Chapter 9 THE MAGNETC FELD ntroduction Magnetic field due to a moving point charge Units Biot-Savart Law Gauss s Law for magnetism Ampère s Law Maxwell s equations for statics Summary NTRODUCTON Last lecture introduced the concept of a magnetic field and the magnetic dipole. t was seen that the magnetic dipole has a north-seeking and a south-seeking pole. Opposite poles attract and like poles repel. Magnetic poles cannot be isolated in contrast to the charges of an electric dipole. Oersted s discovery that an electric current produces a magnetic field circling clockwise around the electric current is especially important in that it shows that electricity and magnetism are directly related. The Lorentz force describes the force acting on a charge in both electric and magnetic fields. F = ( E + v B) The Lorentz force was used to derive the force acting on an electric current in a magnetic field. The unusual feature of the magnetic force acting on moving charges is that the force is perpendicular to both B and v Summing the magnetic forces for the moving charges in an infinitessimal volume and using the fact that j = v leads to the magnetic force is F = j B For a conductor of cross sectional area and length the volume =. Then since = j,then the magnetic force on a circuit carrying a current in a magnetic field is F = B = dl B The general features of the magnetic field produced by electric currents has been discussed qualitatively. The next stage is to derive the magnitude as well as the direction of the magnetic field produced by moving Figure 1 The magnetic field due to a point charge q moving with velocity v. charges and currents. The discussion of the electrostatic force started with the experimental facts, condensed into Coulomb s Law plus superposition, and from these derived Maxwell s Equations for electrostatics. The concepts of flux and circulation are required to express fully the laws of electromagnetism. For electrostatics the concept of flux led to Gauss s law while the concept of circulation led to the proof that the electric field is conservative allowing use of the concept of electric potential. For magnetism it is logical also to start with the experimental facts for the magnetic force and from these derive the flux and circulation of the vector B field. Then it is possible to make a comparison of the difference between the laws for electrostatics and magnetism. There are two approaches to introducing the basic experimental facts of the magnetic force. One approach is to use Coulomb s Law plus Einstein s Theory of Relativity to derive directly the magnetic field produced by a moving point charge. A second approach is to start with the Biot Savart Law to define the magnetic field due to an infinitessimal element of current. These will be introduced and then will be used to derive Gauss s law describing the flux of the magnetic field, while the concept of circulation will lead to Ampère s Law. MAGNETC FELD OF A MOVNG PONT CHARGE Coulomb s Law provides the basis for calculating the electric field. An equivalent law is needed for calculation of magnetic fields. The most fundamental relation is the magnetic field produced by a moving point charge. For a point charge moving with a velocity v the induced magnetic field B is: B = 0 4 v br 67
Figure circuit. Magnetic field due to an electric current in a Figure 3 Geometry for the magnetic field due to an infinitely long straight current. Note that B is proportional to and v as well as varying inversely with the square of the distance. The direction of B is perpendicular to both v and r and is a maximum when r is perpendicular to v The magnetic field circles clockwise around the vector v This formula can be derived from experimental data. However, also it can be derived from Coulomb s law using Einstein s Theory of Relativity. UNTS The constant has been chosen such that 0 4 10 7 =4 10 7 exactly and is called the permeability of free space. That is, 0 4 is chosen to be exactly 10 7 in the S system of units as will be discussed later. The factor 4 was inserted to simplify Ampère s law as will be discussed later. Since, in the MKS system, the Lorentz force is defined innewtonsandvelocityin meters/second, defining the constant 0 fixes the unit of charge, the coulomb. However, in Coulomb s Law the Coulomb unit of charge was definedintermsof 1 the constant 4 Clearly, the constants and are related. The magnetic field due to a moving charge is the simplest system for definition of the B field, but in practice it is more useful to use the expression for the B field due to an element of a circuit carrying an electric current, which is called the Biot Savart law. BOT SAVART LAW One month after Oersted s discovery of the magnetic field produced by an electric current, Biot and Savart determined experimentally the magnetic field due to a long straight wire. This result also can be derived by summing over the magnetic field due to the moving charges in a wire assuming that the Principle of Superposition applies. For charges per unit volume, the net magnetic field B due to an element of volume is times the field due to each charge. That is: B = 0 v br 4 = 0 j br 4 since j = v. The magnetic field at a distance r from an element of a conductor of cross sectional area and length carrying current, can be calculated since the volume element is =, andknowing that = j Thus: B = 0 br 4 ntegrating over a closed circuit gives B = 0 br 4 Another convention writes = l where l carries the information as to the direction of the current flow. ntegrating over a complete closed circuit, using this notation, gives that the total magnetic field at the point r from a circuit carrying current, is: B = 0 dl br 4 This is called the Biot Savart Law. t is equivalent to Coulomb s Law in that one can compute, by integration, the magnetic field due to any shape currentcarrying circuit. Let us consider an application of the Biot Savart law. Note that the Biot Savart Law has been written assuming a right-handed coordinate system. Field due to an infinitely long straight current Orient the wire along the axis as shown in figure 3. From symmetry, the magnetic field at a given distance from the wire must look the same anywhere in the 68
Figure 4 The magnetic field produced by a long straight electric current. plane. Thus the magnetic field will be computed at the point = and =0. The contribution to the field from the element l is given by: Figure 5 loop. Magnetic field on the axis of a circular current Since then and B = 0 4 sin φ b k = 0 4 cos θ b k = = cos = cos ntegrating over the complete wire gives: B = 0 4 = 0 4 b k B = 0 R b cos cos cos k b cos where the cross product is included to carry the information of the direction of the magnetic field. That is, the magnetic field circulates clockwise around the current as shown. Note that for a 5 current, the field at =5 is 10 4 1 Field on the axis of a circular current loop A current loop is called a magnetic dipole. t is of interest to calculate the magnetic fieldontheaxisof the magentic dipole. Calculation of the cross product l r can be written in terms of cartesian coordinates. As shown in figure 5, assume that the circle is in the plane. Since and are perpendicular, then the Biot Savart law can be used to evaluate the magnetic field due to the element or the current loop d B = 0 dl br 4 = 0 dl br 4 ( + ) Then the components of d B along the and axes are and = cos = 0 4 = sin = 0 4 ( + ) ( + ) 1 ( + ) ( + ) 1 By symmetry for the whole loop only the component is non-zero, thus = 0 4 ( + ) 3 = 0 4 ( + ) 3 But the circumference of a circle equals, thus B = 0 ( + ) 3 This gives the field on the axis of the magnetic dipole by the brute force approach. Superposition of the field can be used to obtain the field for a magnetic dipole having turns to be B = 0 b i ( + ) 3 Figure 6 illustrates the magnetic field along the axis of a circular loop of turns current. b i 69
Figure 8 Gaussian surface for a long straight current. Figure 6 Graph of magnetic field along the axis of a current loop on turns. When then the field falls off as 1 3 Figure 7 Helmholtz coils used to cancel the earth s magnetic field at the center of the current loops. Note that at the distance = the field falls to one half of the maximum value at =0. This is exploited by the Helmholtz arrangement of two such magnetic dipoles, shown in figure 7, where two identical coils separated by one radius produce a very uniform magnetic field along the axis near the centre of the coils. The off-axis field for the magnetic dipole is more complicated and will be derived later. The Biot Savart law can be used to calculate, by brute force, the field around any current-carrying circuit. Unfotunately, evaluating the integrals for most geometries can be challenging. GAUSS S LAW FOR MAGNETSM t was seen that Gauss s Law for electrostatics is of considerable theoretical importance as well as being powerful for calculating electric fields for symmetric systems. That is, the net electric flux out of a closed surface is: Φ = 1 E S = 0 τ Remember that this is a statement of the fact that Coulomb s law states that the electric field has a r dependence. t states that the net flux out of a closed surface equals the enclosed charge times a constant 1 0 This is independent of the shape or size of the closed surface because the the 1 dependence of Coulomb s law. Gauss s Law for magnetism is given by computing the net flux of the magnetic field out of a closed Gaussian surface. Consider the special case of a concentric cylinder surrounding a long straight current shown in figure 8. As calculated with the Biot Savart Law, the magnetic field is tangential to the surface of the concentric cylinder around the current, and also is tangential to the ends of the cylinder. Thus there is no net flux out of the cylinder, that is, " # B S = br br br + b = 0 since the surface vectors are perpendicular to the cross product over all of the surface of the cylinder. t can be shown using the Biot Savart law that this is a general property of magnetostatics. Thus the most general form of Gauss s Law for magnetostatics is: B S =0 Φ = The net magnetic flux is independent of the size or shape of the closed surface, as expected since the field due to a point charge with velocity v has a 1 dependence. Moreover the net flux is zero because the B field is tangential to br 70
Figure 9 Lines of intersecting a Gaussian surface. Any flux tube entering the surface must exit if the lines of are continuous. Gauss s Law for magnetism is a statement that there are no magnetic monopoles, that is, there are no sources or sinks of magnetic field and therefore the lines of B are continuous. Since the lines of B are continuous, then the number entering a closed surface must equal the number leaving the surface as illustrated in figure 9. Gauss s law for magnetism is useful for limiting the form of the magnetic field. For example, there cannot be a radial component to B around a current element. However, Gauss s law for magnetism is not useful for calculating the strength of the B field. For that purpose one has to turn to the relation for the circulation of the magnetic field. AMPÈRE S LAW n electrostatics it was found that the circulation of E around a closed loop is zero. That is: E l =0 The statement that circulation of the electric field is zero reflects the fact that the electric field of a point charge is radial. t states that the electric field is conservative which allows use of the powerful concept of electric potential. n magnetism, one can use the Biot Savart law to relate the circulation of B around a closed loop to the current flowing through the loop, leading to Ampère s law as shown below. Concentric circle around long straight conductor For simplicity, consider the special case of a magnetic field around a long straight current. TheBiotSavart law gave that: B = br The circulation, given by the line integral for a concentric circle of radius takeninthedirectionof Figure 10 Concentric circular line integral around a long straight current. Figure 11 Arbitrary shaped closed loop enclosing long straight conductor. B,is B dl = = 0 Thus, for this special case, we obtain Ampère s Law: B dl = 0 (Enclosed current) where the current is assumed to flow in the direction given by the right-hand rule relative to the direction of the closed line integral. Arbitrary closed loop around long straight conductor The Biot Savart law can be used to prove that this is true for any current distribution through any surface having the closed loop as a boundary. Consider an arbitrary shaped closed loop enclosing a long straight conductor as shown in figure 11. Note that for an element of line at a radius fromthelinecurrent: B dl = cos = since B is tangential. Therefore for a long straight conductor one obtains: B dl = B dl = 71
Figure 1 Closed loop 1 enclosing the current carrying conductor and the closed loop not enclosing the current carrying conductor. charge inside the volume bounded by 1 +,then the net current flowing into the volume must equal the net outflow of current. That is, R j S is the same for both surfaces. Thus the net current flowing though the closed loop is independent of the shape of the surface bounded by the closed loop. Ampère s law is of considerable theoretical importance beyond that of the Biot Savart law from which it was derived. Also Ampère s Law provides an easy way to compute the magnetic field for systems possessing symmetry. Unfortunately, there are only a limited set of cases where is is possible to use symmetry to find a curve for which B l is constant. MAXWELL S EQUATONS FOR STATCS Figure 13 Closed loop and surface bounded by closed loop as used by Ampère s Law. since the factor cancels. Note that the integral H = if the closed loop encloses the origin, e.g. 1,that is B dl = 0 = 0 (Enclosed current) f the conductor is outside the closed loop in figure 1, e.g. then the angle integral equals zero.again it is assumed that the current flowsinadirectiongivenby the right-hand rule for the line integral. The more general form of Ampère s Law is written in terms of the current density using the fact that = j S leading to the relation: B dl = 0 j S wherethesurfaceisboundedbytheclosedloopc.t is important that the direction of the line integral and S be given by the right-hand rule. Note that this proof implicitly assumes that the magnetic fields produced by different currents superpose, that is the Principle of Superposition has been assumed. There is an infinite number of surfaces that can be drawn that are bounded by one closed loop. As shown in figure 13, take surfaces 1 and both bounded by. f there are no sources or sinks of t is interesting to compare and contrast the flux and circulation equations we have derived for static electric and magnetic fields. Flux Circ. Electrostatics Magnetostatics H R H ES = 1 BS =0 H H j El =0 Bl = 0 S These are the Maxwell equations for statics. The flux relations show that the electrostatic field has nonzero flux for a Gaussian surface enclosing charge because charges are sources and sinks of the electric field, whereas the magnetic field has zero flux out of a Gaussian surface because there are no sources or sinks of magnetic field. The circulation relations show that the static electric field has zero circulation, because the electric field for a point charge is radial, whereas the circulation of the magnetostatic field has a non-zero circulation if it encloses an electric current. Thus, in contrast to the static electric field which is circulation free, the magnetostatic field is flux free. For statics the electric field and magnetic field are unrelated by the Maxwell equations. t will be shown later that the circulation equations lead to coupling of the magnetic and electric fields for time-dependent systems. SUMMARY The magnetic field due to a moving charge is: B = 0 v br 4 TheBiotSavartLawgivesthefield at a point r from a circuit carrying current as: 7
B = 0 dl br 4 n the S system of units, the distances are in meters, force in Newtons while the constant has been chosen to be 0 4 10 7 Gauss s law for magnetism gives that the total magnetic flux out of a closed surface is: B S =0 Φ = The circulation of the magnetic field leads to Ampère s law.: B dl = 0 j S This is especially useful for calculating magnetic fields for systems possessing symmetry. t is interesting to compare and contrast the flux and circulation equations we have derived for static electric and magnetic fields. Flux Circ Electrostatics Magnetostatics H ES = 1 R H BS =0 H E l =0 H B l = 0 j S These are the Maxwell equations for statics. The flux relations show that the electrostatic field has nonzero flux for a Gaussian surface enclosing charge because charges are sources and sinks of the electric field, whereas the magnetic field has zero flux out of a Gaussian surface because there are no sources or sinks of magnetic field. The circulation relations show that the static electric field has zero circulation, because the electric field for a point charge is radial, whereas the circulation of the magnetostatic field has a non-zero circulation if it encloses an electric current. Thus in contrast to the static electric field which is circulation free, the magnetostatic field is flux free. For statics the electric field and magnetic field are unrelated by the Maxwell equations. t will be shown later that the circulation equations lead to coupling of the magnetic and electric fields for time-dependent systems. The next lecture will discuss further applications of Ampère s law, magnetic forces and then review what we have done so far this term. Reading assignment: Giancoli, Chapter 8.1 8.6. 73