Lecture 26: Time Varying Electric and Magnetic Fields - Prof. Phillip Duxbury, Study notes of Physics

Download Lecture 26: Time Varying Electric and Magnetic Fields - Prof. Phillip Duxbury and more Study notes Physics in PDF only on Docsity! PHY481 - Lecture 26 Sections 8.3, 10.2.1, 11.5 of PS, Section 7.3, 9.2 of Griffiths A. The effect of time varying electric and magnetic fields Electrostatics is the study of time independent (static) electric fields. Magnetostatics is the study of time independent (static) magnetic fields. The source of static electric fields is the fundamental charge, q of elementary particles, for example the electron charge e. The sources of magnetic fields are: (i) The fundamental or intrinsic magnetic moment of elementary particles - the magnitude of the magnetic moment of elementary particles, g q 2m s, where g is the g-factor, m is the mass and s is the spin of the particle. e.g. for the electron s = h̄/2 and q = e. (ii) Steady state currents (DC). Note that the intrinsic magnetic moment formula g q 2m s applies to elementary particles, ie quarks, leptons, photons, gluons etc. For example the neutron is neutral but it has a magnetic moment because it is a composite particle made up of quarks. In fact the magnetic moment of the neutron is negative so that the neutron spin aligns in the opposite direction to the magnetic field. The fact the neutron has a finite magnetic moment and yet is not charged is very important to its use in scattering studies of materials, particularly polymeric materials, biological materials and magnetic materials. The effects of time varying electric and magnetic fields are included through two quite simple physical observations: 1. A time varying magnetic flux leads to an induced electric field (Faraday) 2. A time varying electric flux leads to an induced magnetic field (Maxwell’s displacement current term) Faraday’s law takes the forms, induced emf =  = −∂φB ∂t or ∫ ~E · d~l = −∂φB ∂t or ~∇∧ ~E = −∂ ~B ∂t (1) This equation states that a time varying magnetic flux induces a voltage or electromotive force. The minus sign on the RHS is understood through Lenz’s law which states that the induced emf acts to oppose the change in flux. Example of Faraday’s law Consider the simplest time-varying magnetic field ~B = (c1 + c2t)k̂ that is uniform in space i.e. does not dependent on x, y, z. Place a circular conducting ring in this magnetic 1 field with its unit normal along the z-direction i.e. the vector area of the ring is ~a = πR2k̂. Faraday’s law states that there is an induced emf in the ring, given by, emf = −∂φB ∂t = − ∂ ∂t ∫ ~B · d~a = −∂(c1 + c2t)πR 2 ∂t = −c2πR2 (2) Some questions: - where is this emf? - does the emf exist if there is no conducting current ring? - what direction is the emf? - which direction does the current flow? Maxwell’s displacement current term is an additional source term added to Ampere’s law of magnetostatics namely,∮ ~B · d~l = µ0i+ µ00 ∂φE ∂t ; ~∇∧ ~B = µ0~j + µ00 ∂ ~E ∂t (Ampere−Maxwell law) (3) Maxwell’s term describes the fact that a time varying electric field induces a magnetic field. Maxwell noticed that when a capacitor is charging, there is a logical inconsistency in Ampere’s law. To understand this inconsistency, consider an initially uncharged capacitor connected to a voltage source at t = 0. For simplicity, we consider a parallel plate capacitor. Current begins to flow in the circuit at t = 0, charging up the capacitor. Now we can construct a loop around the wire in the circuit. However this loop does not really enclose the current in the wire. The loop can pass through the capacitor without cutting the wire. Therefore when the capacitor is charging, Amperes law would state that there is no enclosed current and hence the magnetic field is zero. This is wrong. There is a magnetic field produced by the current. Maxwell resolved this difficulty by adding a new term which includes the effect of the electric field which builds up between the capacitor plates. His idea was to related this electric field to the current flowing the circuit. From Gauss’ law, we have, dφE dt = 1 0 dq dt (4) or the ”Maxwell displacement current” is, id = 0 dφE dt (5) This relates the current flowing into the capacitor to the electric field between the plates. Maxwell realized that if Amperes law is modified to,∮ ~B · d~l = µ0(i+ id) = µ0i+ µ00 dφE dt (6) 2