Module 3: Electromagnetism

Transcription

1 Module 3: Electromagnetism Lecture - Magnetic Field Objectives In this lecture you will learn the following Electric current is the source of magnetic field. When a charged particle is placed in an electromagnetic field, it experiences a force, called Lorentz force. Motion of a charged particle in a magnetic field and concept of cyclotron frequency. Find the trajectory of a charged particle in crossed electric and magnetic field. MAGNETIC FIELD Electric charges are source of electric fields. An electric field exerts force on an electric charge, whether the charge happens to be moving or at rest. One could similarly think of a magnetic charge as being the source of a magnetic field. However, isolated magnetic charge ( or magnetic monopoles) have never been found to exist. Magnetic poles always occur in pairs ( dipoles) - a north pole and a south pole. Thus, the region around a bar magnet is a magnetic field. What characterizes a magnetic field is the qualitative nature of the force that it exerts on an electric charge. The field does not exert any force on a static charge. However, if the charge happens to be moving (excepting in a direction parallel to the direction of the field) it experiences a force in the magnetic field. It is not necessary to invoke the presence of magnetic poles to discuss the source of magnetic field. Experiments by Oersted showed that a magnetic needle gets deflected in the region around a current carrying conductor. The direction of deflection is shown in the figure below 1

2 Thus a current carrying conductor is the source of a magnetic field. In fact, a magnetic dipole can be considered as a closed current loop. Electric Current and Current Density Electric current is the rate of flow of charges in electrical conductor. In a conductor the charges may have random motion. However, the net drift velocity of the charges is zero, giving a zero net current. In the presence of an external force field, the charges move with a net non-zero drift velocity, which gives a current. The direction of current has been defined, conventionally, as the direction in which the positive charges move. In case of metallic conductors, the current is caused by flow of negatively charged electrons, whose direction of motion is opposite to the direction of current. In electrolytes, however, the current is due to flow of both positive and negative ions. The current density at a position is defined as the amount of charge crossing a unit crosssectional area per unit time. In terms of the net drift velocity of the charges (taken opposite to the net drift velocity of electrons),, where is the volume density of the mobile charges. The integral of the current density over a surface defines electric current, which is a scalar. The unit of electric current is Ampere (= coulomb/sec) and that of the current density is A/m. For a thin wire with a small cross sectional area, the current density may be taken as uniform. In this case, where is the component of the area vector parallel to the direction of current. 2

3 Lecture - Electric Current and Current Density The Right Hand Rule The direction of magnetic field due to a current carrying conductor is given by the right hand rule. If one clasps the conductor with one's right hand in such a way that the thumb points in the direction of the current (i.e. in the direction opposite to the direction of electron flow) then, the direction in which the fingers curl gives the direction of the magnetic field due to such a conductor. The Lorentz Force We know that an electric field exerts a force on a charge. In the presence of a magnetic field, a charge experiences an additional force where is the velocity of the charge. Note that There is no force on a charge at rest. A force is exerted on the charge only if there is a component of the magnetic field perpendicular to the direction of the velocity, i.e. the component of the magnetic field parallel to does not contribute to. 3

4 , which shows that the magnetic force does not do any work. In the case where both and are present, the force on the charge is given by This is called Lorentz force after H.E. Lorentz who postulated the relationship. It may be noted that the force expression is valid even when and are time dependent Unit of Magnetic Field From the Lorentz equation, it may be seen that the unit of magnetic field is Newtonsecond/coulomb-meter, which is known as a Tesla (T). (The unit is occasionally written as Weber/m as the unit of magnetic flux is known as Weber). However, Tesla is a very large unit and it is common to measure in terms of a smaller unit called Gauss, It may be noted that is also referred to as magnetic field of induction or simply as the induction field. However, we will use the term ``magnetic field". Motion of a Charged Particle in a Uniform Magnetic Field Let the direction of the magnetic field be taken to be z- direction, we can write the force on the particle to be The problem can be looked at qualitatively as follows. We can resolve the motion of the charged particle into two components, one parallel to the magnetic field and the other perpendicular to it. Since the motion parallel to the magnetic field is not affected, the velocity component in the z- direction remains constant. 4

5 where is the initial velocity of the particle. Let us denote the velocity component perpendicular to the direction of the magnetic field by. Since the force (and hence the acceleration) is perpendicular to the direction of velocity, the motion in a plane perpendicular to is a circle. The centripetal force necessary to sustain the circular motion is provided by the Lorentz force where the radius of the circle is called the Larmor radius, and is given by The time taken by the particle to complete one revolution is The cyclotron frequency is given by 5

6 Motion in a crossed electric and magnetic fields The force on the charged particle in the presence of both electric and magnetic fields is given by Let the electric and magnetic fields be at right angle to each other, so that, If the particle is initially at rest no magnetic force acts on the particle. As the electric field exerts a force on the particle, it acquires a velocity in the direction of. The magnetic force now acts sidewise on the particle. For a quantitative analysis of the motion, let be taken along the x-direction and along z- direction. As there is no component of the force along the z-direction, the velocity of the particle remains zero in this direction. The motion, therefore, takes place in x-y plane. The equations of motion are (1) (2) As in the earlier case, we can solve the equations by differentiating one of the equations and substituting the other, which, as before, has the solution with. Substituting this solution into the equation for, we get 6

7 Since at, the constant, so that The constant may be determined by substituting the solutions in eqn. (1) which gives Since the equation above is valid for all times, the constant terms on the right must cancel, which gives. Thus we have The equation to the trajectory is obtained by integrating the above equation and determining the constant of integration from the initial position (taken to be at the origin), The equation to the trajectory is which represents a circle of radius 7

8 whose centre travels along the negative y direction with a constant speed The trajectory resembles that of a point on the circumference of a wheel of radius, rolling down the y-axis without slipping with a speed. The trajectory is known as a cycloid. 8

9 Lecture - Force on a Current Carrying Conductor Objectives In this lecture you will learn the following Calculate the force on a current carrying conductor in a magnetic field. Find the torque on a current loop. Define magnetic dipole and its magnetic moment. Determine potential energy of a magnetic dipole Force on a Current Carrying Conductor A conductor has free electrons which can move in the presence of a field. Since a magnetic field exerts a force on a charge moving with a velocity, it also exerts a force on a conductor carrying a current. Consider a conducting wire carrying a current. The current density at any point in the wire is given by' where is the number density of electrons having a charge each and is the average drift velocity at that point. Consider a section of length of the wire. If is the cross sectional area of the section oriented perpendicular to the direction of, the force on the electrons in this section is where is the amount of charge in the section 9

10 Thus the force on the conductor in this section is If represents a vector whose magnitude is the length of the segment and whose direction is along the direction of, we may rewrite the above as The net force on the conductor is given by summing over all the length elements. If denotes a unit vector in the direction of the current, then Torque on a Current Loop in a Uniform Magnetic Field Though the net force on a closed current loop in a uniform magnetic field is zero, it experiences a torque. Consider a rectangular current loop PQRS of length and width. The loop is in a uniform magnetic field which acts parallel to the x-axis. The loop, which is pivoted about an axis OO', carries a current 10

11 along the direction shown in the figure. The plane of the loop (i.e. the normal to the loop) makes an angle to the direction of the field. We take the shorter sides PQ and RS (as well as the pivot axis OO') to be perpendicular to the field direction, OO' being taken as the y-axis. The longer sides QR and QS make an angle with the field direction. Since the force on a current segment is, the force is directed perpendicular to both and to the direction of the current in these segments. The force has a magnitude and is directed oppositely on the two sides. These forces are labelled and in the figure. (The forces are actually distributed along the lengths and the cancellation occurs for the forces acting on symmetrically placed elements on these two arms.) Further, since the lines of action of the forces acting on corresponding elements on these two sides are the same, there is no torque. The forces acting on the sides PQ and RS (labelled and respectively) are also equal and opposite 11

12 and have magnitude. However, these forces do not act along the same line. The force on PQ acts parallel to axis while that on RS acts parallel to axis. Note that axis is not in the plane of the loop. The situation can be better visualized by redrawing the figure in the plane containing one of the longer sides and. The current enters the branch SR at S marked with and reenters from QP at P, marked with. The directions of the forces on the branches SR and QP are shown. The magnitude of the torque about the pivot is where is the area of the loop. The result above is independent of the shape of the loop. The following example for a circular loop gives an identical result. Potential Energy of a Magnetic Dipole A current loop does not experience a net force in a magnetic field. It however, experiences a torque. This is very similar to the behaviour of an electric dipole in an electric field. A current 12

13 loop, therefore, behaves like a magnetic dipole. We define the magnetic dipole moment of a current loop to be a vector of magnitude and direction perpendicular to the plane of the loop (as determined by right hand rule). If the loop has turns,. In a magnetic field, the dipole experiences a torque The form of torque suggests that in a magnetic field the dipole tends to align parallel to the field. If the orientation of the dipole is at some angle to the field, there must be some potential energy stored in the dipole. This is because, if we wish to bring the dipole from to some arbitrary angle, we have to oppose the torque due to the field and do work in the process. The work done is given by This amount of work is stored as the additional potential energy of the dipole. In analogy with the case of electric dipole in an electric field, the potential energy of the magnetic dipole in a magnetic field is given by The energy is lowest when and are along the same direction and is maximum when they are anti-parallel. 13

14 Lecture - Biot- Savarts' Law Objectives In this lecture you will learn the following Study Biot-Savart's law Calculate magnetic field of induction due to some simple current configurations. Define magnetic moment of a current loop. Find force between two current carrying conductors. Biot- Savarts' Law Biot-Savarts' law provides an expression for the magnetic field due to a current segment. The field at a position due to a current segment is experimentally found to be perpendicular to and. The magnitude of the field is proportional to the length and to the current and to the sine of the angle between and. inversely proportional to the square of the distance of the point P from the current element. Mathematically, In SI units the constant of proportionality is, where is the permeability of the free space. The value of is The expression for field at a point P having a position vector with respect to the current element is 14

15 For a conducting wire of arbitrary shape, the field is obtained by vectorially adding the contributions due to such current elements as per superposition principle, is along the path of the current flow. where the integration 15

16 Lecture - Ampere's Law Objectives In this lecture you will learn the following Establish Ampere' law in integral form. Calculate the magnetic field for certain current configuration using Ampere's law. Derive the differential form of Ampere's law. Ampere's Law Biot-Savart's law for magnetic field due to a current element is difficult to visualize physically as such elements cannot be isolated from the circuit which they are part of. Andre Ampere formulated a law based on Oersted's as well as his own experimental studies. Ampere's law states that `` the line integral of magnetic field around any closed path equals which threads the surface bounded by such closed path.. Mathematically, times the current In spite of its apparent simplicity, Ampere's law can be used to calculate magnetic field of a current distribution in cases where a lot of information exists on the behaviour of. The field must have enough symmetry in space so as to enable us to express the left hand side of (1) in a functional form. The simplest application of Ampere' s law consists of applying the law to the case of an infinitely long straight and thin wire. Ampere's Law in Differential Form 16

17 We may express Ampere's law in a differential form by use of Stoke's theorem, according to which the line integral of a vector field is equal to the surface integral of the curl of the field, The surface is any surface whose boundary is the closed path of integration of the line integral. In terms of the current density, we have, where is the total current through the surface. Thus, Ampere's law is equivalent to which gives 17

18 You may recall that in the case of electric field, we had shown that the divergence of the field to be given by. In the case of magnetic field there are no free sources (monopoles). As a result the divergence of the magnetic field is zero The integral form of above is obtained by application of the divergence theorem Thus the flux of the magnetic field through a closed surface is zero. 18

19 19

20 Lecture - Vector Potential Objectives In this lecture you will learn the following Define vector potential for a magnetic field. Understand why vector potential is defined in a gauge. Calculate vector potential for simple geometries. Define electromotive force and state Faraday's law of induction Vector Potential For the electric field case, we had seen that it is possible to define a scalar function called the ``potential" whose negative gradient is equal to the electric field :. The existence of such a scalar function is a consequence of the conservative nature of the electric force. It also followed that the electric field is irrotational, i.e.. For the magnetic field, Ampere's law gives a non-zero curl Since the curl of a gradient is always zero, we cannot express function as it would then violate Ampere's law. as a gradient of a scalar However, we may introduce a vector function such that This would automatically satisfy since divergence of a curl is zero. is known as vector potential. Recall that a vector field is uniquely determined by specifying its divergence and curl. As is a physical quantity, curl of is also so. However, the divergence of the vector potential has no physical meaning and consequently we are at liberty to specify its 20

21 divergence as per our wish. This freedom to choose a vector potential whose curl is and whose divergence can be conveniently chosen is called by mathematicians as a choice of a gauge. If is a scalar function any transformation of the type gives the same magnetic field as curl of a gradient is identically zero. The transformation above is known as gauge invariance. (we have a similar freedom for the scalar potential of the electric field in the sense that it is determined up to an additive constant. Our most common choice of is one for which at infinite distances.) A popular gauge choice for is one in which which is known as the ``Coulomb gauge". It can be shown that such a choice can always be made. Biot-Savart's Law for Vector Potential 21

22 Biot-Savart's law for magnetic field due to a current element may be used to obtain an expression for the vector potential. Since the element does not depend on the position vector of the point at which the magnetic field is calculated, we can write the change in sign is because. Thus the contribution to the vector potential from the element is The expression is to be integrated over the path of the current to get the vector potential for the system The existence of a vector potential whose curl gives the magnetic field directly gives as the divergence of a curl is zero. The vector identity can be used to express Ampere's law in terms of vector potential. Using a Coulomb gauge in which, the Ampere's law is equivalent to which is actually a set of three equations for the components of, viz., 22

23 which are Poisson's equations. Electromagnetic Induction : We have seen that studies made by Oersted, Biot-Savart and Ampere showed that an electric current produces a magnetic field. Michael Faraday wanted to explore if this phenomenon is reversible in the sense whether a magnetic field could be source for a current in a conductor. However, no current was found when a conductor was placed in a magnetic field. Faraday and (Joseph) Henry, however, found that if a current loop was placed in a time varying magnetic field or if there was a relative motion between a magnet and the loop a transient current was established in the conducting loop. They concluded that the source of the electromotive force driving the current in the conductor is not the magnetic field but the changing magnetic flux associated with the loop. The change in flux could be effected by (i) a time varying magnetic field or by (ii) motion of the conductor in a magnetic field or (iii) by a combined action of both of these. The discovery is a spectacular milestone in the sense that it led to important developments in Electrical engineering like invention of transformer, alternator and generator. Shortly after Faraday's discovery, Heinrich Lenz found that the direction of the induced current is such that it opposes the very cause that produced the induced current (i.e. the magnetic field associated with the induced current opposes the change in the magnetic flux which caused the induced current in the first place). Lenz's law is illustrated in the following. In the figures the loops are perpendicular to the plane of the page. The direction of induced current is as seen towards the loop from the right. Note that the magnetic field set up by the 23

24 induced current tends to increase the flux in the case where the magnet is moving away from the loop and tends to decrease it in the case where it is moving towards the loop, Mathematically, Faraday's law is stated thus : the electromotive force is proportional to the rate of change of magnetic flux. In SI units, the constant of proportionality is unity. where is the flux associated with the circuit and the minus sign is a reminder of the direction of the current as given by Lenz's law. If the loop contains turns, the equation becomes Though the flux is a scalar, one can fix its sign by considering the sign of the area vector which is fixed by the usual right hand rule. The dot product of and then has a sign. 24

25 25

26 Lecture - Electromagnetic Induction Objectives In this lecture you will learn the following Study in detail the principle of electromagnetic induction. Understand the significance of Lenz's law. Solve problems involving motional emf. Motional Emf : Consider a straight conductor AB moving along the positive x-direction with a uniform speed. The region is in a uniform magnetic field pointing into the plane of the page, i.e. in direction. 26

27 The fixed positive ions in the conductor are immobile. However, the negatively charged electrons experience a Lorentz force, i.e. a force along the direction. This pushes the electrons from the end A to the end B, making the former positive with respect to the latter. Thus an induced electric field is established in the conductor along the positive direction. The acceleration of electrons would stop when the electric field is built to a strength which is strong enough to annul the magnetic force. This electric field is the origin of what is known as motional emf. The motion of charges finally stops due to the resistance of the conductor If the conductor slides along a stationary U- shaped conductor, the electrons find a path and a current is established in the circuit. The moving conductor thereby becomes a seat of the motional emf. We may calculate the emf either by considering the work that an external agency has to do to keep the sliding conductor move with a uniform velocity or by direct application of Faraday's law. If the induced current is, a force acts on the wire in the negative x direction. In order to maintain the uniform velocity, an external agent has to exert an equal and opposite force on the sliding conductor. Since the distance moved in time is, the work done by the external agency is where is the amount of charge moved by the seat of emf along the direction of the current. The emf is an electric potential difference. Thus, the emf is equivalently the work done in moving a unit charge. Thus, This emf corresponds to the potential difference beteen the ends A and B. An alternate derivation of the above is to consier the flux linked with closed circuit. Taking the origin at the extreme left end of the circuit, the area of the circuit in the magnetic field is where is the distance of the sliding rod from the fixed end. The flux linked with the circuit is, therefore,. The rate at which flux changes is therefore given by 27

28 If is not perpendicular to the plane of the circuit, we will need to take the perpendicular component of in the above formula. Note that, in the illustration above, the magnetic flux linked with the circuit is increasing with time in the negative z direction. The direction of the induced current is, therefore, such that the magnetic field due to the current is along the positve z-direction, which will oppose increase of flux. Hence the current, as seen from above, is in the anticlockwise direction. 28

29 Lecture - Time Varying Field Objectives In this lecture you will learn the following Relate time varying magnetic field with emf generated. Define mutual inductance and calculate it in simple cases. Define self inductance. Calculate energy stored in a magnetic field. Time Varying Field Even where there is no relative motion between an observer and a conductor, an emf (and consequently an induced current for a closed conducting loop) may be induced if the magnetic field itself is varying with time as flux change may be effected by change in magnetic field with time. In effect it implies that a changing magnetic field is equivalent to an electric field in which an electric charge at rest experiences a force. Consider, for example, a magnetic field whose direction is out of the page but whose magnitude varies with time. The magnetic field fills a cylindrical region of space of radius. Let the magnetic field be time varying and be given by Since does not depend on the axial coordinate as well as the azimuthal angle, the electric field is also independent of these quantities. Consider a coaxial circular path of radius which encloses a time varying flux. By symmetry of the problem, the electric field at every point of the cicular path must have the same magnitude and must be tangential to the circle. Thus the emf is given by 29

30 By Faraday's law Equating these, we get for, For, the flux is, so that and the electric field foir is Mutual Inductance According to Faraday's law, a changing magnetic flux in a loop causes an emf to be generated in that loop. Consider two stationary coils carrying current. The first coil has turn and carries a 30

31 current. The second coil contains turns. The current in the first coil is the source of a magnetic field in the region around the coil. The second loop encloses a flux, where is the surface of one turn of the loop. If the current in the first coil is varied,, and consequently will vary with time. The variation of causes an emf to be developed in the second coil. Since is proportional to, so is. The emf, which is the rate of change of flux is, therefore, proportional to, where is a constant, called the mutual inductance of the two coils, which depends on geometrical factors of the two loops, their relative orientation and the number of turns in each coil. Analogously, we can argue that if the second loop carries a current which is varied with time, it generates an induced emf in the first coil given by 31

32 For instance, consider two concentric solenoids, the outer one having turns per unit length and inner one with turns per unit length. The solenoids are wound over coaxial cylinders of length each. If the current in the outer solenoid is, the field due to it is, which is confined within the solenoid. The flux enclosed by the inner cylinder is If the current in the outer solenoid varies with time, the emf in the inner solenoid is so that If, on the other hand, the current in the inner solenoid is varied, the field due to it which is non-zero only within the inner solenoid. The flux enclosed by the outer solenoid is, therefore, If is varied, the emf in the outer solenoid is giving One can see that 32

33 This equality can be proved quite generally from Biot-Savart's law. Consider two circuits shown in the figure. The field at, due to current in the loop (called the primary ) is where. We have seen that can be expressed in terms of a vector potential, where, by Biot-Savart's law The flux enclosed by the second loop, (called the secondary ) is 33

34 Clearly, It can be seen that the expression is symmetric between two loops. Hence we would get an identical expression for. This expression is, however, of no significant use in obtaining the mutual inductance because of rather difficult double integral. Thus a knowledge of mutual inductance enables us to determine, how large should be the change in the current (or voltage) in a primary circuit to obtain a desired value of current (or voltage) in the secondary circuit. Since, we represent mutual inductance by the symbol. The emf in the secondary circuit is given by, where is the variable current in the primary circuit. Units of is that of Volt-sec/Ampere which is known as Henry (h) Self Inductance : Even when there is a single circuit carrying a current, the magnetic field of the circuit links with the circuit itself. If the current happens to be time varying, an emf will be generated in the circuit to oppose the change in the flux linked with the circuit. The opposing voltage acts like a second voltage source connected to the circuit. This implies that the primary source in the circuit has to do additional work to overcome this back emf to establish the same current. The induced current has a direction determined by Lenz's law. If no ferromagnetic materials are present, the flux is proportional to the current. If the circuit contains turns, Faraday's law gives 34

35 where is known as Self Inductance of the circuit. By definition, is a positive quantity. From the above it follows, on integrating, Since when, the constant is zero and we get The self inductance can, therefore, interpreted as the amount of flux linked with the circuit for unit current. The emf is given by Energy Stored in Magnetic Field Just as capacitor stores electric energy, an inductor can store magnetic energy. To see this consider an L-R circuit in which a current is established. If the switch is thrown to the position such that the battery gets disconnected from the circuit at, the current in the circuit would decay. As the inductor provides back emf, the circuit is described by With the initial condition, the solution of the above is As the energy dissipated in the circuit in time is, the total energy dissipated from the time the battery is disconnected is 35

36 Thus the energy initially stored must have been. If an inductor carries a current, it stores an energy. Thus the toroidal inductor discussed earlier stores an energy when it carries a current. We eill now show that this is also equal to the volume integral of Consider the magnetic field in the toroid at a distance from the axis. We have seen that the magnetic field is given by. Thus the value of at this distance is. Considering the toroid to consist of shells of surface area and thickness, the volume of the shell is. The volume integral of is therefore, which is exactly the expression for the stored energy derived earlier. 36