Chapter 22: Magnetism PDF

Chapter 22: Magnetism Mrs. Pooja Brahmaiahchari Magnets All magnets attract iron, such as that in a refrigerator door. However, magnets may attract or repel other magnets. Experimentation shows that all magnets have two poles. If freely suspended, one pole will point toward the north...

Chapter 22: Magnetism Mrs. Pooja Brahmaiahchari Magnets All magnets attract iron, such as that in a refrigerator door. However, magnets may attract or repel other magnets. Experimentation shows that all magnets have two poles. If freely suspended, one pole will point toward the north. The two poles are thus named the north magnetic pole and the south magnetic pole.  It is a universal characteristic of all magnets that like poles repel and unlike poles attract. Further experimentation shows that it is impossible to separate north and south poles in the manner that + and − charges can be separated. Magnetic atoms have both a north pole and a south pole, as do many types of subatomic particles, such as electrons, protons, and neutrons. Ferromagnets Only certain materials, such as iron, cobalt, nickel, and gadolinium, exhibit strong magnetic effects. Such materials are called ferromagnetic, after the Latin word for iron, ferrum. A group of materials made from the alloys of the rare earth elements are also used as strong and permanent magnets; a popular one is neodymium. Not only do ferromagnetic materials respond strongly to magnets (the way iron is attracted to magnets), they can also be magnetized themselves—that is, they can be induced to be magnetic or made into permanent magnets.  When a magnet is brought near a previously unmagnetized ferromagnetic material, it causes local magnetization of the material with unlike poles closest. The regions within the material called domains act like small bar magnets. Within domains, the poles of individual atoms are aligned. Each atom acts like a tiny bar magnet. Domains are small and randomly oriented in an unmagnetized ferromagnetic object. This induced magnetization can be made permanent if the material is heated and then cooled, or simply tapped in the presence of other magnets.  Conversely, a permanent magnet can be demagnetized by hard blows or by heating it in the absence of another magnet. There is a well-defined temperature for ferromagnetic materials, which is called the Curie temperature, above which they cannot be magnetized. The Curie temperature for iron is 1043 K (770oC), which is well above room temperature Electromagnets Early 19th century, it was discovered that electrical currents cause magnetic effects. The first significant observation was by the Danish scientist Hans Christian Oersted (1777–1851), who found that a compass needle was deflected by a current carrying wire. This was the first significant evidence that the movement of charges had any connection with magnets. Electromagnetism is the use of electric current to make magnets. These temporarily induced magnets are called electromagnets.  Figure shows that the response of iron filings to a current-carrying coil and to a permanent bar magnet. The patterns are similar. In fact, electromagnets and ferromagnets have the same basic characteristics. For example, they have north and south poles that cannot be separated and for which like poles repel and unlike poles attract.  Combining a ferromagnet with an electromagnet can produce particularly strong magnetic effects. Whenever strong magnetic effects are needed, such as lifting scrap metal, or in particle accelerators, electromagnets are enhanced by ferromagnetic materials. Limits to how strong the magnets can be made are imposed by coil resistance (it will overheat and melt at sufficiently high current), and so superconducting magnets may be employed. These are still limited, because superconducting properties are destroyed by too great a magnetic field.  Figure shows a few uses of combinations of electromagnets and ferromagnets. Ferromagnetic materials can act as memory devices, because the orientation of the magnetic fields of small domains can be reversed or erased. Magnetic information storage on videotapes and computer hard drives are among the most common applications. This property is vital in our digital world. Current: The Source of All Magnetism Electric current is the source of all magnetism is the fact that it is impossible to separate north and south magnetic poles. A current loop always produces a magnetic dipole—that is, a magnetic field that acts like a north pole and south pole pair. Since isolated north and south magnetic poles, called magnetic monopoles, are not observed, currents are used to explain all magnetic effects.  Crucial to the statement that electric current is the source of all magnetism is the fact that it is impossible to separate north and south magnetic poles. A current loop always produces a magnetic dipole—that is, a magnetic field that acts like a north pole and south pole pair. Since isolated north and south magnetic poles, called magnetic monopoles, are not observed, currents are used to explain all magnetic effects. If magnetic monopoles did exist, then we would have to modify this underlying connection that all magnetism is due to electrical current. There is no known reason that magnetic monopoles should not exist—they are simply never observed—and so searches at the subnuclear level continue. Magnetic Fields and Magnetic Field Lines Since magnetic forces act at a distance, we define a magnetic field to represent magnetic forces. The pictorial representation of magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. As shown in Figure, the direction of magnetic field lines is defined to be the direction in which the north end of a compass needle points. The magnetic field is traditionally called the B-field. The properties of magnetic field lines can be summarized by these rules: 1. The direction of the magnetic field is tangent to the field line at any point in space. A small compass will point in the direction of the field line. 2. The strength of the field is proportional to the closeness of the lines. It is exactly proportional to the number of lines per unit area perpendicular to the lines (called the areal density). 3. Magnetic field lines can never cross, meaning that the field is unique at any point in space. 4. Magnetic field lines are continuous, forming closed loops without beginning or end. They go from the north pole to the south pole. Magnetic Field Strength What is the mechanism by which one magnet exerts a force on another? The answer is related to the fact that all magnetism is caused by current, the flow of charge. Magnetic fields exert forces on moving charges, and so they exert forces on other magnets, all of which have moving charges. Right Hand Rule 1 Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the magnetic force F on a charge moving at a speed v in a magnetic field of strength B is given by 𝐹 = 𝑞𝑣 𝐵 sin 𝜃 where 𝜃 is the angle between the directions of v and B. This force is often called the Lorentz force.  In fact, this is how we define the magnetic field strength —in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength is called the tesla (T) after the eccentric but brilliant inventor Nikola Tesla (1856–1943). Solving for B, we get: 𝐹 𝐵= 𝑞𝑣 sin 𝜃 The tesla is, 1𝑁 1𝑁 1𝑇 = = 𝐶. 𝑚/𝑠 𝐴. 𝑚  Another smaller unit, called the gauss (G), where 1G = 10-4 T, is sometimes used. The strongest permanent magnets have fields near 2T; superconducting electromagnets may attain 10 T or more. The Earth’s magnetic field on its surface is only about 5 x 10-5 T or 0.5G  RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of v, the fingers in the direction of B, and a perpendicular to the palm points in the direction of F. One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge. Force on a Moving Charge in a Magnetic Field: Examples and Applications Magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic charged particles in outer space, some of which approach the Earth. They can be forced into spiral paths by the Earth’s magnetic field. Protons in giant accelerators are kept in a circular path by magnetic force.  Magnetic force is always perpendicular to velocity, so that it does no work on the charged particle. The particle’s kinetic energy and speed thus remain constant. The direction of motion is affected, but not the speed. This is typical of uniform circular motion. Because the magnetic force supplies the centripetal force 𝐹𝑐 , we have, 𝑚𝑣 2 𝑞𝑣𝐵 = 𝑟 Solving for r yields, 𝑚𝑣 𝑟= 𝑞𝐵 Here, r is the radius of curvature of the path of a charged particle with mass m and charge q , moving at a speed v perpendicular to a magnetic field of strength B.  When a charged particle moves along a magnetic field line into a region where the field becomes stronger, the particle experiences a force that reduces the component of velocity parallel to the field. This force slows the motion along the field line and here reverses it, forming a “magnetic mirror.”  Charged particles approaching magnetic field lines may get trapped in spiral orbits about the lines rather than crossing them, as seen above. Some cosmic rays, for example, follow the Earth’s magnetic field lines, entering the atmosphere near the magnetic poles and causing the southern or northern lights through their ionization of molecules in the atmosphere. Some incoming charged particles become trapped in the Earth’s magnetic field, forming two belts above the atmosphere known as the Van Allen radiation belts after the discoverer James A. Van Allen, an American astrophysicist. The Hall Effect The magnetic field also affects charges moving in a conductor. One result is the Hall effect, which has important implications and applications.  Moving electrons feel a magnetic force toward one side of the conductor, leaving a net positive charge on the other side. This separation of charge creates a voltage 𝜀, known as the Hall emf, across the conductor. The creation of a voltage across a current carrying conductor by a magnetic field is known as the Hall effect, after Edwin Hall, the American physicist who discovered it in 1879.  The Hall emf yields, 𝜀 = 𝐵𝑙𝑣 (𝐵 , 𝑣 𝑎𝑛𝑑 𝑙, 𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟) where 𝜀 is the Hall effect voltage across a conductor of width l through which charges move at a speed v. One of the most common uses of the Hall effect is in the measurement of magnetic field strength. Such devices, called Hall probes, can be made very small, allowing fine position mapping. Hall probes can also be made very accurate, usually accomplished by careful calibration. Another application of the Hall effect is to measure fluid flow in any fluid that has free charges (most do).  A magnetic field applied perpendicular to the flow direction produces a Hall emf 𝜀 as shown. Note that the sign of 𝜀 depends not on the sign of the charges, but only on the directions of B and v. The magnitude of the Hall emf is 𝜀 = 𝐵𝑙𝑣 , where l is the pipe diameter, so that the average velocity v can be determined from 𝜀 providing the other factors are known. 1. A large water main is 2.50 m in diameter and the average water velocity is 6.00 m/s. Find the Hall voltage produced if the pipe runs perpendicular to the Earth’s 5.00 x 10-5 T field. Magnetic Force on a Current-Carrying Conductor Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor is transmitted to the conductor itself.  The force on a current-carrying wire in a magnetic field is 𝐹 = 𝐼𝑙𝐵 sin 𝜃. Its direction is given by RHR-1. Is the equation for magnetic force on a length l of wire carrying a current I in a uniform magnetic field B. Magnetic force on current-carrying conductors is used to convert electric energy to work. Magnetohydrodynamics (MHD) is the technical name given to a clever application where magnetic force pumps fluids without moving mechanical parts. 2. A wire carrying 30A current has the length of 12cm between the pole faces of the magnet at an angle of 60o. The magnetic field is approximately uniform at 0.90 T. What is the magnitude of the force on wire? Torque on a Current Loop: Motors and Meters Motors are the most common application of magnetic force on current-carrying wires. Motors have loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts torque on the loops, which rotates a shaft. Electrical energy is converted to mechanical work in the process.  Torque is defined as 𝜏 = 𝑟𝐹𝑠𝑖𝑛𝜃, where F is the force, r is the distance from the pivot that the force is applied, and θ is the angle between r and F. Right hand rule 1 gives the forces on the sides to be equal in magnitude and opposite in direction, so that the net force is again zero. However, each force produces a clockwise torque. Since r = w/2, the torque on each vertical segment is (w/2)F sinθ, and the two add to give the total torque. 𝑤 𝑤 𝜏= 𝐹 𝑠𝑖𝑛𝜃 + 𝐹 𝑠𝑖𝑛𝜃 = 𝑤 𝐹 𝑠𝑖𝑛𝜃 2 2  Now, each vertical segment has a length l that is perpendicular to B, so that the force on each is F = IlB. Entering F into the expression for torque yields 𝜏 = 𝑤𝐼𝑙𝐵𝑠𝑖𝑛𝜃 If we have a multiple loop of N turns, we get N times the torque of one loop. Finally, note that the area of the loop is A = wl ; the expression for the torque becomes 𝜏 = 𝑁𝐼𝐴𝐵 𝑠𝑖𝑛𝜃 This is the torque on a current-carrying loop in a uniform magnetic field. 3. A circular coil of wire has a radius of 30cm and contains 50 loops. The current is 8.0A and coil is placed in magnetic field of 5.0 T. What is the maximum torque exerted on coil by magnetic field? 4. A rectangular coil (40cm x 50cm) contains 200 loops and has current of 15A. What is the magnitude of magnetic field required to produce a maximum torque of 1200N.m? Magnetic Fields Produced by Currents: Ampere’s Law The right hand rule 2 (RHR-2) emerges from this exploration and is valid for any current segment—point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it. (a) Compasses placed near a long straight current-carrying wire indicate that field lines form circular loops centered on the wire. (b) Right hand rule 2 states that, if the right hand thumb points in the direction of the current, the fingers curl in the direction of the field. This rule is consistent with the field mapped for the long straight wire and is valid for any current segment.  The magnetic field strength (magnitude) produced by a long straight current-carrying wire is found by experiment to be, 𝜇0 𝐼 𝐵= 2𝜋𝑟 where I is the current, r is the shortest distance to the wire, and the constant 𝜇0 = 4π x 10 -7 T. m/A is the permeability of free space. 5. A vertical wire caries a current of 45A due to south. Calculate magnitude and direction of magnetic field 2cm to right of the wire. 6. A wire carries a current of 10A. At what distance from the wire will magnetic field of 8.0 x 10-4 T be produced? Ampere’s Law and Others Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment. The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law. Integral calculus is needed to sum the field for an arbitrary shape current. This results in a more complete law, called Ampere’s law, which relates magnetic field and current in a general way. Ampere’s law in turn is a part of Maxwell’s equations, which give a complete theory of all electromagnetic phenomena. Magnetic Field Produced by a Current- Carrying Circular Loop There is a simple formula for the magnetic field strength at the center of a circular loop. It is 𝜇0 𝐼 𝐵= 𝑎𝑡 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑙𝑜𝑜𝑝 2𝑅 Magnetic Field Produced by a Current- Carrying Solenoid A solenoid is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero.  The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the magnetic field strength inside a solenoid is simply 𝐵 = 𝜇0 𝑛𝐼 𝑖𝑛𝑠𝑖𝑑𝑒 𝑎 𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 where n is the number of loops per unit length of the solenoid. 7. A solenoid has a length of 15cm and a total of 800 turns of wire. Calculate strength of magnetic field at its center if the solenoid carries a current of 5.0 A. Magnetic Force between Two Parallel Conductors The force between two long straight and parallel conductors separated by a distance can be found by applying what we have developed in preceding sections. Figure shows the wires, their currents, the fields they create, and the subsequent forces they exert on one another. (a) The magnetic field produced by a long straight conductor is perpendicular to a parallel conductor, as indicated by RHR-2. (b) A view from above of the two wires shown in (a), with one magnetic field line shown for each wire. RHR-1 shows that the force between The parallel conductors is attractive when the currents are in the same direction. A similar analysis shows that the force is repulsive between currents in opposite directions.  The field due 𝐼1 to at a distance r is given to be 𝜇0 𝐼1 𝐵1 = 2𝜋𝑟 This field is uniform along wire 2 and perpendicular to it, and so the force F2 it exerts on wire 2 is given by 𝐹 = 𝐼𝑙𝐵𝑠𝑖𝑛 𝜃with sinθ =1 : 𝐹2 = 𝐼2 𝑙 𝐵1 By Newton’s third law, the forces on the wires are equal in magnitude, and so we just write for the magnitude of F2. Substituting the expression for into the last equation and rearranging terms gives 𝐹 𝜇0 𝐼1 𝐼2 = 𝑙 2𝜋𝑟  One ampere of current through each of two parallel conductors of infinite length, separated by one meter in empty space free of other magnetic fields, causes a force of exactly 2 X 10-7 N/m on each conductor. Applications of Magnetism Mass Spectrometry The curved paths followed by charged particles in magnetic fields can be put to use. A charged particle moving perpendicular to a magnetic field travels in a circular path having a radius. 𝑚𝑣 𝑟= 𝑞𝐵 It was noted that this relationship could be used to measure the mass of charged particles such as ions. A mass spectrometer is a device that measures such masses. Most mass spectrometers use magnetic fields for this purpose, although some of them have extremely sophisticated designs.  Mass spectrometry today is used extensively in chemistry and biology laboratories to identify chemical and biological substances according to their mass-to-charge ratios. In medicine, mass spectrometers are used to measure the concentration of isotopes used as tracers. Usually, biological molecules such as proteins are very large, so they are broken down into smaller fragments before analyzing. Recently, large virus particles have been analyzed as a whole on mass spectrometers. Sometimes a gas chromatograph or high-performance liquid chromatograph provides an initial separation of the large molecules, which are then input into the mass spectrometer. Cathode Ray Tubes—CRTs What do non-flat-screen TVs, old computer monitors, x-ray machines, have in common? All of them accelerate electrons, making them different versions of the electron gun. Many of these devices use magnetic fields to steer the accelerated electrons. Figure shows the construction of the type of cathode ray tube (CRT) found in some TVs, oscilloscopes, and old computer monitors. Two pairs of coils are used to steer the electrons, one vertically and the other horizontally, to their desired destination. Cathode Ray Tubes—CRTs The cathode ray tube (CRT) is so named because rays of electrons originate at the cathode in the electron gun. Magnetic coils are used to steer the beam in many CRTs. In this case, the beam is moved down. Another pair of horizontal coils would steer the beam horizontally. Magnetic Resonance Imaging Magnetic resonance imaging (MRI) is one of the most useful and rapidly growing medical imaging tools. MRI is based on an effect called Nuclear Magnetic Resonance (NMR) in which an externally applied magnetic field interacts with the nuclei of certain atoms, particularly those of hydrogen (protons). These nuclei possess their own small magnetic fields, similar to those of electrons and the current loops.  When placed in an external magnetic field, such nuclei experience a torque that pushes or aligns the nuclei into one of two new energy states— depending on the orientation of its spin. Transitions from the lower to higher energy state can be achieved by using an external radio frequency signal to “flip” the orientation of the small magnets. The specific frequency of the radio waves that are absorbed and reemitted depends sensitively on the type of nucleus, the chemical environment, and the external magnetic field strength. Therefore, this is a resonance phenomenon in which nuclei in a magnetic field act like resonators that absorb and reemit only certain frequencies. Hence, the phenomenon is named nuclear magnetic resonance (NMR).  NMR has been used for more than 50 years as an analytical tool. It was formulated in 1946 by F. Bloch and E. Purcell, with the 1952 Nobel Prize in Physics going to them for their work. Over the past two decades, NMR has been developed to produce detailed images in a process now called magnetic resonance imaging (MRI). The 2003 Nobel Prize in Medicine went to P. Lauterbur and P. Mansfield for their work with MRI applications. The largest part of the MRI unit is a superconducting magnet that creates a magnetic field, typically between 1 and 2 T in strength, over a relatively large volume. MRI images can be both highly detailed and informative about structures and organ functions. With excellent spatial resolution, MRI can provide information about tumors, strokes, shoulder injuries, infections, etc.  MRI is less effective than x rays for detecting breaks in bone. MRI images are also expensive compared to simple x-ray images and tend to be used most often where they supply information not readily obtained from x rays. Another disadvantage of MRI is that the patient is totally enclosed with detectors close to the body for about 30 minutes or more, leading to claustrophobia. Most of the brain scans today use fMRI. called “functional MRI” (fMRI), has allowed us to map the functioning of various regions in the brain responsible for thought and motor control. Other Medical Uses of Magnetic Fields Currents in nerve cells and the heart create magnetic fields like any other currents. These can be measured but with some difficulty since their strengths are about 10-6 to 10-8 less than the Earth’s magnetic field. Recording of the heart’s magnetic field as it beats is called a Magnetocardiogram (MCG). while measurements of the brain’s magnetic field is called a magnetoencephalogram (MEG). THANK YOU