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1 EEE 498/598 Overview of Electrical Engineering Lecture 7: Magnetostatics: Amperes Law Of Force; Magnetic Flux Density; Lorentz Force; Biot-savart Law; Applications Of Amperes Law In Integral Form; Vector Magnetic Potential; Magnetic Dipole; Magnetic Flux

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Lecture 7 2 Lecture 7 Objectives To begin our study of magnetostatics with Amperes law of force; magnetic flux density; Lorentz force; Biot-Savart law; applications of Amperes law in integral form; vector magnetic potential; magnetic dipole; and magnetic flux. To begin our study of magnetostatics with Amperes law of force; magnetic flux density; Lorentz force; Biot-Savart law; applications of Amperes law in integral form; vector magnetic potential; magnetic dipole; and magnetic flux.

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Lecture 7 3 Overview of Electromagnetics Maxwells equations Fundamental laws of classical electromagnetics Special cases Electro- statics Magneto- statics Electro- magnetic waves Kirchoffs Laws Statics: Geometric Optics Transmission Line Theory Circuit Theory Input from other disciplines

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Lecture 7 4 Magnetostatics Magnetostatics is the branch of electromagnetics dealing with the effects of electric charges in steady motion (i.e, steady current or DC). Magnetostatics is the branch of electromagnetics dealing with the effects of electric charges in steady motion (i.e, steady current or DC). The fundamental law of magnetostatics is Amperes law of force. The fundamental law of magnetostatics is Amperes law of force. Amperes law of force is analogous to Coulombs law in electrostatics. Amperes law of force is analogous to Coulombs law in electrostatics.

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Lecture 7 5 Magnetostatics (Contd) In magnetostatics, the magnetic field is produced by steady currents. The magnetostatic field does not allow for In magnetostatics, the magnetic field is produced by steady currents. The magnetostatic field does not allow for inductive coupling between circuits inductive coupling between circuits coupling between electric and magnetic fields coupling between electric and magnetic fields

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Lecture 7 6 Amperes Law of Force Amperes law of force is the law of action between current carrying circuits. Amperes law of force is the law of action between current carrying circuits. Amperes law of force gives the magnetic force between two current carrying circuits in an otherwise empty universe. Amperes law of force gives the magnetic force between two current carrying circuits in an otherwise empty universe. Amperes law of force involves complete circuits since current must flow in closed loops. Amperes law of force involves complete circuits since current must flow in closed loops.

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Lecture 7 7 Amperes Law of Force (Contd) Experimental facts: Experimental facts: Two parallel wires carrying current in the same direction attract. Two parallel wires carrying current in the same direction attract. Two parallel wires carrying current in the opposite directions repel. Two parallel wires carrying current in the opposite directions repel. I1I1 I2I2 F 12 F 21 I1I1 I2I2 F 12 F 21

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Lecture 7 8 Amperes Law of Force (Contd) Experimental facts: Experimental facts: A short current- carrying wire oriented perpendicular to a long current-carrying wire experiences no force. A short current- carrying wire oriented perpendicular to a long current-carrying wire experiences no force. I1I1 F 12 = 0 I2I2

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Lecture 7 9 Amperes Law of Force (Contd) Experimental facts: Experimental facts: The magnitude of the force is inversely proportional to the distance squared. The magnitude of the force is inversely proportional to the distance squared. The magnitude of the force is proportional to the product of the currents carried by the two wires. The magnitude of the force is proportional to the product of the currents carried by the two wires.

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Lecture 7 10 Amperes Law of Force (Contd) The direction of the force established by the experimental facts can be mathematically represented by The direction of the force established by the experimental facts can be mathematically represented by unit vector in direction of force on I 2 due to I 1 unit vector in direction of I 2 from I 1 unit vector in direction of current I 1 unit vector in direction of current I 2

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Lecture 7 11 Amperes Law of Force (Contd) The force acting on a current element I 2 dl 2 by a current element I 1 dl 1 is given by The force acting on a current element I 2 dl 2 by a current element I 1 dl 1 is given by Permeability of free space 0 = F/m

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Lecture 7 12 Amperes Law of Force (Contd) The total force acting on a circuit C 2 having a current I 2 by a circuit C 1 having current I 1 is given by The total force acting on a circuit C 2 having a current I 2 by a circuit C 1 having current I 1 is given by

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Lecture 7 13 Amperes Law of Force (Contd) The force on C 1 due to C 2 is equal in magnitude but opposite in direction to the force on C 2 due to C 1. The force on C 1 due to C 2 is equal in magnitude but opposite in direction to the force on C 2 due to C 1.

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Lecture 7 14 Magnetic Flux Density Amperes force law describes an action at a distance analogous to Coulombs law. Amperes force law describes an action at a distance analogous to Coulombs law. In Coulombs law, it was useful to introduce the concept of an electric field to describe the interaction between the charges. In Coulombs law, it was useful to introduce the concept of an electric field to describe the interaction between the charges. In Amperes law, we can define an appropriate field that may be regarded as the means by which currents exert force on each other. In Amperes law, we can define an appropriate field that may be regarded as the means by which currents exert force on each other.

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Lecture 7 15 Magnetic Flux Density (Contd) The magnetic flux density can be introduced by writing The magnetic flux density can be introduced by writing

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Lecture 7 16 Magnetic Flux Density (Contd) where where the magnetic flux density at the location of dl 2 due to the current I 1 in C 1

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Lecture 7 17 Magnetic Flux Density (Contd) Suppose that an infinitesimal current element Idl is immersed in a region of magnetic flux density B. The current element experiences a force dF given by Suppose that an infinitesimal current element Idl is immersed in a region of magnetic flux density B. The current element experiences a force dF given by

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Lecture 7 18 Magnetic Flux Density (Contd) The total force exerted on a circuit C carrying current I that is immersed in a magnetic flux density B is given by The total force exerted on a circuit C carrying current I that is immersed in a magnetic flux density B is given by

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Lecture 7 19 Force on a Moving Charge A moving point charge placed in a magnetic field experiences a force given by A moving point charge placed in a magnetic field experiences a force given by B v Q The force experienced by the point charge is in the direction into the paper.

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Lecture 7 20 Lorentz Force If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by The Lorentz force equation is useful for determining the equation of motion for electrons in electromagnetic deflection systems such as CRTs. The Lorentz force equation is useful for determining the equation of motion for electrons in electromagnetic deflection systems such as CRTs.

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Lecture 7 21 The Biot-Savart Law The Biot-Savart law gives us the B -field arising at a specified point P from a given current distribution. The Biot-Savart law gives us the B -field arising at a specified point P from a given current distribution. It is a fundamental law of magnetostatics. It is a fundamental law of magnetostatics.

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Lecture 7 22 The Biot-Savart Law (Contd) The contribution to the B -field at a point P from a differential current element Idl is given by The contribution to the B -field at a point P from a differential current element Idl is given by

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Lecture 7 23 The Biot-Savart Law (Contd)

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Lecture 7 24 The Biot-Savart Law (Contd) The total magnetic flux at the point P due to the entire circuit C is given by The total magnetic flux at the point P due to the entire circuit C is given by

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Lecture 7 25 Types of Current Distributions Line current density (current) - occurs for infinitesimally thin filamentary bodies (i.e., wires of negligible diameter). Line current density (current) - occurs for infinitesimally thin filamentary bodies (i.e., wires of negligible diameter). Surface current density (current per unit width) - occurs when body is perfectly conducting. Surface current density (current per unit width) - occurs when body is perfectly conducting. Volume current density (current per unit cross sectional area) - most general. Volume current density (current per unit cross sectional area) - most general.

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Lecture 7 26 The Biot-Savart Law (Contd) For a surface distribution of current, the B-S law becomes For a surface distribution of current, the B-S law becomes For a volume distribution of current, the B-S law becomes For a volume distribution of current, the B-S law becomes

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Lecture 7 27 Amperes Circuital Law in Integral Form Amperes Circuital Law in integral form states that the circulation of the magnetic flux density in free space is proportional to the total current through the surface bounding the path over which the circulation is computed. Amperes Circuital Law in integral form states that the circulation of the magnetic flux density in free space is proportional to the total current through the surface bounding the path over which the circulation is computed.

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Lecture 7 28 Amperes Circuital Law in Integral Form (Contd) By convention, dS is taken to be in the direction defined by the right-hand rule applied to dl. Since volume current density is the most general, we can write I encl in this way. S dldl dSdS

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Lecture 7 29 Amperes Law and Gausss Law Just as Gausss law follows from Coulombs law, so Amperes circuital law follows from Amperes force law. Just as Gausss law follows from Coulombs law, so Amperes circuital law follows from Amperes force law. Just as Gausss law can be used to derive the electrostatic field from symmetric charge distributions, so Amperes law can be used to derive the magnetostatic field from symmetric current distributions. Just as Gausss law can be used to derive the electrostatic field from symmetric charge distributions, so Amperes law can be used to derive the magnetostatic field from symmetric current distributions.

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Lecture 7 30 Applications of Amperes Law Amperes law in integral form is an integral equation for the unknown magnetic flux density resulting from a given current distribution. Amperes law in integral form is an integral equation for the unknown magnetic flux density resulting from a given current distribution. known unknown

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Lecture 7 31 Applications of Amperes Law (Contd) In general, solutions to integral equations must be obtained using numerical techniques. In general, solutions to integral equations must be obtained using numerical techniques. However, for certain symmetric current distributions closed form solutions to Amperes law can be obtained. However, for certain symmetric current distributions closed form solutions to Amperes law can be obtained.

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Lecture 7 32 Applications of Amperes Law (Contd) Closed form solution to Amperes law relies on our ability to construct a suitable family of Amperian paths. Closed form solution to Amperes law relies on our ability to construct a suitable family of Amperian paths. An Amperian path is a closed contour to which the magnetic flux density is tangential and over which equal to a constant value. An Amperian path is a closed contour to which the magnetic flux density is tangential and over which equal to a constant value.

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Lecture 7 33 Magnetic Flux Density of an Infinite Line Current Using Amperes Law Consider an infinite line current along the z-axis carrying current in the +z-direction: I

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Lecture 7 34 Magnetic Flux Density of an Infinite Line Current Using Amperes Law (Contd) (1) Assume from symmetry and the right-hand rule the form of the field (2) Construct a family of Amperian paths circles of radius where

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Lecture 7 35 Magnetic Flux Density of an Infinite Line Current Using Amperes Law (Contd) (3) Evaluate the total current passing through the surface bounded by the Amperian path

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Lecture 7 36 Magnetic Flux Density of an Infinite Line Current Using Amperes Law (Contd) Amperian path I x y

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Lecture 7 37 Magnetic Flux Density of an Infinite Line Current Using Amperes Law (Contd) (4) For each Amperian path, evaluate the integral magnitude of B on Amperian path. length of Amperian path.

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Lecture 7 38 Magnetic Flux Density of an Infinite Line Current Using Amperes Law (Contd) (5) Solve for B on each Amperian path

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Lecture 7 39 Applying Stokess Theorem to Amperes Law Because the above must hold for any surface S, we must have Differential form of Amperes Law

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Lecture 7 40 Amperes Law in Differential Form Amperes law in differential form implies that the B -field is conservative outside of regions where current is flowing. Amperes law in differential form implies that the B -field is conservative outside of regions where current is flowing.

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Lecture 7 41 Fundamental Postulates of Magnetostatics Amperes law in differential form Amperes law in differential form No isolated magnetic charges No isolated magnetic charges B is solenoidal

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Lecture 7 42 Vector Magnetic Potential Vector identity: the divergence of the curl of any vector field is identically zero. Vector identity: the divergence of the curl of any vector field is identically zero. Corollary: If the divergence of a vector field is identically zero, then that vector field can be written as the curl of some vector potential field. Corollary: If the divergence of a vector field is identically zero, then that vector field can be written as the curl of some vector potential field.

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Lecture 7 43 Vector Magnetic Potential (Contd) Since the magnetic flux density is solenoidal, it can be written as the curl of a vector field called the vector magnetic potential. Since the magnetic flux density is solenoidal, it can be written as the curl of a vector field called the vector magnetic potential.

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Lecture 7 44 Vector Magnetic Potential (Contd) The general form of the B-S law is The general form of the B-S law is Note that Note that

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Lecture 7 45 Vector Magnetic Potential (Contd) Furthermore, note that the del operator operates only on the unprimed coordinates so that Furthermore, note that the del operator operates only on the unprimed coordinates so that

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Lecture 7 46 Vector Magnetic Potential (Contd) Hence, we have Hence, we have

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Lecture 7 47 Vector Magnetic Potential (Contd) For a surface distribution of current, the vector magnetic potential is given by For a surface distribution of current, the vector magnetic potential is given by For a line current, the vector magnetic potential is given by For a line current, the vector magnetic potential is given by

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Lecture 7 48 Vector Magnetic Potential (Contd) In some cases, it is easier to evaluate the vector magnetic potential and then use B = A, rather than to use the B-S law to directly find B. In some cases, it is easier to evaluate the vector magnetic potential and then use B = A, rather than to use the B-S law to directly find B. In some ways, the vector magnetic potential A is analogous to the scalar electric potential V. In some ways, the vector magnetic potential A is analogous to the scalar electric potential V.

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Lecture 7 49 Vector Magnetic Potential (Contd) In classical physics, the vector magnetic potential is viewed as an auxiliary function with no physical meaning. In classical physics, the vector magnetic potential is viewed as an auxiliary function with no physical meaning. However, there are phenomena in quantum mechanics that suggest that the vector magnetic potential is a real (i.e., measurable) field. However, there are phenomena in quantum mechanics that suggest that the vector magnetic potential is a real (i.e., measurable) field.

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Lecture 7 50 Magnetic Dipole A magnetic dipole comprises a small current carrying loop. A magnetic dipole comprises a small current carrying loop. The point charge ( charge monopole ) is the simplest source of electrostatic field. The magnetic dipole is the simplest source of magnetostatic field. There is no such thing as a magnetic monopole (at least as far as classical physics is concerned). The point charge ( charge monopole ) is the simplest source of electrostatic field. The magnetic dipole is the simplest source of magnetostatic field. There is no such thing as a magnetic monopole (at least as far as classical physics is concerned).

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Lecture 7 51 Magnetic Dipole (Contd) The magnetic dipole is analogous to the electric dipole. The magnetic dipole is analogous to the electric dipole. Just as the electric dipole is useful in helping us to understand the behavior of dielectric materials, so the magnetic dipole is useful in helping us to understand the behavior of magnetic materials. Just as the electric dipole is useful in helping us to understand the behavior of dielectric materials, so the magnetic dipole is useful in helping us to understand the behavior of magnetic materials.

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Lecture 7 52 Magnetic Dipole (Contd) Consider a small circular loop of radius b carrying a steady current I. Assume that the wire radius has a negligible cross-section. Consider a small circular loop of radius b carrying a steady current I. Assume that the wire radius has a negligible cross-section. x y b

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Lecture 7 53 Magnetic Dipole (Contd) The vector magnetic potential is evaluated for R >> b as The vector magnetic potential is evaluated for R >> b as

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Lecture 7 54 Magnetic Dipole (Contd) The magnetic flux density is evaluated for R >> b as The magnetic flux density is evaluated for R >> b as

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Lecture 7 55 Magnetic Dipole (Contd) Recall electric dipole Recall electric dipole The electric field due to the electric charge dipole and the magnetic field due to the magnetic dipole are dual quantities. The electric field due to the electric charge dipole and the magnetic field due to the magnetic dipole are dual quantities.

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Lecture 7 56 Magnetic Dipole Moment The magnetic dipole moment can be defined as The magnetic dipole moment can be defined as Direction of the dipole moment is determined by the direction of current using the right-hand rule. Magnitude of the dipole moment is the product of the current and the area of the loop.

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Lecture 7 57 Magnetic Dipole Moment (Contd) We can write the vector magnetic potential in terms of the magnetic dipole moment as We can write the vector magnetic potential in terms of the magnetic dipole moment as We can write the B field in terms of the magnetic dipole moment as We can write the B field in terms of the magnetic dipole moment as

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Lecture 7 58 Divergence of B -Field The B-field is solenoidal, i.e. the divergence of the B-field is identically equal to zero: The B-field is solenoidal, i.e. the divergence of the B-field is identically equal to zero: Physically, this means that magnetic charges (monopoles) do not exist. Physically, this means that magnetic charges (monopoles) do not exist. A magnetic charge can be viewed as an isolated magnetic pole. A magnetic charge can be viewed as an isolated magnetic pole.

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Lecture 7 59 Divergence of B -Field (Contd) N S N S N S No matter how small the magnetic is divided, it always has a north pole and a south pole. No matter how small the magnetic is divided, it always has a north pole and a south pole. The elementary source of magnetic field is a magnetic dipole. The elementary source of magnetic field is a magnetic dipole. I N S

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Lecture 7 60 Magnetic Flux The magnetic flux crossing an open surface S is given by The magnetic flux crossing an open surface S is given by S B C Wb/m 2 Wb

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Lecture 7 61 Magnetic Flux (Contd) From the divergence theorem, we have From the divergence theorem, we have Hence, the net magnetic flux leaving any closed surface is zero. This is another manifestation of the fact that there are no magnetic charges. Hence, the net magnetic flux leaving any closed surface is zero. This is another manifestation of the fact that there are no magnetic charges.

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Lecture 7 62 Magnetic Flux and Vector Magnetic Potential The magnetic flux across an open surface may be evaluated in terms of the vector magnetic potential using Stokess theorem: The magnetic flux across an open surface may be evaluated in terms of the vector magnetic potential using Stokess theorem:

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