EM Waves and Faraday's Law: Induced EMF and Current in Conducting Wires - Prof. Phillip Du, Study notes of Physics

Download EM Waves and Faraday's Law: Induced EMF and Current in Conducting Wires - Prof. Phillip Du and more Study notes Physics in PDF only on Docsity! PHY481 - Lecture 27 Sections 10.1-10.3 PS, 7.2 of Griffiths In the last lecture we went through the most remarkable consequence of including dynamics in the study of electricity and magnetism - EM waves. It is important to emphasize that the although the relations between time varying magnetic flux and induced emf (Faraday’s law) and between time varying electric flus and induced magnetic field (Maxwell displacement term), both involve loops, these terms exist regardless of whether there is a material loop in the experiment. Today we look at some of the remarkable effects of Faraday’s law that occur when there are materials in the experiment. A. Faraday’s law and the Lorentz force law: A moving conducting rod in a constant field Faraday’s law and the Lorentz force law are closely related as can be seen by considering a conducting wire moving through a constant magnetic field. Consider a uniform and constant magnetic field, B, directed along the z-axis. Now consider moving a conducting rod, of length l, which is directed along the y-direction at constant speed v along the x-direction. First we use Faraday’s law to show that a motional emf is developed between the ends of this wire. To find this emf, consider a rectangular loop which is composed of a side of length L lying on the y-axis (centered at the origin), the moving piece of wire, and the two sides which join them to form the rectangle. These two joining sides have length L = vt, where we assume that the conducting rod starts at the origin at time t = 0. The rate of change of the flux is given by, E = −dφB dt = −BldL dt = −Blv (1) This emf is induced around the loop, however the only conducting part of the loop is the piece of wire. The charges in the piece of wire move until the voltage drop between the ends of the wire just balance the induced emf, ie. Vwire = Blv, so that the voltage inside the rod is zero. If the rod was made of an insulating material, the charges would not move and there would be a voltage across the rod. The behavior described above can also be understood from the Lorentz force law. Con- sider the wire to be composed of negatively charge carriers. The motion of these carriers in the magnetic field leads to the Lorentz force law FB = −evB. The charges build up at the 1 ends of the wire until the induced electric field produces a force on the conductors which just balances the magnetic force. We then have, eE = e Vwire l = evB so that Vwire = Blv (2) as found using Faraday’s law. A more general derivation of from the Lorentz force law as is given in PS 10.1.2. An important deduction from this example is that when an observer moves through a constant magnetic field, the observer sees an electric field. The effect of transformation to a moving co-ordinate system is then very interesting and is believed to be the way in which Einstein first started thinking about relativity. B. A conducting loop and magnetic drag Now consider a square loop of wire which lies in the x-y plane, and where each side has length l. Consider that the half space x < 0 constains a constant and uniform magnetic, B, directed along the positive z-axis. Now consider the situation in which the square loop is initially within the B field and it is drawn out of the B field at velocity v along the x-axis. If the loop has resistance R find the induced current. The rate of change of flux is given by, E = −dφB dt = Blv (3) The induced emf is then Blv. The current in the loop is thus, i = Blv R (4) The direction of the current is to oppose the changing flux. Since the flux is decreasing, the current flows counterclockwise in the x-y plane. This produces an induced flux in the z-direction which opposes the changing flux induced by the motion of the loop out of the uniform B field. Note that the induced current is small if the resistance of the loop is large, while the induced current is large if the resistance is small. In cases where an induced current flows, eg. a conducting loop, there is dissipation in the loop. This energy loss must be equal to the work done by an external force, but what is the origin of the force - it is the Lorentz force and the most convenient form in this problem is 2