Electrical Engineering (EE) > Magnetostatics and Ohms Law - Maxwell's Equations

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Page 1 Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Objectives In this course you will learn the following Magnetic Field and Magnetic Flux Density. Conduction Current Density J. Page 2 Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Objectives In this course you will learn the following Magnetic Field and Magnetic Flux Density. Conduction Current Density J. Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Magnetic Field and Magnetic Flux Density The magnetic field can be defined as the force experienced by a unit magnetic monopole. However, there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law. Consider an infinitesimally small piece of a wire carrying current in it. The length of the piece is say . The current moment is then defined by the product . Since the piece of the wire can be oriented in any direction the length is a vector quantity and therefore should be denoted by . The current is a scalar quantity making the current moment is a vector quantity. Without loss of generality let us assume that the current element is located at the origin of the co-ordinate system as shown in Fig. According to the Biot - Savart's law the magnetic field ( field due to infinitesimally small current element) at a point is given as Where is the unit vector in the direction of the line joining the current element and the observation point . Since the vectors and lie in the plane of the paper, the cross-product of the two has a direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards, the direction of is upwards and the direction of the magnetic field will be going into the paper as indicated by at point . If we take some other observation point on the left of the current element, by right hand screw rule, the magnetic field will come out of the paper as indicated by . If one streches his imagination a little, he can then see that the magnetic field forms a circular loop around the axis of the current element. The magnitude of the magnetic field is directly proportional to the current moment and is inversely proportional to the square of the distance between the source and the observation point. Page 3 Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Objectives In this course you will learn the following Magnetic Field and Magnetic Flux Density. Conduction Current Density J. Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Magnetic Field and Magnetic Flux Density The magnetic field can be defined as the force experienced by a unit magnetic monopole. However, there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law. Consider an infinitesimally small piece of a wire carrying current in it. The length of the piece is say . The current moment is then defined by the product . Since the piece of the wire can be oriented in any direction the length is a vector quantity and therefore should be denoted by . The current is a scalar quantity making the current moment is a vector quantity. Without loss of generality let us assume that the current element is located at the origin of the co-ordinate system as shown in Fig. According to the Biot - Savart's law the magnetic field ( field due to infinitesimally small current element) at a point is given as Where is the unit vector in the direction of the line joining the current element and the observation point . Since the vectors and lie in the plane of the paper, the cross-product of the two has a direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards, the direction of is upwards and the direction of the magnetic field will be going into the paper as indicated by at point . If we take some other observation point on the left of the current element, by right hand screw rule, the magnetic field will come out of the paper as indicated by . If one streches his imagination a little, he can then see that the magnetic field forms a circular loop around the axis of the current element. The magnitude of the magnetic field is directly proportional to the current moment and is inversely proportional to the square of the distance between the source and the observation point. The cross-product also suggests that the strength of the magnetic field is maximum when and are perpendicular to each other, that is, for equal to . As reduces the angle between and reduces consequently reducing the strength of the magnetic field reduces. For extreme case of the vectors and are colinear making cross product and hence the magnetic field identically zero. The magnetic field therefore is identically zero at the axis of the current element. The magnetic field is related to through a medium characteristic parameter called permeability of the medium as The permeability of vacuum is denoted by and its value is Henery/meter. A ratio of permeability of a medium to that of the vacuum is called the relative permeability giving Page 4 Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Objectives In this course you will learn the following Magnetic Field and Magnetic Flux Density. Conduction Current Density J. Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Magnetic Field and Magnetic Flux Density The magnetic field can be defined as the force experienced by a unit magnetic monopole. However, there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law. Consider an infinitesimally small piece of a wire carrying current in it. The length of the piece is say . The current moment is then defined by the product . Since the piece of the wire can be oriented in any direction the length is a vector quantity and therefore should be denoted by . The current is a scalar quantity making the current moment is a vector quantity. Without loss of generality let us assume that the current element is located at the origin of the co-ordinate system as shown in Fig. According to the Biot - Savart's law the magnetic field ( field due to infinitesimally small current element) at a point is given as Where is the unit vector in the direction of the line joining the current element and the observation point . Since the vectors and lie in the plane of the paper, the cross-product of the two has a direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards, the direction of is upwards and the direction of the magnetic field will be going into the paper as indicated by at point . If we take some other observation point on the left of the current element, by right hand screw rule, the magnetic field will come out of the paper as indicated by . If one streches his imagination a little, he can then see that the magnetic field forms a circular loop around the axis of the current element. The magnitude of the magnetic field is directly proportional to the current moment and is inversely proportional to the square of the distance between the source and the observation point. The cross-product also suggests that the strength of the magnetic field is maximum when and are perpendicular to each other, that is, for equal to . As reduces the angle between and reduces consequently reducing the strength of the magnetic field reduces. For extreme case of the vectors and are colinear making cross product and hence the magnetic field identically zero. The magnetic field therefore is identically zero at the axis of the current element. The magnetic field is related to through a medium characteristic parameter called permeability of the medium as The permeability of vacuum is denoted by and its value is Henery/meter. A ratio of permeability of a medium to that of the vacuum is called the relative permeability giving Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Conduction Current Density J In situations where there is a variation in the current magnitude it is rather inadequate to define just the total current flow in the system. One can then use the current density as the primary parameter which is defined as the current flow per unit area. Consider a cylinder made of a conducting material with an arbitrary cross-section . Let us assume that the current density exists in the cylinder over a length ` '. If the conductivity of the cylinder is its resistivity is . The resistance of the cylinder between its two ends A and B is then given by For simplicity assuming that the current density is constant all over the cross-section, the total current in the conductor is Due to the current flow in the conductor, there is a voltage drop between point A and B. If we assume the conducting cylinder to be of infinitesimal length we can assume the electric field associated with the voltage drop to be constant between A and B. Then one can write the voltage to be Now from Ohm's law we have Substituting for , and we get The magnitude of the conduction current density is proportional to the electric field strength E. From the Ohm's law we also know that the current flows in the direction in which the potential drop is maximum. That is, the current flows in the direction in which the potential has maximum change. This direction is nothing but the direction of the gradient of . The direction of gradient of V is same as that of the electric field . Therefore we can conclude that not only the magnitude of is proportional to but its direction is also same as that of . The relation then can be written for vector and as This is the general form of the Ohms law. Page 5 Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Objectives In this course you will learn the following Magnetic Field and Magnetic Flux Density. Conduction Current Density J. Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Magnetic Field and Magnetic Flux Density The magnetic field can be defined as the force experienced by a unit magnetic monopole. However, there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law. Consider an infinitesimally small piece of a wire carrying current in it. The length of the piece is say . The current moment is then defined by the product . Since the piece of the wire can be oriented in any direction the length is a vector quantity and therefore should be denoted by . The current is a scalar quantity making the current moment is a vector quantity. Without loss of generality let us assume that the current element is located at the origin of the co-ordinate system as shown in Fig. According to the Biot - Savart's law the magnetic field ( field due to infinitesimally small current element) at a point is given as Where is the unit vector in the direction of the line joining the current element and the observation point . Since the vectors and lie in the plane of the paper, the cross-product of the two has a direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards, the direction of is upwards and the direction of the magnetic field will be going into the paper as indicated by at point . If we take some other observation point on the left of the current element, by right hand screw rule, the magnetic field will come out of the paper as indicated by . If one streches his imagination a little, he can then see that the magnetic field forms a circular loop around the axis of the current element. The magnitude of the magnetic field is directly proportional to the current moment and is inversely proportional to the square of the distance between the source and the observation point. The cross-product also suggests that the strength of the magnetic field is maximum when and are perpendicular to each other, that is, for equal to . As reduces the angle between and reduces consequently reducing the strength of the magnetic field reduces. For extreme case of the vectors and are colinear making cross product and hence the magnetic field identically zero. The magnetic field therefore is identically zero at the axis of the current element. The magnetic field is related to through a medium characteristic parameter called permeability of the medium as The permeability of vacuum is denoted by and its value is Henery/meter. A ratio of permeability of a medium to that of the vacuum is called the relative permeability giving Module 3 : Maxwell's Equations Lecture 20 : Magnetostatics and Ohms Law Conduction Current Density J In situations where there is a variation in the current magnitude it is rather inadequate to define just the total current flow in the system. One can then use the current density as the primary parameter which is defined as the current flow per unit area. Consider a cylinder made of a conducting material with an arbitrary cross-section . Let us assume that the current density exists in the cylinder over a length ` '. If the conductivity of the cylinder is its resistivity is . The resistance of the cylinder between its two ends A and B is then given by For simplicity assuming that the current density is constant all over the cross-section, the total current in the conductor is Due to the current flow in the conductor, there is a voltage drop between point A and B. If we assume the conducting cylinder to be of infinitesimal length we can assume the electric field associated with the voltage drop to be constant between A and B. Then one can write the voltage to be Now from Ohm's law we have Substituting for , and we get The magnitude of the conduction current density is proportional to the electric field strength E. From the Ohm's law we also know that the current flows in the direction in which the potential drop is maximum. That is, the current flows in the direction in which the potential has maximum change. This direction is nothing but the direction of the gradient of . The direction of gradient of V is same as that of the electric field . Therefore we can conclude that not only the magnitude of is proportional to but its direction is also same as that of . The relation then can be written for vector and as This is the general form of the Ohms law.Read More

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