Physics 122B Electricity and Magnetism

Physics 122B Electricity and Magnetism Lecture 21 (Knight:33.1-33.4) Electromagnetic Induction Martin Savage

Lecture 21 Announcements • Lecture HW due tonight at 10 PM. Physics 122B - Lecture 21

Induced Magnetic Dipoles When an unmagnetized ferromagnetic material is placed in an externally applied magnetic field, magnetic domains in the material that are aligned with the field are energetically favored. This causes such aligned domains to grow, and for domains that are nearly aligned to rotate their magnetic moments to match the field direction. The net result is that a magnetic dipole moment is induced in the material, with a new south pole close to the north pole of the external magnet. If, when the field is removed, some fraction of the magnetic dipole moment remains, the material has become a permanent magnet. Physics 122B - Lecture 21

Hysteresis* Some ferromagnetic materials can be permanently magnetized, and “remember” their history of magnetization. The “hysteresis curve” shows the response of a ferromagnetic material to an external applied field. As the external field is applied, the material at first has increased magnetization, but then reaches a limit at (a) and saturates. When the external field drops to zero at (b), the material retains about 60% of its maximum magnetization. Partially magnetized Saturated Unmagnetized Physics 122B - Lecture 21

Aligned: Anti-aligned: Energy Difference: Nuclear Magnetism A single proton (like the one in every hydrogen nucleus) has a charge (+e) and an intrinsic angular momentum (“spin”). If we (naively) imagine the proton’s charge circulating in a loop, it should have a magnetic dipole moment μ. And indeed it does. • In an external B-field: • Classically:There will be torques unless m is aligned along B or against it. • QM: The proton spin has only 2 projections onto B. In magnetic resonance imaging, this energy difference is used to determine the local ``environment’’ of protons in, say, tissue using strong magnetic fields and high-frequency electromagnetic waves. Physics 122B - Lecture 21

Magnetic Resonance Imaging As mentioned previously, the behavior of the intrinsic spins and magnetic moments of nuclei in a magnetic field allows the spatial imaging of the positions of specific nuclei, which can produce a high-resolution image of the interior of the human body and other objects. This is called magnetic resonance imaging or MRI. The technique requires a very strong and homogeneous magnetic field. Large solenoids, often superconducting, are used for this purpose. The magnetic fields generated range up to a few tesla. The B-field is “swept” by auxiliary coils, so that the conditions for resonance are met at successive points in the volume of interest. Physics 122B - Lecture 21

Question Which magnet configurations will produce this induced magnetization? (a) Magnets 1&2; (b) Magnets 1&3; (c) Magnets 1&4;(c) Magnets 2&3; (e) Magnets 2&4; Physics 122B - Lecture 21

Chapter 32 - Summary (1) Physics 122B - Lecture 21

Chapter 32 - Summary (2) Physics 122B - Lecture 21

A Second Prelude toMaxwell’s Equations Suppose you come across a vector field (flow, E, B) that looks something like this. What are the identifiable structures in this field? 1. An “outflow” structure: 2. An “inflow” structure: 3. An “clockwise circulation” structure: 4. An “counterclockwise circulation” structure: Maxwell’s Equations will tell us that the “flow” structures are charges (+ and -) and the “circulation” structures are energy flows in the field. Physics 122B - Lecture 21

The History of Induction In 1831, Joseph Henry, a Professor of Mathematics and Natural Philosophy at the Albany Academy in New York, discovered magnetic induction. In July, 1832 he published a paper entitled “On the Production of Currents and Sparks of Electricity from Magnetism” describing his work. Because Henry published after Michael Faraday, his did not receive much credit for this discovery, which actually preceded Faraday’s. Joseph Henry (1797-1878) Michael Faraday's ideas about conservation of energy led him to believe that since an electric current could cause a magnetic field, a magnetic field should be able to produce an electric current. He demonstrated this principle of induction in 1831 and published his results immediately. The principle of induction was a landmark in applied science, for it made possible the dynamo, or generator, which produces electricity by mechanical means. Michael Faraday (1791-1867) Physics 122B - Lecture 21

Faraday’s Discovery Faraday had wound two coils around the same iron ring. He was using a current flow in one coil to produce a magnetic field in the ring, and he hoped that this field would produce a current in the other coil. Like all previous attempts to use a static magnetic field to produce a current, his attempt failed to generate a current. However, Faraday noticed something strange. In the instant when he closed the switch to start the current flow in the left circuit, the current meter in the right circuit jumped ever so slightly. When he broke the circuit by opening the switch, the meter also jumped, but in the opposite direction. The effect occurred when the current was stopping or starting, but not when the current was steady. Faraday had invented the picture of lines of force, and he used this to conclude that the current flowed only when lines of force cut through the coil. Physics 122B - Lecture 21

Faraday Investigates Induction Was it necessary to move the magnet? Faraday placed the coil in the field of a permanent magnet. He found that there was a momentary current when the coil was moved. Faraday replaced the upper coil with a bar magnet. He found that there was a momentary current when the bar magnet was moved in or out of the coil. Faraday placed one coil above the other, without the iron ring. Again there was a momentary current when the switch opened or closed. Conclusion: There is a current in the coil if and only if the magnetic field passing through the coil is changing. Physics 122B - Lecture 21

Motional EMF Consider a length l of conductor moving to the right in a magnetic field that is into the diagram. Positive charges in the conductor will experience an upward force and negative charges a downward force. The net result is that charges will “pile up” at the two ends of the conductor and create an electric field E. When the force produced by E becomes large enough to balance the magnetic force, the movement of charges will stop and the system will be in equilibrium. Physics 122B - Lecture 21 This is also true ``locally’’

Separating Charge and EMF Physics 122B - Lecture 21

Question The square conductor moves upward through a uniform magnetic field that is directed out of the diagram. Which of the figures shows the correct distribution of charges on the conductor? Physics 122B - Lecture 21

Example: A Battery Substitute A 6.0 cm long flashlight battery has an EMF of 1.5 V. With what speed must a 6.0 cm wire move through a 0.10 T magnetic field to create the same EMF? Physics 122B - Lecture 21

Example: Potential Difference along a Rotating Bar A metal bar of length l rotates with angular velocity w about a pivot at one end. A uniform magnetic field B is perpendicular to the plane of rotation. What is the potential difference between the ends of the bar? Physics 122B - Lecture 21

Induced Current in a Circuit The figure shows a conducting wire sliding with speed v along a U-shaped conducting rail. The induced emf E will create a current I around the loop. Physics 122B - Lecture 21

Force and Induction We have assumed that the sliding conductor moves with a constant speed v. It turns out that a current carrying wire in a magnetic field experiences a force Fmag, so we must supply a counter-force Fpull to make this happen. Physics 122B - Lecture 21

Energy Considerations Therefore, the work done in moving the conductor is equal to the energy dissipated in the resistance. Energy is conserved. Whether the wire is moved to the right or to the left, a force opposing the motion is observed. Physics 122B - Lecture 21

Example: Lighting A Bulb The figure shows a circuit including a 3 V 1.5 W light bulb connected by ideal wires with no resistance. The right wire is pulled with constant speed v through a perpendicular 0.10 T magnetic field. (a) What speed must the wire have to light the bulb to full brightness? (b) What force is needed to keep the wire moving? Physics 122B - Lecture 21

Eddy Currents Suppose that a rigid square copper loop is between the poles of a magnet. If the loop moves, as long as no conductors are in the field of the magnet there will be no current and no forces. But when one side of the loop enters the magnetic field, a current flow will be induced and a force will be produced. Therefore, a force will be required to pull the loop out of the magnetic field, even though copper is not a magnetic material. However, if we cut the loop, there will be no force. Physics 122B - Lecture 21

Eddy Currents (2) Another way of looking at the system is to consider the magnetic field produced by the current in the loop. The current loop is effectively a dipole magnet with a S pole near the N pole of the magnet, and vice versa. The attractive forces between these poles must be overcome by an external force to pull the loop out of the magnet. Physics 122B - Lecture 21

Eddy Currents (3) Now consider a sheet of conductor pulled through a magnetic field. There will be induced current, just as with the wire, but there are now no well-defined current paths. As a consequence, two “whirlpools” of current will circulate in the conductor. These are called eddy currents. A magnetic braking system. Physics 122B - Lecture 21

Question What is the ranking of the forces in the figure? (a) F1=F2=F3=F4; (b) F1<F2=F3>F4; (c) F1=F3<F2=F4; (d) F1=F4<F2=F3; (e) F1<F2<F3=F4; Physics 122B - Lecture 21

Air Flow and Flux The amount of air flow through the loop depends on the orientation of the loop with respect to the air-flow direction. Physics 122B - Lecture 21

Magnetic Flux • The number of arrows passing through the loop depends on two factors: • (1) The density of arrows, which is proportional to B • The effective areaAeff = A cos q of the loop • We use these ideas to define the magnetic flux: Physics 122B - Lecture 21

Area Vector Define the area vector A of a loop such that it has the loop area as its magnitude and is perpendicular to the plane of the loop. If a current is present, the area vector points in the direction given by the thumb of the right hand when the fingers curl in the direction of current flow. If the area is part of a closed surface, the area vector points outside the enclosed volume. With this definition: Physics 122B - Lecture 21

Example: A Circular Loop Rotating in a Magnetic Field The figure shows a 10 cm diameter loop rotating in a uniform 0.050 T magnetic field. What is the magnitude of the flux through the loop when the angle is q=00, 300, 600, and 900? Physics 122B - Lecture 21

Magnetic Fluxin a Nonuniform Field So far, we have assumed that the loop is in a uniform field. What if that is not the case? The solution is to break up the area into infinitesimal pieces, each so small that the field within it is essentially constant. Then: Physics 122B - Lecture 21

Example: Magnetic Flux froma Long Straight Wire The near edge of a 1.0 cm x 4.0 cm rectangular loop is 1.0 cm from a long straight wire that carries a current of 1.0 A, as shown in the figure. What is the magnetic flux through the loop? Physics 122B - Lecture 21

Lenz’s Law (1) Heinrich Friedrich Emil Lenz (1804-1865) In 1834, Heinrich Lenz announced a rule for determining the direction of an induced current, which has come to be known as Lenz’s Law. Here is the statement of Lenz’s Law: There is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux. Physics 122B - Lecture 21

Lenz’s Law (2) If the field of the bar magnet is already inthe loop and the bar magnet is removed, theinduced current is in the direction that triesto keep the field constant. If the loop is a superconductor, a persistentstanding current is induced in the loop, and thefield remains constant. Superconductingloop Physics 122B - Lecture 21

Six Induced Current Scenarios Physics 122B - Lecture 21

Example: Lenz’s Law 1 - + - + The switch in the circuit shown has been closed for a long time. What happens to the lower loop when the switch is opened? Physics 122B - Lecture 21

Example: Lenz’s Law 2 + - The figure shows two solenoids facing each other. When the switch for coil 1 is closed, does the current in coil 2 flow from right to left or from left to right? Physics 122B - Lecture 21

Example: A Rotating Loop A loop of wire is initially in the xy plane in a uniform magnetic field in the x direction. It is suddenly rotated 900 about the y axis, until it is in the yz plane. In what direction will be the induced current in the loop? Initially there is no flux through the coil. After rotation the coil will be threaded by magnetic flux in the x direction. The induced current in the coil will oppose this change by producing flux in the –x direction. Let your thumb point on the –x direction, and your fingers will curl clockwise. Therefore, the induced current will be clockwise, as shown in the figure. Physics 122B - Lecture 21

End of Lecture 21 • Before the next lecture, read Knight, sections 33.5 through 33.7. • Lecture HW is due tonight at 10 PM. . Physics 122B - Lecture 21

Physics 122B Electricity and Magnetism