MAGNETOSTATICS Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The motion of a single charge in the vicinity of a current-carrying conductor can be analyzed, and then express the total force on the particle as a sum of the magnetic forces exerted on it by the individual electrons moving in the conductor. MAGNETOSTATICS • So lets consider the two point charges Q1 and Q2 moving in a vacuum with the velocities v1 and v2, respectively, with respect to an observer. Let us also suppose that v1 and v2 are less than the velocity of light. v1 Q1 U12(unit vector) Q2 v2 MAGNETOSTATICS • The magnetic force Fm12 exerted by the charge Q1 on the charge Q2 is • πΉπ12 = π1 π2 πΎπ π2 (π£1 × (π£1 × π’12) ………………….. (1) • where πΎπ depends on the property of the space. • πΎπ = π0 4π , π0 is the absolute permeability • The unit of velocity in SI units is m/s, and one coulomb per second is one ampere. Hence the unit of π0 is N/(A)2 which can be written as henry/metre, its notation being H/m MAGNETOSTATICS • Comparing the force between the moving charges with the Coulomb’s law for the force between the static charge. That is • • π0 π1 π2 πΉπ12 = (π£1 2 4π π 1 π1 π2 πΉ12 = 4ππ0 π2 × (π£1 × π’12) • we find that the product (π0 π0 )–1 has the dimension (velocity)2 1 −12 Now we know, π0 = 8.854 × 10 = 9 36π×10 • And the value of π0 is exactly given byπ0 = 4π × 10−17 H/m MAGNETOSTATICS • Air also has practically the same permeability • (π0 π0 )–1≅ 9 × 1016 = (3 × 109 )2 • 1 π0 π 0 = 3 × 109 ≡ ππ π ππππ ππ πππβπ‘ ππ π£πππ’π’π • It should be noted that the magnitude of the magnetic force Fm12 given by Eq. (1) is a function of both the magnitude and the direction of the velocities of the two charges. The largest possible magnitude of this force would be • πΉπ12 = π0 π1 π2 4π π2 (π£1 × π£2) MAGNETOSTATICS • The ratio of the maximal possible magnetic force and electric force is • πΉπ12 πΉπ12 = ππ ππ π£1 π£2 = π£1 π£2 πΆπ2 • Let consider the direction of the magnetic force exerted by the charge π2 on the π1 which is obtained by interchanging the subscripts 1 and 2 in the equation (1) πΉπ21 = π0 π1 π2 4π π2 (π£1 × (π£1 × π’21), although π21 = −π12 In general(π£1 × (π£1 × π’12) ≠ −(π£1 × (π£1 × π’21), MAGNETOSTATICS • In the expression for πΉπ12 , given by equation (1), if instead of π2 , there is a charge moving with the velocity V, then the force πΉπ can be written as • • • • π0 π1 π2 πΉπ = (π£1 × (π£1 × 2 4π π π0 π1 π£1 × π12 πΉπ = π2 π2 × 4π π2 π0 π1 π£1 × π12 πΉπ = ππ × ( ) 4π π2 π0 π1 π£1 × π12 BUT π΅ = ( ) 4π π2 π12 ) MAGNETOSTATICS • πΉπ = ππ × π΅ • Where B is the magnetic flux density vector produced by the moving charge Q • The unit of magnetic flux density is newton-second per coulombmetre. It is also called tesla (T) or webers per square metre (Wb/m2) LORENTZ FORCE • Now, instead of only two charges, let us assume that there are n charges Q1,Q2,Q3…….,Qn, moving with velocity V1,V2,V3…….Vn, respectively.(with respect to the same coordinate system), then by the principle of superposition of the magnetic forces, the total magnetic force on Q due to those n charges will be • πΉπ = ππ × π0 4π π1 π£1 × π1 π2 +β―+ ππ ππ × ππ π2 • So the magnetic flux density vector B due to the n moving charge is •π΅= π0 ( ) 4π π π π π × ππ π π=1 ππ2 LORENTZ FORCE • To generalize, suppose in the region there are same static charges as well and Q is subject to an electric field πΉπ which will be • πΉπ = ππΈ where E is the electric field intensity due to the static charges. • Hence the total force on the charge Q is • πΉ = πΉπ + πΉπ • πΉ = ππΈ + ππ × π΅ • πΉ = π πΈ + π + π΅ , The total force F is known as the Lorentz Force. BIOT AND SAVART • Biot and Savart experimentally observed that the magnitude of magnetic field ππ΅ at a point P at a distance r from the small elemental length taken on a conductor carrying current varies • directly as the strength of the current I • directly as the magnitude of the length element π π • directly as the sine of the angle (say,θ) between ππ and π • inversely as the square of the distance between the point P and length element ππ • This is expressed as πΌππ sin π π2 πΌ ππ ππ΅ = π 2 sin π π π Where π = 0 4π • ππ΅ ∝ • • • ππ΅ = π0 πΌ ππ sin π 4π π 2 • π = unit vector in the direction from differential current element to P • According to the cross product rule ππ × π = ππ π sin π = ππ sin π • ππ΅ = π0 πΌ ππ× π 4π π 2 • But π = π/π • ππ΅ = π0 πΌ ππ× π 4π π 3 BIOT AND SAVART • To obtain the entire magnetic field intensity, the equation has to be integrated as •π΅= ππ΅ = π0 πΌ 4π ππ× π π2 Ampere’s Circuital Law • Ampère’s circuital law is used to calculate magnetic field at a point whenever there is a symmetry in the problem. This is similar to Gauss’s law in electrostatics. • By using this law, complex problems are solved in magnetostatics. This law can be used to find the magnetic field intensity due to any current distributions. • According the Ampere circuital law, the line integral of magnetic field intensity π΅ around a closed path is equal to the direct current enclosed by that path • π΅ β ππ = π0 πΌ Ampere’s Circuital Law • Where I is the current enclosed by the closed path. Note that the line integral does not depend on the shape of the path or the position of the conductor with the magnetic field Magnetic field due to the current carrying wire of infinite length using Ampère’s law • Consider a straight conductor of infinite length carrying current I and the direction of magnetic field lines is shown in Figure below. Since the wire is geometrically cylindrical in shape and symmetrical about its axis, we construct an Ampèrian loop in the form of a circular shape at a distance r from the centre of the conductor as shown in Figure below. From the Ampère’s law, we get Magnetic field due to the current carrying wire of infinite length using Ampère’s law • π π΅. ππ = π0 πΌ • where ππ is the line element along the Amperian loop (tangent to the circular loop). Hence, the angle between magnetic field vector and line element is zero. Therefore • π π΅. ππ = π0 πΌ • where I is the current enclosed by the Ampèrian loop. Due to the symmetry, the magnitude of the magnetic field is uniform over the Ampèrian loop, we can take B out of the integration. •π΅ π ππ = π0 πΌ • For a circular loop, the circumference is 2πr, which implies Magnetic field due to the current carrying wire of infinite length using Ampère’s law •π΅ 2ππ ππ π = π0 πΌ • π΅ β 2ππ = π0 πΌ •π΅= •π΅= 2ππ , In π0 πΌ 2ππ π π0 πΌ vector form, the magnetic field is • where n is the unit vector along the tangent to the Ampèrian loop Magnetic flux and magnetic flux density • Magnetic flux is a measurement of the total magnetic field which passes through a given area. It is a useful tool for helping describe the effects of the magnetic force on something occupying a given area. The measurement of magnetic flux is tied to the particular area chosen. We can choose to make the area any size we want and orient it in any way relative to the magnetic field • Magnetic flux is usually measured with a flux meter. The SI unit of magnetic flux is Weber (Wb) Magnetic flux and magnetic flux density • In order to calculate the magnetic flux, we consider the field-line image of a magnet or the system of magnets. The magnetic flux through a plane of the area, A, that is placed in a uniform magnetic field of magnitude, B is given as the scalar product of the magnetic field and the area A. Here, the angle at which the field lines pass through the given surface area is also important. If the field lines intersect the area at a glancing angle, that is, • when the angle between the magnetic field vector and the area vector is nearly equal to 90α΅, then the resulting flux is very low. • When the angle is equal to 0α΅, the resulting flux is maximum Magnetic flux and magnetic flux density • ΙΈ = π΅. π΄ = π΅π΄πΆπππ • here θ is the angle between vector A and vector B • Magnetic flux density(B) is defined as the force acting per unit current per unit length on a wire placed at right angles to the magnetic field πΉ πΌ • π΅ = , SI unit is Tesla (abbreviated as T) Magnetic field • When electric charges are moving, a force in addition to that described by Coulomb's law is exerted on them. To account for this additional force, we defined another force field, analogous to the Efield definition in the previous section. This second force field is called the magnetic-flux-density (B-field) vector, B. It is defined in terms of the force exerted on a small test charge, q. The magnitude of B is defined as • B = Fm/qv Magnetic field • where Fm is the maximum force on q in any direction, and v is the velocity of q. The units of B are webers per square meter. The B-field is more complicated than the E-field in that the direction of force exerted on q by the B-field is always perpendicular to both the velocity of the particle and to the B-field. This force is given by • F = q(v x B) Magnetic field • The quantity in parentheses is called a vector cross product. The direction of the vector cross product is perpendicular to both v and B and is in the direction that a right-handed screw would travel if v were turned into B. When a moving charge, q, is placed in a space where both an E-field and a B-field exist, the total force exerted on the charge is given by the sum of the two equations we saw. • F = q(E + v x B) • This is called the Lorentz force equation Quasi-Static Fields • An important class of electromagnetic fields is quasi-static fields. These fields have the same spatial patterns as static fields but vary with time. For example, if the charges that produce the E-fields were to vary slowly with time, the field patterns would vary correspondingly with time but at any one instant would be similar to the static-field patterns. • Similar statements could be made for the static B-fields. • Thus when the frequency of the source charges or currents is low enough, the fields produced by the sources can be considered quasi-static fields; the field patterns will be the same as the static-field patterns but will change with time. Analysis of quasi-static fields is thus much easier than analysis of fields that change more rapidly with time. Energy in static and quasi-static magnetic field • Because of the force exerted by an electric field on a charge placed in that field, the charge possesses potential energy. If a charge were placed in an E-field and released, its potential energy would be changed to kinetic energy as the force exerted by the E-field on the charge caused it to move. • Moving a charge from one point to another in an E-field requires work by whatever moves the charge. This work is equivalent to the change in potential energy of the charge. The potential energy of a charge divided by the magnitude of the charge is called electric-field potential. E-field potential is a scalar field Quasi-static magnetic fields • Quasi-static magnetic field problems can be somewhat decoupled from electric parameters, however, it is common to require the inclusion of electric scalar potential. Consider the magnetic curl equation • in quasi-static magnetic problems it is usual to neglect the rate of change of electric flux: • In this case, the current density will be considered as the sum of an applied current density (e.g. due to a stranded current carrying winding, where the gradient of electric scalar potential over the conducting area is negligible) plus a current density due to induced electric fields. This is a common situation in electric machines, especially in induction motors where there are stranded stator coils and solid rotor conductors. Quasi-static magnetic fields • substituting for magnetic vector potential give the "curl-curl" equation: • Now consider the electric curl equation and substitute for magnetic vector potential • an electric scalar potential may be defined such that Quasi-static magnetic fields • and substituted in to the curl-curl equation to give
