a test charge at a given location. It is caused by every other charge in the Universe pushing and pulling on the test charge. Dropping all the messy subscripts, we have: πΉπΉβ πΈπΈ = ππ Which reads: “The electric field is defined at every point in space as the net coulomb force per unit charge at that point in space caused by ALL OTHER CHARGES IN THE UNIVERSE.” CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS An electric charge of +1 C is placed at the origin. What is the electric field vector (both magnitude and direction) at every point in space in: a) Cartesian Coordinates b) Cylindrical Coordinates c) Spherical Coordinates CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS An electric charge of +1 C is placed at π§π§ = +ππ/2. Another electric charge of −1 C is placed at π§π§ = −ππ/2. What is the electric field vector (both magnitude and direction) at every point in space in: a) Cartesian Coordinates b) Cylindrical Coordinates c) Spherical Coordinates CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS An infinitely long wire has a uniform charge density of ππππ = 1 C/m. Orient your axes such that the wire is along the π§π§-axis. What is the electric field vector (both magnitude and direction) at every point in space in: a) Cartesian Coordinates b) Cylindrical Coordinates c) Spherical Coordinates CHAPTER 2 – COULOMB’S LAW AND ELECTRIC FIELDS Three electric charges of +1 C each are placed at π₯π₯, π¦π¦, π§π§ = +ππ/2 (one charge at each point). An additional three electric charges of −1 C each are placed at π₯π₯, π¦π¦, π§π§ = −ππ/2 (one charge at each point). What is the electric field vector (both magnitude and direction) at every point in space in: a) Cartesian Coordinates b) Cylindrical Coordinates c) Spherical Coordinates
