MINISTRY OF EDUCATION AND TRAINING NONG LAM UNIVERSITY FACULTY OF CHEMICAL ENGINEERING AND FOOD TECHNOLOGY Course: Physics Module 4: Electricity and magnetism Instructor: Dr Nguyen Thanh Son Academic year: 2022-2023 Contents Module 4: Electricity and magnetism 4.1 Electromagnetic concepts and law of conservation of electric charge 4.1.1 Electromagnetic concepts 4.1.2 Law of conservation of electric charge 4.2 Electric current 4.2.1 Electric current 4.2.2 Electric current density 4.3 Magnetic interaction - Ampère’s law 4.3.1 Magnetic interaction 4.3.2 Ampère’s law for magnetic field 4.4 Magnetic intensity 4.4.1 Magnetic intensity 4.4.2 Relationship between magnetic intensity and magnetic induction 4.5 Electromagnetic induction 4.5.1 Magnetic flux 4.5.2 Faraday’s law of electromagnetic induction 4.6 Magnetic energy 4.6.1 Energy stored in a magnetic field 4.6.2 Magnetic energy density Physic Module 4: Electricity and magnetism 4.1 Electromagnetic concepts and law of conservation of electric charge 4.1.1 Electromagnetic concepts • A magnetic field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles (such as permanent magnets) When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field Magnetic fields surround and are created by electric currents, magnetic dipoles, and changing electric fields Magnetic fields also have their own energy, with an energy density proportional to the square of the field magnitude • Magnetic field forms one aspect of electromagnetic field A pure electric field in one reference frame will be viewed as a combination of both an electric field and a magnetic field in a moving reference frame Together, electric and magnetic fields make up electromagnetic field, which is best known for underlying light and other electromagnetic waves • Electromagnetism describes the relationship between electricity and magnetism Electromagnetism is essentially the foundation for all of electrical engineering We use electromagnets to generate electricity, store memory on computers, generate pictures on a television screen, diagnose illnesses, and in just about every other aspect of our lives that depends on electricity • Electromagnetism works on the principle that an electric current through a wire generates a magnetic field We already know that a charge in motion creates a current If the movement of the charge is restricted in such a way that the resulting current is constant in time, the field thus created is called a static magnetic field Since the current is constant in time, the magnetic field is also constant in time The branch of science relating to constant magnetic field is called magnetostatics, or static magnetic field In this case, we are interested in the determination of (a) magnetic field intensity, (b) magnetic flux density, (c) magnetic flux, and (d) the energy stored in the magnetic field ♦ Linking electricity and magnetism • There is a strong connection between electricity and magnetism With electricity, there are positive and negative charges With magnetism, there are north and south poles Similar to electric charges, like magnetic poles repel each other, while unlike poles attract Physic Module 4: Electricity and magnetism • An important difference between electricity and magnetism is that in electricity it is possible to have individual positive and negative charges In magnetism, north and south poles are always found in pairs Single magnetic poles, known as magnetic monopoles, have been proposed theoretically, but a magnetic monopole has never been observed • In the same way that electric charges create electric fields around them, north and south poles will set up magnetic fields around them Again, there is a difference While electric field lines begin on positive charges and end on negative charges, magnetic field lines are closed loops, extending from the south pole to the north pole and back again (or, equivalently, from the north pole to the south pole and back again) With a typical bar magnet, for example, the field goes from the north pole to the south pole outside the magnet, and back from south to north inside the magnet • Electric fields come from electric charges So magnetic fields, but from moving charges, or currents, which are simply a whole bunch of moving charges In a permanent magnet, the magnetic field comes from the motion of the electrons inside the material, or, more precisely, from something called the electron spin The electron spin is a bit like the Earth spinning on its axis • The magnetic field is a vector; the same way the electric field is The electric field at a particular point is in the direction of the force that a positive charge would experience if it were placed at that point The magnetic field at a point is in the direction of the force that a north pole of a magnet would experience if it were placed there In other words, the north pole of a compass points in the direction of the magnetic field that exerts a force on the compass • The symbol for magnetic field induction or magnetic flux density is the letter B The SI unit of B is the tesla (T) • One of various manifestations of the linking between electricity and magnetism is electromagnetic induction (see Section 4.5) This involves generating a voltage (an induced electromotive force) by changing the magnetic field that passes through a coil of wire • In other words, electromagnetism is a two-way link between electricity and magnetism An electric current creates a magnetic field, and a magnetic field, when it changes, creates a voltage The discovery of this link led to the invention of transformer, electric motor, and generator It also explained what light is and led to the invention of radio Physic Module 4: Electricity and magnetism 4.1.2 Law of conservation of electric charge • Electric charge • There are two kinds of charge, positive and negative • Like charges repel; unlike charges attract • Positive charge results from having more protons than electrons; negative charge results from having more electrons than protons • Charge is quantized, meaning that charge comes in integer multiples of the elementary charge e • Charge is conserved • Probably everyone is familiar with the first three concepts, but what does it mean for charge to be quantized? Charge comes in multiples of an indivisible unit of charge, represented by the letter e In other words, charge comes in multiples of the charge of the electron or the proton A proton has a charge of +e, while an electron has a charge of -e The amount of electric charge of any object is only available in discrete units These discrete units are exactly equal to the amount of electric charge that is found on the electron or the proton • Electrons and protons are not the only things that carry charge Other particles (positrons, for example) also carry charge in multiples of the electronic charge Putting "charge is quantized" in terms of an equation, we say: q = ne (79) where q is the symbol used to represent electric charge, while n is a positive or negative integer (n = 0, ±1, ±2, ±3, …), and e is the elementary charge, 1.60 x 10-19 coulombs • Table of elementary particle masses and charges: Physic Module 4: Electricity and magnetism ♦ The law of conservation of electric charge • The law of conservation of charge states that the net charge of an isolated system remains constant This law is inherent to all processes known to Physics • In other words, electric charge conservation is the principle that electric charges can neither be created nor destroyed The quantity of electric charge of an isolated system is always conserved • If a system starts out with an equal number of positive and negative charges, there is nothing we can to create an excess of one kind of charge in that system unless we bring in some charge from outside the system (or remove some charge from the system) Likewise, if something starts out with a certain net charge, say +100 e, it will always have +100 e unless it is allowed to interact with something external to it ♦ Electrostatic charging • Forces between two electrically-charged objects can be extremely large Most things are electrically neutral; they have equal amounts of positive and negative charge If this was not the case, the world we live in would be a much stranger place We also have a lot of control over how things get charged This is because we can choose the appropriate material to use in a given situation • Metals are good conductors of electric charge, while plastics, wood, and rubber are not They are called insulators Charge does not flow nearly as easily through insulators as it does through conductors; that is the reason why wires you plug into a wall socket are covered with a protective rubber coating Charge flows along the wire, but not through the coating to you • In fact, materials are divided into three categories, depending on how easily they will allow charge (i.e., electrons) to flow along them These are: • conductors, metals, for example, • semi-conductors, silicon is a good example, and • insulators, rubber, wood, plastics, for example • Most materials are either conductors or insulators The difference between them is that in conductors, the outermost electrons in the atoms are so loosely bound to their atoms that they are Physic Module 4: Electricity and magnetism free to travel around In insulators, on the other hand, the electrons are much more tightly bound to their atoms, and are not free to flow Semi-conductors are a very useful intermediate class, not as conductive as metals but considerably more conductive than insulators By adding certain impurities to semi-conductors in the appropriate concentrations, the conductivity can be wellcontrolled • There are three ways that objects can be given a net charge These are: Charging by friction - this is useful for charging insulators If you rub one material with another (say, a plastic ruler with a piece of paper towel), electrons have a tendency to be transferred from one material to the other For example, rubbing glass with silk or saran wrap generally leaves the glass with a positive charge; rubbing PVC rod with fur generally gives the rod a negative charge Charging by conduction - useful for charging metals and other conductors If a charged object touches a conductor, some charge will be transferred between the object and the conductor, charging the conductor with the same sign as the charge on the object Charging by induction - also useful for charging metals and other conductors Again, a charged object is used, but this time it is only brought close to the conductor, and does not touch it If the conductor is connected to ground (ground is basically anything neutral that can give up electrons to, or take electrons from, an object), electrons will either flow on to it or away from it When the ground connection is removed, the conductor will have a charge opposite in sign to that of the charged object • Electric charge is a property of the particles that make up an atom The electrons that surround the nucleus of the atom have a negative electric charge The protons which partly make up the nucleus have a positive electric charge The neutrons which also make up the nucleus have no electric charge The negative charge of the electron is exactly equal and opposite to the positive charge of the proton For example, two electrons separated by a certain distance will repel one another with the same force as two protons separated by the same distance, and, likewise, a proton and an electron separated by the same distance will attract one another with a force of the same magnitude • In practice, charge conservation is a physical law that states that the net change in the amount of electric charge in a specific volume of space is exactly equal to the net amount of charge Physic Module 4: Electricity and magnetism flowing into the volume minus the amount of charge flowing out of the volume In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region Mathematically, we can state the law as q(t2) = q(t1) + qin – qout (80) where q(t) is the quantity of electric charge in a specific volume at time t, q in is the amount of charge flowing into the volume between time t1 and t2, and qout is the amount of charge flowing out of the volume during the same time period • Another statement for this law is the net electric charge of an isolated system remains constant • The simple version of (80) is q = constant for an isolated system (80’) • The SI unit of electric charge is the coulomb (C) 4.2 Electric current 4.2.1 Electric current • Electric current is the flow of electric charge, as shown in Figure 51 The moving electric charges may be either electrons or ions or both • Whenever there is a net flow of charge through some region, an electric current is said to exist To define current more precisely, suppose that the charges are moving perpendicular to a surface of area A, as shown in Figure 51 This area could be the cross-sectional area of a wire, for example • The electric current intensity I is the rate at which charge flows through this surface If ∆Q is the amount of charge that passes through Physic Module 4: Electricity and magnetism Figure 51 Charges in motion through an area A The time rate at which charge flows through the area is defined as the current intensity I The direction of the current is the direction in which positive charges flow when free to so this area in a time interval ∆t, the average current intensity Iave is equal to the charge that passes through A per unit time: Iave = ∆Q/∆t (81) • If the rate at which charge flows varies in time, then the current varies in time; we define the instantaneous current intensity I as the differential limit of average current: I = lim ∆t → ∆Q dQ = ∆t dt (82) I = Q/t (82’) • If I is constant, (82) becomes where Q is the quantity of electric charge passing through the cross-sectional area in the time t • The SI unit of electric current intensity is the ampère (A): A = C/1 s That is, A of current is equivalent to C of charge passing through the surface area in s • If the ends of a conducting wire are connected to form a loop, all points on the loop are at the same electric potential, and hence the electric field is zero within and at the surface of the conductor Because the electric field is zero, there is no net transport of charge through the wire, and therefore there is no electric current • If the ends of the conducting wire are connected to a battery, all points on the loop are not at the same potential The battery sets up a potential difference between the ends of the loop, creating an electric field within the wire The electric field exerts forces on the electrons in the wire, causing them to move around the loop and thus creating an electric current It is common to refer to a moving charge (positive or negative) as a mobile charge carrier For example, the mobile charge carriers in a metal are electrons ♦ Electric current direction • The charges passing through the surface, as shown in Figure 51, can be positive or negative, or both It is conventional to assign the electric current direction the same direction as the flow of positive charge In electrical conductors, such as copper or aluminum, the electric Physic Module 4: Electricity and magnetism current is due to the motion of negatively charged electrons Therefore, when we speak of electric current in an ordinary conductor, the direction of the current is opposite to that of the flow of electrons However, if we are considering a beam of positively charged protons in an accelerator, the current is in the direction of motion of the protons In some cases - such as those involving gases and electrolytes, for instance - the electric current is the result of the flow of both positive and negative charges • An electric current can be represented by an arrow The sense of the electric current arrow is defined as follows: If the current is due to the motion of positive charges, the current arrow is parallel to the charge velocity If the current is due to the motion of negative charges, the current arrow is antiparallel to the charge velocity Example: During minutes a 5.0 A current is set up in a metal wire, how many (a) coulombs and (b) electrons pass through any cross section across the wire’s width? ANS: (a) Q = It = 1.2x103 C (b) N = Q/e = 7.5 x 10 21 Solution (a) From (82’) we have Q = It; plugging numbers leads to Q = It = 1.2x103 C (b) N = Q/|qe| = Q/e = 7.5 x 1021 (e = 1.60 x 10 -19 C) 4.2.2 Electric current density • Electric current density J is a vector quantity whose magnitude is the ratio of the magnitude of electric current flowing in a conductor to the cross-sectional area perpendicular to the current flow and whose direction points in the direction of the current • In other words, J is a vector quantity, and the scalar product of which with the cross-sectional area vector A is equal to the electric current intensity By magnitude it is the electric current intensity divided by the cross-sectional area If the electric current density is constant then I= J.A (scalar product of J and A ) If the current density is not constant, then Physic Module 4: Electricity and magnetism 10 (83) Magnetic field created by a long straight coil of wire (solenoid) carrying an electric current • A long straight coil of wire can be used to generate a nearly uniform magnetic field similar to that of a bar magnet Such coils, called solenoids, have an enormous number of practical applications The field can be greatly strengthened by the addition of an iron core Such cores are typical in electromagnets • In Equation 95 for the magnetic field B inside a solenoid carrying an electric current, n is the number of turns per unit length, sometimes called the "turn density" The expression is an idealization to an infinitely long solenoid, but provides a good approximation to the magnetic field created by a long solenoid carrying an electric current Physic Module 4: Electricity and magnetism 23 Solenoid field from Ampère’s law • Taking a rectangular path about which to evaluate ∫ B.ds such that the length of the side parallel to the solenoid field is L gives a contribution BL inside the coil The field is essentially perpendicular to the other sides of the path, giving negligible contribution If the end is taken so far from the coil that the field is negligible, then the length inside the coil is the dominant contribution • This admittedly idealized case for Ampère’s law gives (95) • This turns out to be a good a proximation for the solenoid field, particularly in the case of an iron core solenoid Figure 61 Magnetic field created by a long straight coil of wire (solenoid) carrying an electric current Example: An air-core solenoid is 100 cm long and has 3000 turns of copper wire It carries an electric current of A Find the magnetic field inside the solenoid Solution The core is air µ = µ0 Using (95), we have B = µnI = µ0 (N/L)I Plugging numbers leads to B = 1.51x10-2 T Recalling Physic Module 4: Electricity and magnetism 24 Magnetic field created by a toroid carrying an electric current • A device called a toroid (see Figure 62) is often used to create a magnetic field with almost uniform magnitude in some enclosed area The device consists of a conducting wire wrapped around a ring (a torus) made of a nonconducting material For a toroid having N closely spaced turns of wire, we calculate the magnetic field in the region occupied by the torus, a distance r from the center • To calculate this field, we must evaluate ∫ B.ds over the circle of radius r, as shown in Figure 62 By symmetry, we see that the field has a constant magnitude on this circle and is tangent to it, so B d s = Bds Furthermore, note that the closed circular path surrounds N loops of wire, each of which carries a current I Therefore, the right side of Equation 93 is µ0NI in this case • Ampère’s law applied to the circle gives B= à0 NI r (96) ã This result shows that B varies as 1/r and hence is nonuniform in the region occupied by the torus However, if r is very large compared with the cross-sectional radius of the torus, then the field is approximately uniform inside the torus • For an ideal toroid, in which the turns are closely spaced, the external magnetic field is zero This can be seen by noting that the net current passing through any circular path lying outside the toroid (including the region of the “hole in the doughnut”) is zero Therefore, from Ampère’s law we find that B = in the regions exterior to the torus Physic Module 4: Electricity and magnetism 25 • Finding the magnetic field inside a toroid is a good example of the power of Ampère’s law The current enclosed by the dashed line is just the number of loops N times the current intensity I in each loop • Ampere’s law then gives the magnetic field by (96) • The toroid is a useful device used in everything from tape heads to tokamaks Figure 62 Magnetic field created by a toroid carrying an electric current From Equation 96 we see that the magnitude of the magnetic field created by a toroid carrying an electric current = permeability x turn density x current intensity If the core is not air of the toroid, in (96) we use µ instead of µ0 where µ is the permeability of the core medium Example An air – core toroidal coil has 3000 turns and carries an electric current with intensity I = 5.0 A The inner and outer diameters are 22 cm and 26 cm, respectively Calculate the magnetic flux intensity B at a distance of r from the center of the coil where r is the mean radius of the coil Solution Physic Module 4: Electricity and magnetism 26 r = (rinner + router )/2 = (22/2 + 26/2)/2 (cm) = 12 cm The core is air µ = µ0 Using (96), we have B = 0.025 T 4.4 Magnetic field intensity or magnetic field strength 4.4.1 Magnetic intensity • There are two vectors namely B and H characterizing a magnetic field The vector field B is known among electrical engineers as magnetic flux density or magnetic induction, or simply magnetic field, as used by physicists The vector field H is known among electrical engineers as the magnetic field intensity or magnetic field strength and is also known among physicists as auxiliary magnetic field or magnetizing field • The magnetic field B has the SI unit of teslas (T), equivalent to webers per square meter (Wb/m²) The vector field H is measured in ampères per meter (A/m) in the SI unit system An older unit of magnetic field strength is the oersted: A/m = 0.01257 oersted • The magnetic fields generated by electric currents and calculated from Ampere's law are characterized by the magnetic field B measured in teslas However, when the generated fields pass through magnetic materials which themselves contribute to internal magnetic fields, ambiguities can arise about what part of the field comes from the external currents and what part comes from the material itself It has been common practice to define another magnetic field quantity, usually called the "magnetic field strength" and designated by H 4.4.2 Relationship between magnetic intensity and magnetic induction • The commonly used form for the relationship between B and H is B = µH (97) where µ is the permeability of the medium and given by µ = Kmµ0 µ0 being the magnetic permeability of free space and Km the relative permeability of the material If the material does not respond to the external magnetic field by producing any magnetization, then Km = Physic Module 4: Electricity and magnetism 27 (98) ã For paramagnetic (à > µ0) and diamagnetic (µ < µ0) materials, the relative permeability is very close to For ferromagnetic materials, µ is much greater than µ0 4.5 Electromagnetic induction 4.5.1 Magnetic flux • The magnetic flux, ΦB, through an element of area perpendicular to the direction of magnetic field is given by the product of the magnetic field and the element’s area More generally, magnetic flux is defined by a scalar product of the magnetic field vector and the area element vector The SI unit of magnetic flux is the weber (Wb) • The magnetic flux through a surface is proportional to the net number of magnetic field lines that pass through the surface The net number of magnetic field lines is the total number passing through in one direction minus the total number passing through in the opposite direction • As illustrated by Figure 63, we divide the surface that has the loop as its border into small elements of area dA For Figure 63 Depicting of the magnetic flux each element we calculate the differential magnetic flux of the magnetic field B through it: dΦB = B.d A = B.dA.cosφ (99) where φ is the angle between the normal unit vector nˆ ( d A = nˆ dA) and the magnetic field vector B at the position of the element • We then integrate all the terms ΦB = ∫ B.dA.cosφ = ∫ B.dA (100) If B is constant over the surface A, then Equation 100 becomes ΦB = B.A = BA cos φ Physic Module 4: Electricity and magnetism 28 (100’) where φ is the angle between the normal unit vector nˆ ( A = nˆ A) of the surface and the magnetic field vector B In this case, note that nˆ or A is perpendicular to the surface of interest Example: A rectangular coil is located in a constant magnetic field whose magnitude is 0.5 T The coil has an area of 2.0 m2 Determine the magnetic flux for the orientations: (a) φ = 0o and (b) φ = 60 o Solution Because the magnetic field is constant, we can use (100’) ΦB = B.A = BA cos φ to determine the magnetic flux through the coil (a) φ = 0o (b) φ = 60o cosφ = 1; plugging numbers leads to ΦB = 1.0 Wb cosφ = 1/2; plugging numbers leads to ΦB = 0.5 Wb 4.5.2 Faraday’s law of electromagnetic induction ♦ Faraday's experiments • These experiments helped formulate what is known as "Faraday's law of electromagnetic induction." • The circuit shown in the left panel of Figure 64 consists of a wire loop connected to a sensitive ammeter (known as a "galvanometer") If we approach the loop with a permanent magnet, we see an electric current being registered by the galvanometer The results can be summarized as follows: i An electric current appears only if there is relative motion between the magnet and the loop ii Faster motion results in a larger current intensity iii If we reverse the direction of motion or the polarity of the magnet, the electric current reverses sign and flows in the opposite direction • The electric current generated is known as "induced current"; the electromotive force (emf) that appears is known as "induced emf"; the whole effect is called "electromagnetic induction." Figure 64 Faraday’s experiments of electromagnetic induction: (Left) A permanent magnet a loop;(Right) Switching Physic approaching Module 4: Electricity and magnetism 29 the electric current in one loop induces an electric current in another loop • In the right panel of Figure 64, we show a second type of experiment in which an electric current is induced in loop when the switch S in loop is either closed or opened When the electric current in loop is constant, no induced electric current is observed in loop • We see that the magnetic field in an electromagnetic induction experiment can be generated either by a permanent magnet or by an electric current in a coil • Faraday summarized the results of his experiments in what is known as Faraday's law of electromagnetic induction: An emf is induced in a loop when the number of magnetic field lines (or magnetic flux) that pass through the loop is changing • We can also express Faraday's law of electromagnetic induction in the following form: The magnitude of the emf induced in a conductive loop is equal to the rate at which the magnetic flux ΦB through the loop changes with time • The corresponding formula is ε=– dΦB dt (101) where ε is the induced emf If the circuit is a coil consisting of N loops of the same area and ΦB is still the flux through each loop, the total induced emf in the coil is given by the expression ε = –N dΦB dt (102) • The negative sign in Equations 101 and 102 is of important physical significance, as described later • The SI unit of emf is the volt (V) dΦB but we can find the change ∆ΦB of dt magnetic flux through a loop, ∆ΦB = ΦB,f – ΦB,i, in a time interval ∆t then we can calculate the average induced emf, denoted by εave, instead of ε and equations (101) and (102) become • Note that there are cases in which we cannot find Physic Module 4: Electricity and magnetism 30 and εave = –∆ΦB/∆t (101’) εave = –N∆ΦB/∆t (102’) respectively ♦ Methods for changing the magnetic flux ΦB through a loop • We see that the magnetic flux ΦB can be changed and an emf is then induced in a circuit in several ways: • The magnitude of B can change with time • The area enclosed by the loop can change with time • The angle φ between the magnetic field vector B and the normal vector nˆ to the loop can change with time • Any combination of the above can be used ♦ Lenz’s law • Faraday’s law of electromagnetic induction (Equation 101 or Equation 102) indicates that the induced emf and the change in flux have opposite algebraic signs This has a very real physical interpretation that has come to be known as Lenz’s law: The polarity of the induced emf is such that it tends to produce an electric current that creates a magnetic flux to oppose the change in the original magnetic flux through the area enclosed by the loop • That is, the induced current tends to keep the original magnetic flux through the circuit from changing This law is actually a consequence of the law of conservation of energy Figure 65 Depicting Lenz’s law • We now concentrate on the negative sign in the equation (Equation 101 or Equation 102) that expresses Faraday's law of electromagnetic induction The direction of the flow of induced current in a loop is accurately predicted by what is known as Lenz's law (or Lenz's rule) • To understand Lenz’s law, we consider an example as shown in Figure 65 In the figure, we show a bar magnet approaching a loop The induced electric current flows in the direction indicated because this current generates an induced magnetic field that has the field lines pointing from left to right The loop is then equivalent to a magnet whose north pole faces the Physic Module 4: Electricity and magnetism 31 corresponding north pole of the bar magnet that is approaching the loop The loop then repels the approaching magnet and thus opposes the change in the original magnetic flux that generates the induced current Example: A coil of conducting wire with 25 turns is wrapped on a square frame 1.80 cm on a side each turn has the same area, equal to that of the frame; the total resistance of the coil is 0.35 Ω A uniform magnetic field is applied perpendicularly to the plane of the coil If the magnitude of the field changes from to 0.5 T in 0.8 s, find (1) the magnitude of the average induced emf in the coil while the field is changing, and (2) the magnitude of the average induced current in the coil while the field is changing Solution (1) A = (1.8 x 10-2)2 m2 = 3.24 x 10-4 m2; using (100’) ∆ΦB = ΦB,f – ΦB,i = BA – = 1.62 x 10-4 Wb (Because the magnetic field is perpendicular to the plane of the coil, the angle φ is 0o cosφ = 1); from (102’) we have |εave| = N∆ΦB/∆t = 25 x 1.62 x 10 -4/0.8 (V) = 5.1 x 10 -3 V = 5.1 mV (2) Iave = |εave|/R = 5.1 mV/0.35 Ω = 14 mA ♦ MOTIONAL ELECTROMOTIVE FORCE • In examples illustrated by Figure 64, we considered cases in which an emf is induced in a stationary circuit placed in a magnetic field when the field changes with time In this section, we describe what is called motional electromotive force, which is the emf induced in a straight conductor moving through a constant magnetic field • Consider a loop of width l, as shown in Figure 66 Part of the loop is located in a region where a uniform magnetic field exists The magnetic flux through the loop is ΦB = Blx When the loop is being pulled out of the magnetic field region with Figure 66 Depicting the producing the motional electromotive force electromotive force constant speed v, the flux ΦB decreases with time t; according to Faraday’s law of electromagnetic induction, there is an induced emf in the loop, given by ε=– dΦB dx = –Bl = –Blv dt dt (103) where l is the width of the loop or the length of the straight conductor; B the magnitude of the uniform magnetic field and v the speed of the conductor Physic Module 4: Electricity and magnetism 32 • If the resistance of the circuit is R, the intensity (magnitude) of the induced current in the loop is I = |ε|/R = Blv/R (104) Example: A metal rod has a length of 1.6 m and is moving at a speed of 5.0 m/s in a direction perpendicular to a 0.8 T magnetic field Find the magnitude of the motional electromotive force ε induced in the rod Solution Plugging numbers into (103) leads to |ε| = |-Blv| = 6.4 V ♦ Self – induction • If we change the current i through an inductor whose inductance is L, this causes a change in the magnetic flux ΦB = Li through the inductor itself Using Faraday's law of Figure 67 Depicting of producing the self-induction electromagnetic induction, we can determine the resulting emf, known as self-induced emf εL εL = – dΦB di = –L dt dt (105) where L is the inductance of the inductor We have assumed that L is constant • Note that there are cases in which we cannot find di/dt but we can find the change ∆i of electric current intensity in the inductor, ∆i = if – ii, in a time interval ∆t then we can calculate the average self-induced emf, denoted by εLave, instead of εL and Equation (105) becomes εLave = –∆i/∆t (105’) • If the inductor is an ideal solenoid of cross-sectional area A with N turns, its inductance is given by L = µ(N2/l)A = µn2Al Physic Module 4: Electricity and magnetism 33 (106) where µ = Kmµ0 (Km is the relative permeability of the core) and n = N/l is the number of turns per unit length or the turn density of the solenoid • The permeability can be changed by putting a soft iron core into the solenoid, greatly increasing the inductance of the solenoid; for soft iron Km is much greater than • The SI unit of L is the henry (H) Example: (a) Calculate the inductance of an air-core solenoid containing 300 turns if the length of the solenoid is 25.0 cm and its cross-sectional area is 4.0 cm2 (b) Calculate the self-induced emf in the solenoid if the electric current intensity through it is decreasing at the rate of 50.0 A/s (Ans (a) 0.181 mH; (b) 9.05 mV Solution (a) Using (106) with µ = Kmµ0 = µ0 (Km = because the core is air) , we have L = 0.181 mH Recalling (b) We have di/dt = –50.0 A/s (decreasing minus sign); plugging values of L and di/dt into (105) leads to εL = 9.05 mV 4.6 Magnetic energy 4.6.1 Energy stored in a magnetic field ♦ RL circuit • Consider a series RL circuit, as shown in Figure 68 When the switch S is closed, the electric current intensity immediately starts to increase The induced emf (or back emf) in the inductor is large as the current intensity is changing Figure 68 A series RL circuit As the current intensity increases toward its maximum value, an emf that opposes the increasing current intensity is induced in the inductor rapidly As time goes on, the current intensity increases more slowly, and the potential difference across the inductor decreases • It takes energy to establish an electric current in an inductor; this energy is carried by the magnetic field inside the inductor Physic Module 4: Electricity and magnetism 34 • Considering the emf needed to establish a particular electric current and the power involved, we find: • As the current intensity through the coil increases, the magnetic field of the coil also increases and electrical energy from the source is converted into magnetic energy and this energy is stored in the coil The magnetic energy UB stored in the coil is given by UB = LI (107) • In capacitors, we found that energy is stored in the electric field between their plates In inductors, energy is similarly stored, only now in the magnetic field inside the inductors Just as with capacitors, where the electric field is created by a charge on the capacitor and electric energy is stored inside the capacitors, we now have a magnetic field created when there is an electric current through the inductor Thus, just as with the capacitors, the magnetic energy is stored inside the inductor • Again, although we introduce the magnetic energy when talking about energy in inductors, it is a generic concept – whenever a magnetic field is created, it takes energy to so, and that energy is stored in the field itself • The SI unit of magnetic energy is the joule (J) 4.6 Magnetic energy density • For simplicity, consider an ideal solenoid whose inductance is given by L = à(N2/l)A = àn2Al (Equation 106) ã The magnetic field inside a solenoid is given by B = µnI As a result, I = B/àn ã Substituting the expressions for L and for I into Equation 107 leads to UB = B2 Al 2à ã Because Al = V is the volume of the solenoid, the energy stored per unit volume in the magnetic field or the magnetic energy density, u B = UB/V, inside the inductor is Physic Module 4: Electricity and magnetism 35 (108) magnetic energy B uB = = volume 2à (109) ã Although this expression was derived for the special case of a solenoid, it is valid for any region of space in which a magnetic field exists regardless of its source From Equation 109, we see that magnetic energy density is proportional to the square of the magnetic field magnitude • The SI unit of magnetic energy density is the joule per cubic meter (J/m3) Example The earth’s magnetic field in a certain region has the magnitude 6.0 x 10 -5 T Find the magnetic energy density in this region (Ans 1.4 x 10 -3 J/m3) Solution Using (109) with µ = Kmµ0 = µ0 (µ = Km = because the region contains air), we have uB = 1.4 x 10-3 J/m3 Recalling REFERENCES 1) Halliday, David; Resnick, Robert; Walker, Jearl (1999), Fundamentals of Physics, 7th ed., John Wiley & Sons, Inc 2) Feynman, Richard; Leighton, Robert; Sands, Matthew (1989), Feynman Lectures on Physics, Addison-Wesley 3) Serway, Raymond; Faughn, Jerry (2003), College Physics, 7th ed., Thompson, Brooks/Cole 4) Sears, Francis; Zemansky Mark; Young, Hugh (1991), College Physics, 7th ed., AddisonWesley 5) Beiser, Arthur (1992), Physics, 5th ed., Addison-Wesley Publishing Company 6) Jones, Edwin; Childers, Richard (1992), Contemporary College Physics, 7th ed., AddisonWesley 7) Alonso, Marcelo; Finn, Edward (1972), Physics, 7th ed., Addison-Wesley Publishing Company 8) Michels, Walter; Correll, Malcom; Patterson, A L (1968), Foundations of Physics, 7th ed., Addison-Wesley Publishing Company 9) Hecht, Eugene (1987), Optics, 2th ed., Addison-Wesley Publishing Company 10) Eisberg, R M (1961), Modern Physics, John Wiley & Sons, Inc 11) Reitz, John; Milford, Frederick; Christy Robert (1993), Foundations of Electromagnetic Theory, 4th ed., Addison-Wesley Publishing Company 12) Priest, Joseph (1991), Energy: Principle, Problems, Alternatives, 4th ed., Addison-Wesley Publishing Company 13) Giambattista Alan; Richardson, B M; Richardson, R C (2004), College Physics, McGrawHill 14) WEBSITES http://ocw.mit.edu/OcwWeb/Physics/8-02TSpring-2005/LectureNotes/index.htm Physic Module 4: Electricity and magnetism 36 http://physics.bu.edu/~duffy/PY106/Charge.html http://science.jrank.org/pages/1729/Conservation-Laws-Conservation-electric-charge.html http://web.pdx.edu/~bseipel/ch31.pdf http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html#c1 http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcon.html#c1 https://courses.lumenlearning.com/physics/ Physic Module 4: Electricity and magnetism 37