Electricity and Magnetism: Induction and Faraday's Law, Exercises of Law

Download Electricity and Magnetism: Induction and Faraday's Law and more Exercises Law in PDF only on Docsity! Mohammed Q. Taha Electricity & Magnetism Induction and Inductor Faraday’s Law AC Circuits The electric current can give rise to a magnetic field, and that a magnetic field can exert a force on a moving charge. I wonder if a magnetic field can somehow give rise to an electric current… When B is parallel to the surface, =90° and B = 0. When B is perpendicular to the surface, =0° and B = BA.  B B⊥=0 A =0 B=0 B⊥ A The unit of magnetic flux is the Tm2, called a weber: 1 Wb = 1 T  m2 . In the following discussion, we switch from talking about surfaces in a magnetic field… …to talking about loops of wire in a magnetic field. Now we can quantify the induced emf described qualitatively in the previous section Experimentally, if the flux through N loops of wire changes by an amount B in a time t, the induced emf is BΔφ = - N . Δt ε This is called Faraday’s law of induction. It is one of the fundamental laws of electricity and magnetism, and an important component of the theory that explains electricity and magnetism. Not an OSE—not quite yet. Ways to induce an emf (continued): • Changing The Angle by change the orientation of the loop in the field Conceptual example: Induction Stove An ac current in a coil in the stove top produces a changing magnetic field at the bottom of a metal pan. The changing magnetic field gives rise to a current in the bottom of the pan. Because the pan has resistance, the current heats the pan. If the coil in the stove has low resistance it doesn’t get hot but the pan does. An insulator won’t heat up on an induction stove. Remember the controversy about cancer from power lines a few years back? Careful studies showed no harmful effect. Nevertheless, some believe induction stoves are hazardous. Conceptual example 21-2 Practice with Lenz’s Law In which direction is the current induced in the coil for each situation shown? (no current)BΔφ = N . Δt ε                                                                                                     (a) Find the change in flux through the coil. Initial: Bi = BA . Final: Bf = 0 . B = Bf - Bi = 0 - BA = - (0.6 T)(0.05 m)2 = -1.5x10-3 Wb . (b) Find the current and emf induced.                                                                                                     The current must flow counterclockwise to induce a downward magnetic field (which replaces the “removed” magnetic field).  initial final The induced emf is BΔφ = N Δt ε ( ) ( ) ( ) -3-1.5×10 Wb = 100 0.1 s ε = 1.5 Vε The induced current is I = = = 15 mA . R 100 Ω ε 1.5 V (c) How much energy is dissipated in the coil? Current flows “only*” during the time flux changes. E = Pt = I2Rt = (1.5x10-2 A) (100 ) (0.1 s) = 2.3x10-3 J .                                                                                                     No flux change. No emf. No current. (No work.) The energy calculated in part (c) is the energy dissipated in the coils while the current is flowing. The amount calculated in part (c) is also the mechanical energy put into the system by the force. Ef – Ei = [ Wother]I→f 0 Ef – Ei = F D F = Ef / D F = (2.3x10-3 J) / (0.05 m) F = 0.046 N See 2 slides back for F and D. To change the magnetic flux we can change: 1. the magnitude B of the magnetic field within the coil 2. the area of the coil, or the portion of that area that lies within the magnetic field (eg expanding the coil or moving it in or out of the field) 3. the angle between the direction of the field B and the area of the coil (eg by rotation of the coil) Induced electric fields As long as the magnetic field is increasing with time, the electric field will be present as circular lines (by symmetry). If the magnetic field is constant with time, there will be no electric field lines. If the magnetic field is decreasing with time, the electric field lines will be circles in the opposite direction A changing magnetic field produces an electric field Another way of stating Faraday’s Law Consider a particle of charge q0 moving around the circular path in figure. The work done in one revolution by the induced electric field is q0, where  is the induced emf, ie the work done per unit charge in moving the test charge around the path. E dl =  B changing Bd E dl dt  − =  Another way of stating Faraday’s Law Faraday’s Law We can now expand the meaning of induced emf. E dl =  This means that an induced emf can exist without the need for a current or a particle; it is the sum (by integration) of E.dl around a closed path. Which of the red loops has the largest induced emf? A self-sustaining electromagnetic wave Direction of propagation at speed Vo ave — y B Copyright © 2004 Pearson Education, Inc., publishing as Addison Wesley TABLE 29.2 Maxwell's Equations laws Mathematical Statement. What ItSays Equation Number How charges produce electric field; field lines begin and (29.2) end on charges. Gauss for E No magnetic charge; magnetic Gauss for B field lines don’t begin or end. (29.3) = d® Changing magnetic flux Faraday Ear = es dt produces electric field. (29.4) iP Electric current and changing => a E + . B-d? = pol + woe9 — electric flux produce magnetic (29.5) Ampére dt field. Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Mutual inductance • occurs when a changing current in one circuit results, via changing magnetic flux, in an induced emf and thus a current in an adjacent circuit. • Mutual inductance occurs because some of the magnetic flux produced by one circuit passes through the other circuit. Inductance The current i in the circuit causes a magnetic field B in the coil and hence a flux through the coil. When the current changes, the flux changes also and a self-induced emf appears. Inductance The magnetic flux through a circuit is related to the current in that circuit and in other circuits nearby. For a coil carrying current I, there is a magnetic field produced around it. The value of B at each point is proportional to the current. Therefore the magnetic flux through the coil is also proportional to I: Constant of proportionality The potential difference across an Inductor (b) (a) The induced current is opposite the solenoid current. Inductor coil i . . The induced magnetic field Solenoid opposes the change in flux. Bx magnetic y i field ye > > Current J + a ‘ A VL Increasing o Copyright © 2004 Pearson Education, Inc., publishing as Addison Wesley current : The induced current carries positive charge carriers to the left and establishes a potential difference across the inductor. Copyright © 2004 Pearson Education, Inc., publishing as Addison Wesley Magnetic Energy a i) R b Resistor with current i: energy is dissipated a i) Bz b Inductor with current i: energy is stored Copyright © 2004 Pearson Education, inc., publishing as Addison Wesley, Magnetic Energy What is the energy density stored in an inductor? Homework: Find this for a solenoid with N turns, area A, and length l. This is how spark plugs work. The car’s generator sends a large current through the coil (inductor). A switch in the distributor is suddenly opened, breaking the current. The induced voltage, usually a few 1000 V, appears across the terminals of the spark plug, creating the spark that ignites the fuel.