11/16/2016 OUTLINES CHAPTER 5 11/16/2016 A brief history of magnetostatics MAGNETOSTATICS The Lorentz Force Law 5 ChapterMagnetostatics The concept of current Lee Chow The Biot-Savart Law Department of Physics University of Central Florida The divergence and curl of B field Orlando, FL 32816 The magnetic vector potential 2 A brief history of magnetostatics In 1820, Hans Oersted discovered the 11/16/2016 relationship between electric current and 11/16/2016 magnetic field, namely a current The Chinese compass was invented carrying wire will deflect the needle of a around 4th century BC. Chapter 5 5 ChapterMagnetostatics compass. 5 ChapterMagnetostatics In the same year, Jean-Baptiste Biot and Felix Savart found that the magnetic In 1600, William Gilbert field varies inversely with the distance published “De Magnete”, one of from the wire and curl around the the first book on electricity current. The equation describing the and magnetism. Gilbert was magnetic field generated by a current regarded by some as the father bears their names. of electricity and magnetism. ′ 3 4 Ampere learned about Oersted’s result in Then in 1831, Michael Faraday proposed 1820 and started to develop a theory to that change in the magnetic flux can also understand the relationship of electricity 11/16/2016 produce electric field----Faraday’s law of 11/16/2016 and magnetism. Ampere showed that two induction. The “field line” concept also parallel wires carrying current will attract came from Faraday. Chapter 5 5 ChapterMagnetostatics 5 ChapterMagnetostatics or repel each other depending on whether the currents are in the same or opposite direction. · In 1826, Ampere discovered the Ampere’s law. James Maxwell in 1864 published his Maxwell’s equations which combined Gauss’s law, Faraday’s law, and the · modified Ampere’s law. Maxwell coined the phrase “the electromagnetic field” for his theory. 5 6 1 11/16/2016 Let’s put things into perspective. Maxwell Equations 11/16/2016 Electrons were discovered by J. J. 11/16/2016 Thomson in 1897. He found that the • ∮ · · cathode rays in a cathode ray tube 5 ChapterMagnetostatics travel further in air than ions. He 5 ChapterMagnetostatics estimated that the mass of cathode rays • ∮ · · to be at least 1000 times lighter than hydrogen. • ∮ · In 1887, Michelson and Morley performed the experiment bears their • ∮ · name. They show that there is no “aether” and pave the way for the development of special relativity. 7 8 The Lorentz force law Example 5.2 Cycloid motion A uniform magnetic field is in the x-direction while When a charge q moving in a magnetic field , it 11/16/2016 11/16/2016 experiences a force called the Lorentz force another uniform electric field is in the z-direction as shown. Find the trajectory of a charge particle starting at Chapter 5 5 ChapterMagnetostatics rest at the origin. 5 ChapterMagnetostatics · Initially at rest at the origin, the charge will experience a force in the z- Since the Lorentz force is always perpendicular to the velocity, therefore no work done by Lorentz force. direction , so initial , , velocity will be in the z- direction. The Lorentz If both E and B are present force will move the charge in the y-direction. 9 10 The solution to the previous equation can be ̂ = ̂ 11/16/2016 11/16/2016 y(t) = Substitute back into (2), we arrive at 5 ChapterMagnetostatics 5 ChapterMagnetostatics Let , (1) sin Apply the initial conditions, at t = 0, , . Take derivative of (1) and then substitute into (2), then and integrate over time once, we end up and 11 12 2 11/16/2016 Current & Current density 11/16/2016 Current in a wire (1D) is defined as 11/16/2016 the amount of charges passing through a point per unit time. Let 5 ChapterMagnetostatics 5 ChapterMagnetostatics Ampere = Coulomb per second If line charge density is and velocity is V, R This is the equation for a circle of radius Since the current is almost always confined in a R with the center at (Rωt, R) conducting wire, we can use the direction of the wire to (Rωt, R) indicate the direction of current 13 14 A 3-D current model (Ideal gas model) The concept of Lorentz force on a “charge” can be extended to “current”, since “current” can be viewed as “charge 11/16/2016 density” times velocity. 11/16/2016 Chapter 5 5 ChapterMagnetostatics 5 ChapterMagnetostatics Since direction of I is the same as dl, and I is usually a scalar constant, 15 16 Example 5.3 When the loop starts to rise, the charge no A rectangular loop of wire, supporting a mass m, hanging longer moving horizontally, it acquires a vertically with one end in a uniform magnetic field as shown. vertical component u, so also tilts left What current would be needed to balance the gravitational force? 11/16/2016 11/16/2016 as shown. is still perpendicular to , no work done on charge q. The vertical component, qwB which can be written as 5 ChapterMagnetostatics 5 ChapterMagnetostatics The horizontal force, , so the work done by What happens if we increase the current? battery to overcome is The magnetic force now will be larger λ than gravitational force mg, and the mass will rise. This is similar to sliding a block Does that means the magnetic force up a frictionless ramp, pushing actual do work? 17 18 horizontally 3 11/16/2016 When charge flows on the surface of a conductor (two dimensional), From definition of current density we describe it by the “surface current density” K as · 11/16/2016 11/16/2016 They are From divergence theorem, we have different Chapter 5 5 ChapterMagnetostatics Chapter 5 5 MagnetostaticsChapter ⋅· When the flow of charge is 3D, we describe it by the volume From conservation of charge, the amount of charge leaving a volume current density, · · = · · ·· Combine with previous equation, we have For magneto-statics · 19 · 20 The magnetic field produced by a steady current, The Biot-Savart Law 11/16/2016 ′ 11/16/2016 Steady current is the source that produce magnetic field in magnetostatics. It plays exactly the same role as stationary charge in electrostatics. The Biot-Savart law Chapter 5 5 ChapterMagnetostatics where is the permeability of the free space. 5 ChapterMagnetostatics plays the exactly same role in magnetostatics as the Coulomb’s law which links the source to the field Electrostatics Magnetostatics . ⁄ (Speed of light) Source Stationary charges Steady current Equation Coulomb’s Law Biot-Savart Law The unit of magnetic field Field* Diverge from a point Curl around a line (MKS unit, or SI unit) · Field*---Electric field produced by one point charge, or magnetic field produced by a current on a straight wire. 21 (cgs unit) 22 · Example 5.5 Find the magnetic field a distance s from a long straight wire · · carrying a steady current I. 11/16/2016 11/16/2016 The Magnetic field is given by · Chapter 5 5 ChapterMagnetostatics 5 ChapterMagnetostatics ′ For infinite straight line, ′ ·α · Now if we have two long parallel wires carrying current · , · I and I the force between them is 1 2, and 23 24 4 11/16/2016 = The divergence and Curl of Force per unit length is given by 11/16/2016 For a straight wire carrying a current I, the magnetic field 11/16/2016 at a distance R is given by 5 ChapterMagnetostatics 5 ChapterMagnetostatics Example 5.6 Find the magnetic field above the center of a The line integral of this magnetic field is circular loop. · ′ This is Ampere’s law. This is true for any shape of closed loop around the current. 25 26 Formal derivation of and · We can show that in cylindrical coordinates We start out with the Biot-Savart Law 11/16/2016 11/16/2016 , ′ Substitute into the equation below 5 ChapterMagnetostatics 5 ChapterMagnetostatics · · ··· · · Apply product rule (4) on page 21 of Griffiths to eq. (1) above From Stoke’s theorem and definition of current density 0 0 · · · · · · Ampere’s Law in Page 57 differential form 27 · 28 First, we can switch from to and change a sign as below Next we’ll take the curl of the Biot-Savart law, ′ 11/16/2016 11/16/2016 ′ · ·′ The integrant can be re-arranged 5 ChapterMagnetostatics Next, we will look at the x-component of the above equation 5 ChapterMagnetostatics and use integration by parts () 0 0 0 0 · · · · ′ · · ′ · · · The second term on the right hand side goes to zero can be seen on next page. 29 (J is zero at large boundary) 30 5 11/16/2016 Applications of Ampere’s Law Example 5.8 Find the magnetic field of an infinite uniform surface current flowing over the xy plane. The Ampere’s law can be written in integral form 11/16/2016 11/16/2016 The way we solve this problem is · 5 ChapterMagnetostatics very similar to the infinite 5 ChapterMagnetostatics uniform surface charge problem of Example 2.4 on page 73. Ampere’s Law in magnetostatics plays exactly the same role as the Gauss’s Law in electrostatics. But Ampere’s Apply Ampere’s Law, we have law is a line integral while the Gauss’s law is a surface integral. · / Electrostatics Coulomb’s Law Gauss’s Law Magnetostatics Biot-Savart Law Ampere’s Law 31 32 Example 5.9 Find the magnetic field of a very long solenoid Magnetostatics and Electrostatics consisting of n closely wound turns per unit length. (a) First we argue that there is no magnetic field in the 11/16/2016 11/16/2016 radial direction.