Moving Charge in Magnetic Field


 Trevor McDonald
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1 Chapter 1 Moving Charge in Magnetic Field Day 1 Introduction Two bar magnets attract when opposite poles (N and S, or and N) are next to each other The bar magnets repel when like poles (N and N, or S and S) are next to each other, Magnetic phenomena were first observed at least 2500 years ago in fragments of magnetized iron ore found near the ancient city of Magnesia (now Manias, is western Turkey). These fragments were examples of what are now called permanent magnets Before the relation of magnetic interactions to moving charges was understood, the interactions of permanent magnets and compass needles were described in terms of magnetic poles. If a barshaped permanent magnet, or bar magnet, is free to rotate, one end points north. This end is called a north pole or Npole; the other end is a south pole or Spole. Opposite pole attract each other, and like poles repel each other. An object that contains iron but is not itself magnetized (that is, it shows no tendency to point north or south) is attracted by either pole of a permanent magnet. The earth itself is a magnet. Its north geographical pole is close to a magnetic south pole, which is why the north pole of a compass needle points north. The earth s magnetic axis is not quite parallel to its geographical axis (the axis of rotation), so a compass reading deviates somewhat from geographic north. This deviation, which varies with location, is called magnetic declination or magnetic variation. Also, the magnetic field is not horizontal at most points on the earth s surface; its angle up or down is called magnetic inclination. At the magnetic poles the magnetic field is vertical Page 1
2 (a), (b) Either pole of a bar magnet attracts an unmagnetized object that contains iron. The concept of magnetic poles may appear similar to that of electric charge, and north and south poles may seem analogous to positive and negative charge. But the analogy can be misleading. While isolated positive and negative charges exist, there is no experimental evidence that a single isolated magnetic pole exists; poles always appear in pairs. If a bar magnet is broken in two, each broken end becomes a pole. The existence of an isolated magnetic pole, or magnetic monopole, would have a sweeping implication for theoretical physics. Extensive searches for magnetic monopoles have been carried out, but so far without success A compass placed at any location in the earth s magnetic field points in the direction of the field line at that location. Representing the earth s field as that of a tilted bar magnet is only a crude approximation of its fairly complex configuration. The field, which is caused by currents in the earth s molten core, changes with time; geologic evidence shows that it reverses direction entirely at irregular intervals of about a half million years. Page 2
3 Breaking a bar magnet. Each piece has a north and south pole, even if the pieces are different sizes. (The smaller the piece, the weaker its magnetism) In Oersted s experiment, a compass is placed directly over a horizontal wire (hire viewed from above). When the compass is placed directly under the wire, the compass swings are reversed. Page 3
4 Magnetic Field 1 A moving charge or a current creates a magnetic field in the surrounding space (in addition to its electric field) 2 The magnetic field exerts a force on any other moving charge or current that is present in the field Like electric field, magnetic field is a vector fieldthat is, a vector quantity associated with each point in space. We will use the symbol for magnetic field. At any position the direction of is defined as that in which the north pole of a compass needle tends to point. The arrows in suggest the direction of the earth s magnetic field; for any magnet, its south pole. points out of its north pole and into The direction of is always perpendicular to the plane containing and. Its magnitude is given by Page 4
5 Where is the magnitude of the charge and ϕ is the angle measured from the direction of to the direction of, as shown in the figure (magnetic force on a moving charged particle) The units of B must be the same as the units of F/qv. Therefore the SI unit of B is equivalent to 1N.s/C.m, or, since one ampere is one coulomb per second (1A = 1C/s), 1N/A, m. This unit is called the tesla (abbreviated T), in honor of Nikola Tesla ( ), the prominent Serbian American scientist and inventor Another unit of B, the gauss (1G = 10 4 T) is also in common use. Instruments for measuring magnetic field are sometimes called gauss meters The magnetic field of the earth is of the order of 10 4 T or 1G. Magnetic fields of the order of 10T occur in the interior of atoms and are important in the analysis of atomic spectra. The largest steady magnetic field that can be produced at present in the laboratory is about 45 T. Some pulsed current electromagnets can produce fields of the order of 120 T for short time intervals of the order of a millisecond. The magnetic field at the surface of a neutron star is believed to be of the order of 10 8 T. Page 5
6 Chapter 2 Magnetic field of Current Element Day 1 Introduction Principle of superposition of magnetic fields: The total magnetic field caused by several moving charges is the vector sum of the fields caused by the individual charges. We begin by calculating the magnetic field caused by a short segment of a current carrying conductor, as shows in Fig. The volume of the segment is a dl, where A is the cross sectional area of the conductor. If there are moving charged particles per unit volume, each of charge q, the total moving charge dq in the segment is The moving charges in this segment are equivalent to a single charge dq, traveling with a velocity equal to the drift velocity. (Magnetic fields due to the random motions of the charges will, on average, cancel out at every point) From the magnitude of the resulting field P is at any field point But from Eq. A equals the current I in the element. So Page 30
7 For these field points, and both lie in the tan colored plane, and is perpendicular to this plane for these field points, and both lie in the orange colored plane, and is perpendicular to this plane (a) Magnetic field vectors due to a current element. (b) Magnetic field lines in a plane containing the current element. The indicates that the current is directed into the plane of the page. Compare this figure to Fig. for the field of a moving point charge. Law of Biot and Savart (pronounced Bee oh and Such var ). We can use this law to find the total magnetic field at any point in space due to the current in a complete circuit. To do this, we integrate over all segments that carry current; symbolically, The field vectors and the magnetic field lines of a current element are exactly like those set up by a positive charge dq moving in the direction of the drift velocity. The field lines are circles in planes perpendicular to and centered on the line of. Their directions are given by the same right hand rule that we introduced for point charges. What we measure experimentally is the total for a complete circuit. But we can still verify these equations indirectly by calculating for various current configurations using and comparing the results with experimental measurements. If matter is present in the space around a current carrying conductor, the field at a field point P in its vicinity will have an additional contribution resulting from the magnetization of the material. We ll have return to this point in Section.However, unless the material is iron or some other ferromagnetic material, the additional field is small and is usually negligible. Additional complications arise if time varying electric or magnetic fields are present or if the material is a super conductor; we ll return to these topics later. Magnetic Field of a Straight Current Carrying Conductor We first use the law of Biot and Savart, to find the field caused by the element of conductor of length dl = dy. From the figure, and The right hand Page 31
8 rule for the vector product shows that the direction of is into the plane of the figure, perpendicular to the plane; furthermore, the directions of the a significant simplification Magnetic Field of a Circular Current Loop Magnetic field of a circular loop. The current in the segment causes the field which lies in the xyplane. The currents in other cause with different components perpendicular to the x axis; these components add to zero. The x  components of the total field at point P combine to give the Page 32
9 Illustration Two circular coils X and Y having equal number of turns and carry equal currents in the same sense and subtend same solid angle at point O. If the smaller coil X is midway between O and Y, then if we represent the magnetic induction due to bigger coil Y at O as B, and due to smaller coil X at O s B x then (a) B y /B x = 1 (b) B y /B x = 2 (c) B y /B x = 2 (d) B y /B x = ¼ Solution Now use binomial (1 + x) n = 1 + nx if x << i then divide. Page 33
10 Illustration Two circular coils of wires made of similar wires but of radius 20 cm and 40 cm are connected in parallel. The ratio of the magnetic fields at their centre is (a) 4 : 1 (b) 1 : 4 (c) 2 : 1 (d) 1 : 2 Solution Page 34
11 Chapter 3 Magnetic Flux and Gauss s law of Magnetism Day 1 Introduction We define the magnetic flux ϕ B through a surface just as we defined electric flux in connection with Gauss s law. We can divide any surface into elements of area da. For each element we determine the component of normal to the surface. (Be careful not to confuse ϕ with.) In general, this component varies from point to point on the surface. We define the magnetic flux through this area as (magnetic flux through a surface) Magnetic flux is a scalar quantity. In the special case in which is uniform over a plane surface with total area A. B and ϕ are the same at all points on the surface and The SI unit of magnetic flux is equal to the unit of magnetic field (1T) times the unit of area (Im 2 ). This unit is called the Weber (1Wb), in honor of the German physicist Wilhelm Weber ( ) The magnetic flux through an area element da is defined to be Page 67
12 In Gauss s law the total electric flux through a closed surface is proportional to the total electric charge enclosed by the surface. For example, if the closed surface encloses an electric dipole, the total electric flux is zero because the total charge is zero. The total magnetic flux through a closed surface would be proportional to the total magnetic charge enclosed. But we have mentioned that no magnetic monopole has ever been observed, despite intensive searches. We conclude that the total magnetic flux through a closed surface is always zero. Symbolically In Gauss s law the total electric flux through a closed surface is proportional to the total electric charge enclosed by the surface. For example, if the closed surface encloses an electric dipole, the total electric flux is zero because the total charge is zero. The total magnetic flux through a closed surface would be proportional to the total magnetic charge enclosed. But we have mentioned that no magnetic monopole has ever been observed, despite intensive searches. We conclude that the total magnetic flux through a closed surface is always zero. Symbolically Caution Unlike electric field lines that begin and end on electric charges, magnetic field lines never have end points; such a point would indicate the presence of a monopole. You might be tempted to draw magnetic field lines that begin at the north pole of a magnet and end at a south pole. But as Fig Shows, the field lines of a magnet actually continue through the interior of the magnet. Like all other magnetic field lines, they form closed loops. For Gauss s law, which always deals with closed surface, the vector area element Eq. always points out of the surface. However, some applications of magnetic flux involve an open surface with a boundary line; there is then an ambiguity of sign in Eq. because of the two possible choices of direction for. In these cases we choose one of the possible choices of direction for. In these cases we choose one of the possible sides of the surface to be the positive side and use that choice consistently. If the element of area da in Eq. is at right angles to the field lines, then calling the area we have That is, the magnitude of magnetic field is equal to flux per unit area across an area at right angles to the magnetic field. For this reason, magnetic field Page 68 is sometimes called magnetic flux density Induction Experiments During the 1830s, several pioneering experiments with magnetically induced emf were carried out in England by Michael Faraday and in the United States by Joseph Henry ( ), later the first director of the Smithsonian Institution. Fig shows several examples. In Fig a coil of wire is connected to a galvanometer. When the nearby magnet is stationary, the meter shows no current. This isn t surprising; there is no source of emf in the circuit. But when we move the magnet either toward or away from the coil, the meter shows current in the circuit, but only while the magnet is
13 moving. If we keep the magnet stationary and move the coil, we again detect a current during the motion. We call this an induced current, and the corresponding emf required to cause this current is called an induced emf. To explore further the common elements in these observations, let s consider a more detailed series of experiments with the situation shown in Fig. We connect a coil of wire to a galvanometer, then place the coil between the poles of an electromagnet whose magnetic field we can very. Here s what we observe 1. When there is no current in the electromagnet, so that, the galvanometer shows no current. 2. When the electromagnet is turned on, there is a momentary current through the meter as increases 3. When levels off at a steady value, the current drops to zero, no matter how large is 4. With the coil in a horizontal plane, we squeeze it so as to decrease the cross sectional area of the coil. The meter detects current only during the deformation, not before or after. When we increase the area to return the coil to its original shape, there is current in the opposite direction, but only while the area of the coil is changing. Page 69
14 (a) A stationary magnet has no effect on a stationary coil of wire. A galvanometer connected to the coil shows zero current. (b) When the magnet and coil move relative to each other, a current is induced in the coil. The current is in one direction if the magnet moves down and the opposite direction if the magnet moves up. (c) We get the same effect as in (b) if we replace the magnet by a second coil carrying a constant current. (d) When the switch is opened or closed, the change in the inside coil s current induces a current in the outer coil 5. If we rotate the coil a few degrees about a horizontal axis, the meter detects current during the rotation, in the same direction as when we decreased the area. When we rotate the coil back, there is a current in the opposite direction during this rotation 6. If we jerk the coil out of the magnetic field, there is a current during the motion, in the same direction as when we decreased the area. Page 70
15 A coil in a magnetic field. When the field is constant and the shape, location, and orientation of the coil do not change, no current is induced in the coil. A current is induced when any of these factors change. 7. If we decrease the number of turns in the coil by unwinding one or more turns, there is a current during the unwinding, in the same direction as when we decreased the area. If we wind more turns onto the coil, there is a current in the opposite direction during the winding 8. When the magnet is turned off, there is a momentary current in the direction opposite to the current when it was turned on 9. The faster we carry out any of these changes, the greater the current 10. If all these experiments are repeated with a coil that has the same shape but different material and different resistance, the current in each case is inversely proportional to the total circuit resistance. This shows that the induced emf that are causing the current do not depend on the material of the coil but only on its shape and the magnetic field Faraday s Law The common element in all induction effects is changing magnetic flux through a circuit. Before stating the simple physical law that summarizes all of the kinds of experiments described in section, let s first review the concept of magnetic flux ϕ B (which we introduced in section). For an infinitesimal area element in a magnetic field, the magnetic flux dϕ B through the area is The magnetic flux through an area element da is defined to be Page 71
16 School Level Para magnetism In an atom, most of the various orbital and spin magnetic moments of the electrons add up to zero. However, in some cases the atom has a net magnetic moment that is of the order of μ B. When such a material is placed in a magnetic field, the field exerts a torque on each magnetic moment, as given by. These torques tend to align the magnetic moments with the field, the position of minimum potential energy, as we discussed in.this position, the directions of the current loops are such as to add to the externally applied magnetic field. We saw that the field produced by a current loop is proportional to the loop s magnetic dipole moment. In the same way, the additional field produced by microscopic electron current loops is proportional to the total magnetic moment per unit volume V in the material. We call this vector quantity the magnetization of the material, denoted by n The additional magnetic field due to magnetization of the material turns out to be equal simply to, where completely surrounds a current carrying conductor, the total magnetic field in the material is Where is the field caused by the current in the conductor. To check that the units in are consistent, note that magnetization is magnetic moment per unit volume. The units of magnetic moment are current times area (A.m 2 ), so the units of magnetization are (A.m 2 )/m 3 = A/m. From the units of the constant μ 0 are T. m/a. So the units of are the same as the units of (T. m/a) (A/m) = T. A material showing the behavior just described is said to be paramagnetic. The result is that the magnetic field material, than it would be if the material were replaced by vacuum. The value of K m is different for different materials; for common paramagnetic solids and liquids at room temperature, K m typically ranges from to All of the equations in this chapter that relate magnetic fields to their sources can be adapted to the situation in which the currentcarrying conductor is embedded in a paramagnetic material. All that need be done is to replace μ 0 by K m μ 0. This product is usually denoted as μ and is called the permeability of the material. The amount by which the relative permeability differs from unity is called the magnetic susceptibility, denoted by X m. Both K m and X m are dimensionless quantities. Values of magnetic susceptibility for several materials are given in Table. For example, for aluminum, X m = and K m = The first group of materials in the table are paramagnetic; we ll discuss the second group of materials, which are called diamagnetic, very shortly Page 105
17 Diamagnetism In some materials the total magnetic moment of all the atomic current loops is zero when no magnetic field is present. But even these materials have magnetic effects because an external field alters electron motions within the atoms, causing additional current loops and induced magnetic dipoles comparable of the induced electric dipoles we studied in section. In this case the additional field caused by these current loops is always opposite in direction to that of the external field. (This behavior is explained by Faraday s law of induction, which we will study. An induced current always tends to cancel the field change that caused it) Such materials are said to be diamagnetic. They always have negative susceptibility, as shown in Table and permeability K m slightly less than unity, typically of the order of to for solids and liquids. Diamagnetic susceptibilities are very nearly temperatureindependent In this drawing adapted from a magnified photo, the arrows shows show the directions of magnetization in the domains of a single crystal of nickel. Domains that are magnetized in the direction of an applied magnetic field grow larger. Ferromagnetism There is a third class of materials, called ferromagnetic materials, which includes iron, nickel, cobalt, and many alloys containing these elements. In these materials. Strong interactions between atomic magnetic moments cause them to line up parallel to each other in regions called magnetic domains, even when no external field is present. Fig show an example of magnetic domain structure. Within each domain, nearly all of the atomic magnetic moments are parallel For many ferromagnetic materials the relation of magnetization to external magnetic field is different when the external field is increasing from when it is decreasing. Fig shows this relation for such a material. When the material is magnetized to saturation and then the external field is reduced to zero, some magnetization remains. This behavior is characteristic of permanent magnets, which retain most of their saturation magnetization when the magnetizing field is removed. To reduce the magnetization to zero requires a magnetic field in reverse direction. This behavior is called hysteresis, and the curves in Fig are called hysteresis loops. Magnetizing and demagnetizing a material that has hysteresis involves the dissipation of energy, and the temperature of the material increases during such a process. Page 106
18 Ferromagnetic materials are widely used in electromagnets, transformer cores and motors and generators, in which it is desirable to have as large a magnetic field as possible for a given current. Because hysteresis dissipates energy, materials that are used in these applications should usually have as narrow a hysteresis loop as possible. Soft iron is often used; it has high permeability without appreciable hysteresis. For permanent magnets a broad hysteresis loop is usually desirable, with large zero field magnetization and large reverse field needed to demagnetize. Many kinds of steel and many alloys, such as Alnico, are commonly A magnetization curve for a ferromagnetic material. The magnetization M approaches its saturation value M sat as the magnetic field B 0 (caused by external currents) becomes large. Hysteresis loops. The materials of both (a) and (b) remain strongly magnetized when is reduced to zero. Since (a) is also hard to demagnetize, it would be good for permanent magnets. Since (b) magnetizes and demagnetizes more easily, it could be used as a computer memory material. The material of (c) would be useful for transformers and other alternating current devices where zero hysteresis would be optimal. Page 107
19 Summary The magnetic field created by a charge q moving with velocity depends on the distance r from the source point (the location of q) to the field point (where is measured). The field is perpendicular to and, the unit vector directed from the source point to the field point. The principle of superposition of magnetic fields states that the total field produced by several moving charges is the vector sum of the fields produced by the individual charges. The law of Biot and Savart gives the magnetic field d created by an element d of a conductor carrying current I. The field is perpendicular to both and the unit vector from the element to the field point. The field created by a finite current carrying conductor is the integral of over the length of the conductor. The magnetic field at a distance r from a long, straight conductor carrying a current I has a magnitude that is inversely proportional to r. the magnetic field lines are circles coaxial with the wire, with directions given by the righthand rule Page 108
20 Two long, parallel, currentcarrying conductors attract if the currents are in the same direction and repel if the currents are in opposite directions. The magnetic force per unit length between the conductors depends on their currents I and I and their separation r. The definition of the ampere is based on this relation. The law of Biot and Savart allots us to calculate the magnetic field produced along the axis of a circular conducting loop of radius a carrying current I. the field depends on the distance x along the axis from the center of the loop to the field point. If there are N loops, the field is multiplied by N. at the center of the loop x = 0. Circular loop (centre of N circular loop) Ampere s law sates that the line integral of around any closed path equals times the net current through the area enclosed by the path. The positive sense of current is determined by a righthand rule. The following table lists fields caused by several current distributions. In each the conductor is carrying current I. Current Distribution Point in Magnetic Field Magnetic Field Magnitude Long, straight conductor Distance r from conductor Circular loop of radius a Long cylindrical conductor of radius R On axis of loop At center of loop Inside conductor, r < R Outside conductor, r > R Page 109 (for N loops, multiply these expressions by N)
21 Long, closely wound solenoid with n turns per unit length, near its midpoint Inside solenoid, near center Outside solenoid Tightly wound toroidal solenoid (toroid) with N turns With the space enclosed by the windings, distance r from symmetry axis Outside the space enclosed by the windings B 0 When magnetic materials are present, the magnetization of the material causes an additional contribution to. For paramagnetic and diamagnetic materials, μ 0 is replaced in magnetic field expressions by. Where μ is the permeability of the material and K m is its relative permeability. The magnetic susceptibility X m is defined as X m = K m 1. Magnetic susceptibilities for paramagnetic materials are small positive quantities; those for diamagnetic materials are small negative quantities. For ferromagnetic materials, K m is much larger than unity and is not constant. Some ferromagnetic materials are permanent magnets, retaining their magnetization even after the external magnetic field is removed. Page 110
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