H217 Electromagnetic Induction - 1. Notes (1718)

9749 H2 Physics Lecture Notes Topic 17: Electromagnetic Induction Page 1 of 15 Topic 17 Electromagnetic Induction Content  Magnetic Flux  Laws of Electromagnetic Induction Learning Outcomes Candidates should be able to: (a) define magnetic flux and the weber (b) recall and solve problems using Φ = BA (c) define magnetic flux linkage (d) infer from appropriate experiments on electromagnetic induction: i. that a changing magnetic flux can induce an e.m.f. ii. that the direction of the induced e.m.f. opposes the change producing it iii. the factors affecting the magnitude of the induced e.m.f. (e) recall and solve problems using Faraday’s law of electromagnetic induction and Lenz’s law (f) explain simple applications of electromagnetic induction

9749 H2 Physics Lecture Notes Topic 17: Electromagnetic Induction Page 2 of 15 17.0 Introduction In the earlier topic, two ways in which electricity and magnetism are related were highlighted: i. an electric current produces a magnetic field, and ii. a magnetic field exerts a force on an electric current or moving electric charge. These discoveries were made in 1820. As nature is often symmetric, the discovery that electric currents produce magnetic fields led scientists to suspect that magnetic fields could produce electric currents. 17.1 Motion Induced Electromotive Force Consider the situation in which a thin straight conductor of length l moving at a constant speed v perpendicular to a uniform magnetic field that is directed into the plane of the page as shown. By Fleming’s Left-Hand Rule, the electrons in the conductor will experience a downward magnetic force FB. This causes the electrons to migrate towards the lower end of the conductor. Thus charge separation occurs within the conductor with the accumulation of negative charge on one end and positive charge on the opposite end. This charge separation will in turn produce an electric field along the rod, which causes the electrons to experience an upward electric force FE. Steady state is reached when there is no further charge separation. This occurs when the magnetic force and electric force acting on the electrons are balanced. i.e. B E induced F F Bqv q l    Hence, the induced e.m.f. across the ends of the conductor is given by induced Blv  x x x x x x x x x x x x x x x x x x x x x x x x v

9749 H2 Physics Lecture Notes Topic 17: Electromagnetic Induction Page 3 of 15 17.2 Magnetic Flux and Magnetic Flux Linkage Ten years after the discovery that an electric current could produce a magnetic field, experiments conducted independently by Joseph Henry in United States and Michael Faraday in England showed that a changing magnetic field could induce an electric current in a circuit. The results of these experiments led to a b asic and important law known as Faraday’s law. But before delving into the Laws of Electromagnetic Induction, the concept of magnetic flux, and consequently magnetic flux linkage, must first be understood. 17.2.1 Magnetic Flux In the earlier topic, the term magnetic flux was introduced and is defined as the product of the magnetic flux density normal to the surface and the area of the surface. The general equation for magnetic flux  is B dA     If the magnetic field is uniform, the integral reduces to  cos B A B A     , where  is the angle between the normal to A and B   . Magnetic flux is a scalar quantity and its SI unit is the weber (Wb), which is defined as the amount of magnetic flux that when reduced to zero in one second induces an e.m.f. of one volt. 17.2.2 Magnetic Flux Linkage The magnetic flux linkage through a loop is the product of the magnetic flux through the loop and the number of turns of wire in the loop. Thus, if the area A is bounded by a coil and the coil has N turns, the magnetic flux linkage through the coil, is  cos N NB A N B A     From the equation, magnetic flux linkage through a coil depends on  the number of turns in the coil N,  the magnitude of magnetic flux density B,  the orientation of the coil with respect to the direction of B, and  the area bounded by the coil A. A change in any of these variables results in a change of magnetic flux linkage.

9749 H2 Physics Lecture Notes Topic 17: Electromagnetic Induction Page 4 of 15 Example 1 A soft iron ring of variable cross-section has two coils wound around it. One coil has 7 turns. The other, of 3 turns, is connected to a d.c. supply. Which quantity is the same inside each coil? A magnetic field B magnetic flux C magnetic flux density D magnetic flux linkage [N09/I/33] Example 2 The three loops of wire shown are all in a region of uniform magnetic field. Loop 1 swings back and forth as the bob on a pendulum. Loop 2 rotates about a vertical axis. Loop 3 oscillates vertically on the end of a spring. Which loop(s) have a magnetic flux that changes with time? Only loop 2 has a changing magnetic flux as its orientation relative to the field changes as it rotates.

9749 H2 Physics Lecture Notes Topic 17: Electromagnetic Induction Page 5 of 15 Example 3 A coil of 10 turns and an enclosed area of 0.35 m2 is placed perpendicular to a magnetic field of flux density 2.0 T as shown in the diagram on the left. Points P and Q are pulled apart until the coil becomes a straight line as shown in the diagram on the right. What is the change in magnetic flux linkage through the coil?      cos 10 2.0cos0 0.35 7.0 Wb 0 Wb 7.0 Wb i o f f i N N B A N N N N               Region of magnetic field Before After P Q P Q

9749 H2 Physics Lecture Notes Topic 17: Electromagnetic Induction Page 6 of 15 17.3 Laws of Electromagnetic Induction 17.3.1 Faraday’s Law Faraday’s Law of electromagnetic induction states that the induced e.m.f. is proportional to the rate of change of the magnetic flux linkage. Mathematically: induced dN dt  The polarity of the induced e.m.f. is governed by another law known as Lenz’s Law. 17.3.2 Lenz’s Law Lenz’s Law of electromagnetic induction states that a current produced by the induced e.m.f. will flow in a direction such that the magnetic field it produces opposes the original change in magnetic flux linkage. Lenz's Law is a consequence of the principle of conservation of energy. Consider moving a magnet towards the face of a solenoid. According to Lenz’s Law, if the magnet’s pole facing the solenoid was North, the induced current in the solenoid would flow such that the end facing the magnet would be a North pole as well. Suppose the induced currents' directions were opposite to those prescribed by Lenz’s Law; the north pole of an approaching magnet would induce a south pole in the near face of the solenoid. The attractive force between these p oles would accelerate the magnet's approach and make the magnetic field increase more quickly. This in turn would increase the current induced in the solenoid, strengthening the magnetic field, which would in turn increase the forces of attraction and hence the acceleration of the magnet. Both the kinetic energy of the magnet and the rate of energy dissipation in the solenoid (due to heat dissipation) would increase. A small energy input would produce a large energy output, thereby violating the law of conservation of energy Hence, the two laws of electromagnetic induction can be summarized succinctly as induced dN dt   with the negative sign indicating that the induced e.m.f. opposes the change in magnetic flux linkage. This is also the relationship that gives rise to the definition of the weber given in the earlier section.

9749 H2 Physics Lecture Notes Topic 17: Electromagn