Faraday s Law of Electromagnetic Induction

Transcription

1 Faraday s Law of Electromagnetic Induction EE 141 Lecture Notes Topic 20 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University of Vermont 2014

2 Motivation

3 Faraday s Law of Electromagnetic Induction Consider a simple closed circuit C enclosing an open surface S in a magnetic field B(r). The magnetic flux Φ m linking the circuit C is defined as Φ m B ˆnd 2 r (1) S where ˆn is the unit normal vector to S in the chosen positive sense. Faraday s law of electromagnetic induction (1831) then states that the induced electromotive force (emf) E E d l (2) C in the positive sense about the closed circuit C (as determined by the choice of ˆn) is related to the change in magnetic flux through the circuit by E = dφ m dt (3)

4 Faraday s Law & Lenz s Law This electromotive force acts as the driving force for the current induced in the closed circuit C, and Faraday s law states that this emf results from any change in the magnetic flux linking that circuit. Notice that the mathematical expression (3) of Faraday s law is a statement of an independent experimental law and is entirely independent of the manner in which the magnetic flux linkage changes. Nevertheless, due care must be taken in defining the respective reference frames that contain the circuit and the magnetic flux source, as is the case in the so-called Faraday s paradox. The negative sign appearing in Faraday s law indicates that the induced emf opposes the change that is producing it. This then leads to Lenz s law which states that a change in magnetic flux induces an emf that produces a current whose magnetic field opposes the original change in magnetic flux.

5 Faraday s Paradox Faraday s paradox results from an experimental arrangement for Faraday s law that consists of a conducting cylindrical disc and a coaxial cylindrical permanent magnet that are separated by a small distance along their common axis. The magnet s B-field is then directed along this axis. In order to measure the effects of this field, an electric circuit is formed through a pair of sliding contacts situated at the center and the rim of the conducting disc. When the disc is rotated with angular frequency ω and the magnet is held fixed, the magnetic Lorentz force F m = qv B produces a radial current that is outwardly directed in the conducting disc, resulting in a measured direct current in the circuit. In this arrangement the device is known as a Faraday generator or Faraday disc. If the disc is held fixed while the magnet is rotated about its axis, no current is measured. If both disc & magnet are rotated together, current is measured.

6 Faraday s Paradox This experiment is described as a paradox because the results appear to violate Faraday s law as the magnetic flux linkage with the conducting disk appears to be the same regardless of which element is rotating so that dφ m /dt = 0 and there should never be an induced emf or current flow. Alternatively, from the viewpoint of the magnetic lines of flux, the induced emf is proportional to the rate at which the magnetic flux lines are cut. If the flux lines originate in the magnet, then they are stationary in the frame of reference of the magnet so that rotating the disc relative to the magnet, either by rotating the disc or the magnet, should then produce an emf, but rotating both of them together should not. The resolution of this apparent paradox is obtained simply by noting that the motion of the magnet is entirely irrelevant in Faraday s law.

7 Self-Inductance From the Biot-Savart law (18.3), the magnetic flux linking an isolated circuit is dependent on the circuit geometry and is linearly related to the current in the circuit 1. For a stationary circuit with fixed shape, the only change in its magnetic flux linkage must then result from changes in the current, so that dφ m /dt = (dφ m /di)(di/dt). Because Φ m is directly proportional to the current I, then dφ m /di is independent of I. This quantity is defined as the inductance L, where L dφ m di (H) (4) With this identification, Faraday s law (3) becomes E = L di dt, (5) where the MKSA unit of inductance is the henry (H Vs/A). 1 If a ferromagnetic material is used, then the relation between Φm and I is nonlinear and dφ m/di is no longer independent of I. In that case dφ m/di is called the incremental inductance and Φ m/i the inductance.

8 Mutual Inductance Consider a system of n isolated closed circuits C j, j = 1,2,...,n, each with current I j. The magnetic flux Φ mi linking the i th circuit is then given by the sum of the individual flux linkages produced by the currents in each of the n circuits, so that Φ mi = n Φ mij, (6) j=1 where Φ mij is the magnetic flux linking the circuit C i due to the current I j flowing in the circuit C j. The induced emf in the closed circuit C i is then given by Faraday s law as E i = dφ m i dt = n j=1 dφ mij dt. (7)

9 Mutual Inductance If each circuit C j is rigid and stationary in the laboratory reference frame, then the only change in each magnetic flux linkage Φ mij is that which is caused by changes in the currents I j, so that dφ mij /dt = (dφ mij /di j )(di j /dt), where dφ mij /di j is independent of the current I j in the linear case. This quantity then defines the mutual inductance M ij dφ m ij di j, i j (H) (8) between circuit i and circuit j. Notice that this expression reduces to the self-inductance of the i th circuit when i = j, where L i = M ii.

10 Mutual Inductance The mutual inductance for a pair of stationary, rigid circuits in a linear magnetic medium is given by M 21 = dφ m21 /di 1 = Φ m21 /I 1. The magnetic flux Φ m21 is obtained from the Biot-Savart law (18.3) and Eq. (20.1) as Φ m21 = µ 4π I 1 S 2 { d l 1 (r 2 r 1 ) C 1 r 2 r 1 3 } ˆndr 2 2. (9) Because d l 1 (r 2 r 1 ) d = C 1 r 2 r l 1 C 1 r 2 r 1, the mutual inductance M 21 linking circuit 2 to the current flowing in circuit 1 is given by M 21 = µ 4π S 2 2 { d l 1 C 1 r 2 r 1 } ˆndr 2 2.

11 Mutual Inductance Notice that, in principle, Neumann s formula is applicable to determining the self-inductance of a circuit. However, due care must be given to the logarithmic singularity at r 2 = r 1 in the resultant contour integral. This singular behavior may be resolved by including the thickness of the current-carrying circuit, a quantity which, in most cases, may be conveniently neglected in determining the mutual inductance. Application of Stokes theorem to the surface integral in this expression results in Neumann s formula for the mutual inducatnce M 21 = µ 4π C 2 d l 1 d l 2 C 1 r 2 r 1 (10) The inherent symmetry of this expression then shows that, in general M ij = M ji. (11)