Physics 54 Lecture March 1, 2012 OUTLINE Micro-quiz problems (magnetic fields and forces) Magnetic dipoles and their interaction with magnetic fields Electromagnetic induction Introduction to electromagnetic induction Production of emf s by a time varying magnetic field Lenz s Law Motional emf s Eddy currents Physics 54 March 1, 2012 1
A couple of Micro-quiz problems Physics 54 March 1, 2012 2
Micro Quiz 1 Physics 54 March 1, 2012 3
Micro Quiz 1 An electron is traveling horizontally east in the magnetic field of the earth near the equator near the equator. The direction of the force on the electron is: N N B W S E W -e v E S The electron is deflected downward: F e = - e v x B Physics 54 March 1, 2012 4
Micro Quiz 2 Physics 54 March 1, 2012 5
Micro Quiz 2 F m = q v x B up B F e = q E down v q v B = q E balance E/B = v independent of q or m Physics 54 March 1, 2012 6
Micro Quiz 3 Physics 54 March 1, 2012 7
Micro Quiz 3 I into page B Electron velocity out of page 2 F e = - e v x B The electrons move to the right and accumulate on surface 2 Physics 54 March 1, 2012 8
Magnetic dipoles Physics 54 March 1, 2012 9
Introduction: monopoles and dipoles The simplest electric object in Nature is a point charge q or electric monopole. An electron is an example of an ideal point charge. The next simplest electric object is an electric dipole. Atoms and molecules can carry permanent or induced electric dipole moments p. Physics 54 March 1, 2012 10
Introduction: monopoles and dipoles No single magnetic single poles have been observed in Nature. There are apparently no magnetic monopoles! The simplest magnetic objects in Nature are magnetic dipoles given the symbol µ. Magnetic dipoles are created by current loops or the rotation of charged objects. Elementary particles, such as electrons, carry a magnetic dipole moment. Physics 54 March 1, 2012 11
A simple iron bar magnet is a magnetic dipole. p µ + break up - In contrast magnetic dipoles can not: S N break up Introduction: monopoles and dipoles The source of current loops in this case are from the atomic structure of the atoms in the magnet. Electric dipoles can be broken apart: + - S N S N electric monopoles = single charges There are no isolated magnetic monopoles Physics 54 March 1, 2012 12
v v Definition of a magnetic dipole A magnetic dipole is created by a closed current loop. Consider a loop with N turns of wire. Let n be a unit vector perpendicular to the wire loop in a direction determined by the current and the right hand rule. Define the magnetic moment µ of this loop to be: µ = I N A n Physics 54 March 1, 2012 13
Torque on a magnetic dipole Recall that for an electric dipole in a uniform electric field: The net force is zero The torque τ = p x E. The potential energy is U = - p E For a magnetic dipole in a uniform magnetic field: The net force is zero The torque is τ = µ x B (See derivation in text). The potential energy is U = - µ B Physics 54 March 1, 2012 14
Example 1 (torque on a wire loop) Find the torque on the current loop. b a current I magnetic field B Physics 54 March 1, 2012 15
Electromagnetic Induction (a new topic: Chapter 34) Physics 54 March 1, 2012 16
Introduction We will now extend electromagnetic theory by introducing a new phenomenon called electromagnetic induction. This is the observation that a time varying magnetic field creates an electromotive force (emf), and at a more basic level an electric field. Electromagnetic induction is described by two of the simplest laws of electromagnetic theory: Faraday s Law and Lenz s Law The applications of electromagnetic induction are far reaching. For example electric motors and generators operate based upon the the principle of electromagnetic induction. Physics 54 March 1, 2012 17
Faraday s Law and induced emf s We will first introduce Faraday s Law in a very practical way. We will use it to relate a time varying magnetic flux to an electromotive force (emf) induced in a wire. Let Φ B = the magnetic flux through the area bounded by a single closed wire loop. Φ B Surface S bounded by the loop Normal defined by curve direction and the right hand rule Physics 54 March 1, 2012 18
Faraday s Law and induced emf s With the definition of magnetic flux given on the previous page, Faraday s Law predicts that the emf induced in the wire loop will be: = dφ B /dt emf induced in wire loop magnetic flux through the loop As stated above the emf is for one loop of wire. If there are N overlapping loops in series then the emf is: = N dφ B /dt where Φ B = the magnetic flux through one loop Physics 54 March 1, 2012 19
The sign of the induced emf is determined by Lenz s Law An easy way to keep track of the sign of the induced emf is formulated by a rule called Lenz s Law. The induced emf produces a current, I, that creates a magnetic field that opposes the change in Φ B. See the next example. Physics 54 March 1, 2012 20
Example 2 A magnet is moved into a wire loop as shown. In which direction will the current flow? Physics 54 March 1, 2012 21
Example 2 (solution using Lenz s Law) B 2 B 1 B 1 current counterclockwise B 2 The magnetic field B 1 caused by the moving magnet causes a magnetic flux increase through the loop. This induces an emf which causes a current I that produces a magnetic field B 2 opposing B 1. Physics 54 March 1, 2012 22
Example 3 A wire loop is a circle of radius r = 50 cm. It has a resistance R = 0.1 Ω. A magnetic field is applied perpendicular to the loop. B(t) B(t) T = 10s B o = 1 Tesla a) Find the direction of the current b) Find I(t) c) Find the heat generated in the coil. B o T Physics 54 March 1, 2012 23 t
Example 4 A loop of wire with resistance R falls under the influence of gravity. I o A. zero The current induced in the wire is: B. non-zero clockwise C. non-zero counterclockwise Physics 54 March 1, 2012 24
Example 5 A magnetic field is constant in space but varies with time: B(t) = [0.3 + 0.5 t 2 ] Tesla (perpendicular to page). b a The magnitude of the emf induced in the above loop is: A. a b t B. a b t 2 C. ab(0.3 + t) D. ab( 0.3 + a b t 2 ) Physics 54 March 1, 2012 25
Example 6 A bar magnet is dropped through a wire loop as shown S N The current induced in the loop will be: A. zero B. clockwise C. counter clockwise D. clockwise and counter clockwise Physics 54 March 1, 2012 26
Motional emf A time-varying magnetic field through a stationary conducting loop generates an emf via Faraday s Law. stationary loop time varying magnetic fux Φ B = d Φ B /dt Faraday s Law also predicts that a conductor moving through a constant magnetic field will generate an emf. constant L v = B o L v magnetic field B o motional emf Physics 54 March 1, 2012 27
Motional emf (con.) The form of the motional emf can be derived as follows. Consider a metal rod sliding on two fixed conducting bars L constant magnetic field B o out of page v x Magnetic flux in loop = Φ B = B o L x and dφ B /dt = B o L dx/dt = B o L v Using Faraday s Law the magnitude of the emf = dφ B /dt and the motional emf = = B o L v Physics 54 March 1, 2012 28
Example 7 The magnetic field B o is out of the page The polarity across the resistor will be: A. top + bottom - B. top - bottom + C. no potential difference Physics 54 March 1, 2012 29
Currents generated in moving conducting plates When a metal plate or conductor is pulled through a nonuniform magnetic field, currents in the form of eddys,i.e. eddy currents, are formed. The time-varying magnetic flux through loops in the metal generate emf s and therefore currents. The currents react back on the magnetic field causing a retarding force. Physics 54 March 1, 2012 30
Currents generated in moving conducting plates Eddy currents therefore can be used to generate a magnetic brake in which the force increases as the speed of the moving conductor increases. In some cases eddy currents are undesirable as they transform energy into heat. (for example in an AC transformer). The eddy currents can be suppressed by cutting or laminating the metal plates. Physics 54 March 1, 2012 31
Example 8 A metal plate falls into a region of magnetic field as shown below. The current when the plate enters the field will be: A. clockwise B. counter clockwise C. zero Physics 54 March 1, 2012 32
Example 8 (solution) The induced eddy currents act as a break to stop the motion of the metal plate Physics 54 March 1, 2012 33
Example 9 (eddy currents) Find the direction of the force on the copper ring due to the induced eddy currents. Physics 54 March 1, 2012 34
End Lecture: Next one on March 13: Will cover the remaining material in Chapter 34 Physics 54 March 1, 2012 35
Example 8 A constant current I wire = I o flows through the long straight wire. The loop of wire moves upward with constant velocity v o so that r(t) = r o + v o t Find the direction and magnitude of the current I(t) in the loop of wire if it has resistance R. Physics 54 March 1, 2012 36