CHETTINAD COLLEGE OF ENGINEERING & TECHNOLOGY NH-67, TRICHY MAIN ROAD, PULIYUR, C.F. 639 114, KARUR DT. DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING COURSE MATERIAL Subject Name: Electromagnetic Fields Class / Sem: BE (ECE) / IV Syllabus:: UNIT II STATIC MAGNETIC FIELD The Biot-Savart Law in vector form Magnetic Field intensity due to a finite and infinite wire carrying a current I Magnetic field intensity on the axis of a circular and rectangular loop carrying a current I Ampere s circuital law and simple applications. Magnetic flux density The Lorentz force equation for a moving charge and applications Force on a wire carrying a current I placed in a magnetic field Torque on a loop carrying a current I Magnetic moment Magnetic Vector Potential. Overview:: The Biot-Savart law and Ampere s circuital laws are important to find the magnetic field intensity and magnetic flux density due to both finite and infinite wire carrying current. The Biot Savart Law is an equation in electromagnetism that describes the magnetic field B generated by an electric current. Ampere s circuital law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. The Lorentz force equation for a moving charge is used to find the force on a point charge due to electromagnetic fields. The other topics discussed are torque on a loop carrying a current, magnetic moment and magnetic vector potential.
Objectives:: To compute the magnetic field generated by a steady current using Biot- Savart Law. To define Ampere s circuital law. To define magnetic field intensity and to determine the MF intensity in an infinite wire, circular loop, long thick wire and solenoid. To Apply Lorentz force equation to find the electric and magnetic fields, for a specified set of forces on a charged particle moving in the field region. To Calculate the magnetic field due to a current distribution by applying superposition in conjunction with the magnetic field due to a current element. To determine the torque on a loop carrying a current I To define magnetic dipole moment and magnetic vector potential. Biot-Savart Law The Biot Savart Law is an equation in electromagnetism that describes the magnetic field B generated by an electric current. The vector field B depends on the magnitude, direction, length, and proximity of the electric current, and also on a fundamental constant called the magnetic constant. The law is valid in the magnetostatic approximation, and results in a B field consistent with both Ampère's circuital law and Gauss's law for magnetism. The Biot Savart law is used to compute the magnetic field generated by a steady current, i.e. a continual flow of charges, for example through a wire, which is constant in time and in which charge is neither building up nor depleting at any point. The equation is as follows: (in SI units), where
I is the current, dl is a vector, whose magnitude is the length of the differential element of the wire, and whose direction is the direction of conventional current, db is the differential contribution to the magnetic field resulting from this differential element of wire, is the magnetic constant, is the displacement unit vector in the direction pointing from the wire element towards the point at which the field is being computed, and is the distance from the wire element to the point at which the field is being computed, the symbols in boldface denote vector quantities. To apply the equation, you choose a point in space at which you want to compute the magnetic field. Holding that point fixed, you integrate over the path of the current(s) to find the total magnetic field at that point. The application of this law implicitly relies on the superposition principle for magnetic fields, i.e. the fact that the magnetic field is a vector sum of the field created by each infinitesimal section of the wire individually. Magnetic Field Intensity Magnetic field strength (H) is the amount of magnetizing force. It is proportional to the length of a conductor and the amount of electrical current passing through the conductor. Magnetic field strength is a vector quantity whose magnitude is the strength of a magnetic field at a point in the direction of the magnetic field at that point. Flux density (B), the amount of magnetism induced in a body, is a function of the magnetizing force (H).
Amphere s Circuital Law In classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is the magnetic analogue of Gauss's law, and one of the four Maxwell's equations that form the basis of classical electromagnetism.
Origial Amphere s Circuital Law: An electric current produces a magnetic field. In its historically original form, Ampère's Circuital law relates the magnetic field to its source, the current density. The equation is not in general correct (see "Maxwell's correction" below), but is correct in the special case where the electric field is constant (unchanging) in time. The law can be written in two forms, the "integral form" and the "differential form". The forms are equivalent, and related by the Kelvin-Stokes theorem.
The generalized law (in SI units), takes the following integral form: where is the vacuum permittivity and E is the electric field. This Ampère-Maxwell law can also be stated in differential form (with the Kelvin-Stokes theorem): There are two simple cases where the magnetic field integrations are easy to carry out, and fortunately they are in geometries that are of practical use. We use the formula for the magnetic field of an infinitely long wire whenever we want to estimate the field near a segment of wire, and we use the formula for the magnetic field at the center of a circular loop of wire whenever we want to estimate the magnetic field near the center of any loop of wire. Infinitely Long Wire: The magnetic field at a point a distance r from an infinitely long wire carrying current I has magnitude and its direction is given by a right-hand rule: point the thumb of your right hand in the direction of the current, and your fingers indicate the direction of the circular magnetic field lines around the wire. Circular Loop: The magnetic field at the center of a circular loop of current-carrying wire of radius R has magnitude and its direction is given by another right-hand rule: curl the fingers of your right hand in the direction of the current flow, and your thumb points in the direction of the magnetic field inside the loop.
Long Thick Wire: Imagine a very long wire of radius a carrying current I distributed symmetrically so that the current density, J, is only a function of distance r from the center of the wire. Ampere's law can be used to find the magnetic field at any radius r. Outside the wire, where, we have just as if all the current were concentrated at the center of the wire. Inside the wire, where r < a, we have where I(r) is the current flowing through the disk of radius r inside the wire; the current outside this disk contributes nothing to the magnetic field at r. Note that this is analogous to the result for symmetric electric fields. Long Solenoid: Imagine a long solenoid of length L with N turns of wire wrapped evenly along its length. Ampere's law can be used to show that the magnetic field inside the solenoid is uniform throughout the volume of the solenoid (except near the ends where the magnetic field becomes weak) and is given by where n = N/L. Toroid: Imagine a toroid consisting of N evenly spaced turns of wire carrying current I. (Imagine winding wire onto a bagel, with the wire coming up through the hole, out around the outside, then up through the hole again, etc..) Ampere's law can be used to show that the magnetic field within the volume enclosed by the toroid is given by where R is the distance from the z-axis in cylindrical coordinates, with the z-axis pointing straight up through the hole in the center of the bagel.
Magnetic Flux Density A vector quantity to measure the strength and direction of the magnetic field around a magnet or an electric current. Magnetic flux density is equal to magnetic field strength times the magnetic permeability in the region in which the field exists. Electric charges moving through a magnetic field are subject to a force described by the equation F = qv B, where q is the amount of electric charge, v is the velocity of the charge, B is the magnetic flux density at the position of the charge, and is the vector product. Magnetic flux density also can be understood as the density of magnetic lines of force, or magnetic flux lines, passing through a specific area. It is measured in units of tesla. Also it is called magnetic flux, magnetic induction. Magnetic Flux Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking into account the strength and the extent of a magnetic field. The SI unit of magnetic flux is the weber (in derived units: volt-seconds), and the unit of magnetic field is the weber per square meter, or tesla. Magnetic flux is defined by a scalar product of the magnetic field and the area element vector. Quantitatively, the magnetic flux through a surface S is defined as the integral of the magnetic field over the area of the surface. where is the magnetic flux B is the magnetic field, S is the surface (area), denotes dot product, ds is an infinitesimal vector, whose magnitude is the area of a differential element of S, and whose direction is the surface normal. (See surface integral for more details.)
Lorentz force equation for a moving charge In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields: F = q ( E + v B) where F is the force (in Newtons) E is the electric field (in volts per meter) B is the magnetic field (in Teslas) q is the electric charge of the particle (in Coulombs) v is the instantaneous velocity of the particle (in meters per second) is the vector cross product and are gradient and curl, respectively or equivalently the following equation in terms of the vector potential and scalar potential: where:a and are the magnetic vector potential and electrostatic potential, respectively. The term qe is called the electric force, while the term qv B is called the magnetic force. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force: F mag = qv B The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the Laplace force.
Force on a current carrying wire When a wire carrying an electrical current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force). By combining the Lorentz force law above with the definition of electrical current, the following equation results, in the case of a straight, stationary wire: F = I L B where F = Force, measured in newtons I = current in wire, measured in amperes B = magnetic field vector, measured in teslas = vector cross product L = a vector, whose magnitude is the length of wire (measured in metres), and whose direction is along the wire, aligned with the direction of conventional current flow.
Alternatively, F = L I B where the vector direction is now associated with the current variable, instead of the length variable. The two forms are equivalent. If the wire is not straight but curved, the force on it can be computed by applying this formula to each infinitesimal segment of wire dl, then adding up all these forces via integration. Formally, the net force on a stationary, rigid wire carrying a current I is F = I dl B (This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.) One application of this is Ampère's force law, which describes how two current-carrying wires can attract or repel each other, since each experiences a Lorentz force from the other's magnetic field. Torque on a current loop The torque on a current-carrying coil, as in a DC motor, can be related to the characteristics of the coil by the "magnetic moment" or "magnetic dipole moment". The torque exerted by the magnetic force (including both sides of the coil) is given by The coil characteristics can be grouped as called the magnetic moment of the loop, and the torque written as
The direction of the magnetic moment is perpendicular to the current loop in the right-hand-rule direction, the direction of the normal to the loop in the illustration. Considering torque as a vector quantity, this can be written as the vector product If you exerted the necessary torque to overcome the magnetic torque and rotate the loop from angle zero to 180 degrees, you would do an amount of rotational work given by the integral Torque on a current carrying loop The fact that a force exists on a wire carrying a current in a magnetic field has an interesting application when we consider a current carrying loop. Consider a rectangular loop carrying a current in a magnetic field as in Fig.
Consider the force = l x on each of the four sides of the loop. On the top and bottom sections this force vanishes, because and are parallel or antiparallel in these cases. The force on the left section will have a magnitude aib in the direction indicated, while the force on the right section will have the same magnitude aib but in the opposite direction. These two forces, since they are oppositely directed, do not give rise to a net linear acceleration, but they do tend to rotate the loop around the vertical axis. There will thus be a net torque on the loop, which is given by = F L (b/2) + F R (b/2) = aibb = BIA where A = ab is the area of the loop. Although we have considered a rectangular loop, the relation arbitrarily shaped loop of area A. = BIA holds for an The fact that current carrying loops experience a net torque in a magnetic field is the principle behind the electric motor, where the electrical energy involved in establishing a current is converted into the mechanical energy of rotating a shaft.
Magnetic Dipole Moment From the expression for the torque on a current loop, the characteristics of the current loop are summarized in its magnetic moment. The magnetic moment can be considered to be a vector quantity with direction perpendicular to the current loop in the right-hand-rule direction. The torque is given by Magnetic Vector Potential There is no general scalar potential for magnetic field B but it can be expressed as the curl of a vector function This function A is given the name "vector potential" but it is not directly associated with work the way that scalar potential is. One rationale for the vector potential is that it may be easier to calculate the vector potential than to calculate the magnetic field directly from a given source current geometry. Its most common application is to antenna theory and the description of electromagnetic waves.
Summary:: The magnetic field generated by a steady current using Biot-Savart Law was computed. Ampere s circuital law was defined. Magnetic field intensity was defined. The Magnetic Field Intensity in an infinite wire, circular loop, long thick wire and solenoid was calculated using Ampere s circuital law. Lorentz force equation and its importance were discussed. The electric and magnetic fields, for a specified set of forces on a charged particle moving in the field region were calculated by applying the Lorentz force equation. The magnetic field due to a current distribution by applying superposition in conjunction with the magnetic field due to a current element was calculated. The torque on a loop carrying a current I was determined. Magnetic dipole moment and magnetic vector potential were discussed in detail. Key Terms:: Biot-Savart law Magnetic Field Intensity Ampere s circuital law Magnetic Flux Density Lorentz Force equation Magnetic moment Magnetic vector potential
Key Term Quiz:: 1. The unit of magnetic field intensity is. 2. Biot-Savarts law expresses the relationship between and. 3. Biot-Savarts law can also be expressed in terms of and. 4. Magnetic field intensity due to a finite wire carrying current I is given by. 5. Magnetic field intensity due to a infinite wire carrying current I is given by. 6. Magnetic field intensity on a circular loop carrying current I is given by. 7. Magnetic field intensity on a rectangular loop carrying current I is given by. 8. The applications of Amphere s circuital law are and. 9. The force on a wire carrying current placed in a magnetic field is given by. 10. The torque on a loop carrying a current I is given by. Multiple Choice:: 1. The unit of magnetic field intensity is (a) volts/meter (b) Weber/meter (c) amperes/meter (d) none of the above 2. The line integral of H about any closed path is equal to the current enclosed by that path. This theorem is called as (a) ampere circuital law (b) biot-savarts law (c) gauss law (d) none of the above 3. The unit of magnetic flux density is (a) H/m (b) Wb/m 2 (c) Wb/m (d) none of the above 4. The value of permeability of free space is (a) 0 (b) infinite (c) 4π x 10-7 (d) none of the above 5. The unit of electric flux is (a) coulombs (b) c/m 2 (c) Weber (d) none of the above
Review Questions:: 1. Derive the equation for Biot-Savart s law in vector form. 2. Derive the expression for magnetic field intensity due to: a. Finite wire carrying a current I b. Infinite wire carrying a current I c. Axis of a circular loop carrying a current I d. Axis of a rectangular loop carrying a current I 3. Explain Ampere s circuital law with simple applications. 4. Explain Lorentz force equations for a moving charge and its applications. 5. Derive the expression for force on a wire carrying a current I placed in a magnetic field. 6. Derive the expression for the torque on a loop carrying a current I. 7. Explain the following terms in detail: a. Magnetic moment b. Magnetic vector potential