Electromagnetics For Engineers – Fawwaz Ulaby (1)

(6.5) Understationaryconditions,thed-ccurrentinthecoilpro- duces a constant magnetic field B, which in turn produces aconstantfluxthroughtheloop.Whenthefluxisconstant, nocurrentisdetectedbythegalvanometer.However,when thebatteryisdisconnected,therebyinterruptingtheflowof currentinthecoil,themagneticfielddropstozero,andthe consequent change in magnetic flux causes a momentary deflection of the galvanometer needle. When the battery isreconnected,thegalvanometeragainexhibitsamomen- tarydeflection,butintheoppositedirection.Thus,current isinducedintheloopwhenthemagneticfluxchanges,and the direction of the current depends on whether the flux is increasing (as when the battery is being connected) or decreasing (as when the battery is being disconnected). It was further discovered that current can also flow in the loop,whilethebatteryisconnectedtothecoil,iftheloopis turnedaroundsuddenlyorwhilemovingitclosertooraway from the coil.

The physical movement of the loop changes the amount of flux linking its surface S, even though the field B due to the coil has not changed. A galvanometer is the predecessor of the voltmeter and ammeter.Whenagalvanometerdetectstheflowofcurrent through the coil, it means that a voltage has been induced across the galvanometer terminals. This voltage is called the electromotive force (emf), Vemf, and the process is called electromagnetic induction. The emf induced in a closed conducting loop of N turns is given by Vemf = −N d dt = −N d dt B · ds (6.6) Even though the results leading to Eq.

(6.6) were also dis- covered independently by Henry, Eq. (6.6) is attributed to Faraday and is known as Faraday’s law. The significance of the negative sign in Eq. (6.6) will be explained in the next section. We note that the derivative in Eq. (6.6) is a total time derivative that operates on the magnetic field B, as well as the differential surface area ds.

Accordingly, an emf can be generated in a closed conducting loop under any of the following three conditions: 1. A time-varying magnetic field linking a stationary loop; the induced emf is then called the transformer emf, V tr emf. 2. A moving loop with a time-varying area (relative to the normal component of B) in a static field B; the induced emf is then called the motional emf, V m emf. 3.

A moving loop in a time-varying field B. The total emf is given by Vemf = V tr emf + V m emf, (6.7) with V m emf = 0 if the loop is stationary (case (1)) and V tr emf = 0 if B is static (case (2)). For case (3), neither term is zero. Each of the three cases will be examined separately in the following sections. 6-2 Stationary Loop in a Time-Varying Magnetic Field The single-turn, conducting, circular loop with con- tour C and surface area S shown in Fig.

6-2(a) is in a time-varying magnetic field B(t). As was stated earlier, the emf induced when S is stationary and the field is time varying is called the transformer emf and is denoted V tr emf. Since the loop is stationary, d/dt in Eq. (6.6) now operates on B(t) only.

The University of Michigan Ann Arbor, MI 48109 Copyright  2025 Fawwaz T. Ulaby This book is published by Michigan Publishing under an agreement with the authors. It is made available free of charge in electronic form to any student or instructor interested in the subject matter. Published in the United States of America by Michigan Publishing. Manufactured in the United States of America ISBN 978-1-60785-911-6 (hardcover) ISBN 978-1-60785-912-3 (electronic) The free ECE Textbook initiative is sponsored by the ECE Department at the University of Michigan.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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