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Magnetostatics 1 EEL 3472 Magnetostatics If charges are moving with constant velocity, a static magnetic (or magnetostatic) field is produced. Thus, magnetostatic fields originate from currents (for instance, direct currents in current-carrying wires). Most of the equations we have derived for the electric fields may be readily used to obtain corresponding equations for magnetic fields if the equivalent analogous quantities are substituted. 2 EEL 3472 Magnetostatics Biot – Savart’s Law dH k The magnetic field intensity dH produced at a point P by the differential current element Idl is proportional to the product of Idl and the sine of the angle between the element and the line joining P to the element and is inversely proportional to the square of the distance R between P and the element. Idl sin R2 IN SI units, k 1 4 , so dH k 3 Idl sin 4R2 EEL 3472 Magnetostatics Using the definition of cross product (A B AB sinA Ben) we can represent the previous equation in vector form as Idl er Idl R dH er R RR 2 3 R 4R 4R The direction of d H can be determined by the right-hand rule or by the right-handed screw rule. If the position of the field point is specified by r and the position of the source point by r dH I(dl er r ) I[dl (r r )] 2 3 4 r r 4 r r where err is the unit vector directed from the source point to the field point, and r r is the distance between these two points. 4 EEL 3472 Magnetostatics Example. A current element is located at x=2 cm,y=0, and z=0. The current has a magnitude of 150 mA and flows in the +y direction. The length of the current element is 1mm. Find the contribution of this element to the magnetic field at x=0, y=3cm, z=0 dl 103 ey r 3 102 ey r 2 * 102 ex (r r) (2ex 3ey ) 102 r r 13 102 dl (r r ) 10 3 ey (2ex 3ey ) 10 2 10 -5 (2ez ) ey ex ez ey ey 0 dH I[dl (r r)] 3 4 r r 0.15(2 10 -5 )ez 4 ( 13 10 2 ) 3 5.09 10 -3 ez A m 5 EEL 3472 Magnetostatics The field produced by a wire can be found adding up the fields produced by a large number of current elements placed head to tail along the wire (the principle of superposition). 6 EEL 3472 Magnetostatics Consider an infinitely long straight wire located on the z axis and carrying a current I in the +z direction. Let a field point be located at z=0 at a radial distance R from the wire. e z R (R Z2 )1 / 2 z 0 r r (R2 Z2 )1 / 2 H 7 dH Idz R 2 2 2 2 1/2 4 ( R Z ) ( R Z ) z 2 I(dl err ) 4 r r 2 Idz(ez err ) 4 r r 2 Idz sin e 4(R 2 Z2 ) 1 IR Z 2 dz 2 3 /2 ( R Z ) 4 R 2 R 2 Z2 I I (1 1) 4R 2R EEL 3472 Magnetostatics 8 EEL 3472 Magnetostatics dH r r I(dl er' r ) 2 4 r r a2 b2 const dl ad (perpendicular to both dl and er 'r ) The horizontal components of H (x-components and ycomponents add to zero cos sin(900 ) 9 a a2 b2 H 2 0 Iad a 2 2 2 4 (a b ) (a b 2 )1/ 2 Ia 2 Ia 2 2 4 (a 2 b 2 ) 3 / 2 2(a 2 b 2 ) 3 / 2 H H ez EEL 3472 Magnetostatics Magnetic Flux Density The magnetic flux density vector is related to the magnetic field intensity H by the following equation B H, T (tesla) or Wb/m2 where is the permeability of the medium. Except for ferromagnetic materials ( such as cobalt, nickel, and iron), most materials have values of very nearly equal to that for vacuum, o 4 10 7 H/m (henry per meter) The magnetic flux through a given surface S is given by B d s, S 10 Wb (weber) EEL 3472 Magnetostatics Magnetic Force One can tell if a magnetostatic field is present because it exerts forces on moving charges and on currents. If a current element Idl is located in a magnetic field B , it experiences a force dF Idl B This force, acting on current element Idl , is in a direction perpendicular to the current element and also perpendicular to B . It is largest when B and the wire are perpendicular. The force acting on a charge q moving through a magnetic field with velocity v is given by F q(v B ) The force is perpendicular to the direction of charge motion, and is also perpendicular to B . 11 EEL 3472 Magnetostatics The Curl Operator This operator acts on a vector field to produce another vector field. Let B(x, y, z) be a vector field. Then the expression for the curl of B in rectangular coordinates is B y B B y B x Bz B B z e ex x ez y z x y z y x where B x , B y , and B z are the rectangular components of The curl operator can also be written in the form of a determinant: 12 ex ey ez B x y z Bx By Bz EEL 3472 Magnetostatics The physical significance of the curl operator is that it describes the “rotation” or “vorticity” of the field at the point in question. It may be regarded as a measure of how much the field curls around that point. The curl of B is defined as an axial ( or rotational) vector whose magnitude is the maximum circulation of B per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum. L B dl curlB rot B B lim an s 0 s max where the area s is bounded by the curve L, and an is the unit vector normal to the surface s and is determined using the right – hand rule. 13 EEL 3472 Magnetostatics 14 EEL 3472 Magnetostatics B y B B y B x Bz B B z ex x ez ey z x y z y x 15 EEL 3472 Magnetostatics Classification of Vector Fields A vector field is uniquely characterized by its divergence and curl (Helmholtz’s theorem). The divergence of a vector field is a measure of the strength of its flow source and the curl of the field is a measure of the strength of its vortex source. 16 EEL 3472 Magnetostatics If A 0 then A is said to be solenoidal or divergenceless. Such a field has neither source nor sink of flux. Since ( F) 0 (for any F ), a solenoidal field A can always be expressed in terms of another vector F : A F If A 0 then A is said to be irrotational (or potential, or conservative). The circulation of A around a closed path is identically zero. Since V 0 (for any scalar V), an irrotational field A can always be expressed in terms of a scalar field V: A V 17 EEL 3472 Magnetostatics Magnetic Vector Potential Some electrostatic field problems can be simplified by relating the electric potential V to the electric field intensity E(E V). Similarly, we can define a potential associated with the magnetostatic field B : B A where A is the magnetic vector potential. Just as we defined in electrostatics dq(r ' ) V(r ) 4 r r ' we can define A(r ) 18 I(r ' )dl' L 4 r r' (electric scalar potential) Wb m (for line current) EEL 3472 Magnetostatics The contribution to A of each differential current element Idl' points in the same direction as the current element that produces it. The use of A provides a powerful, elegant approach to solving EM problems (it is more convenient to find B by first finding A in antenna problems). The Magnetostatic Curl Equation Basic equation of magnetostatics, that allows one to find the current when the magnetic field is known is H J where H is the magnetic field intensity, and J is the current density (current per unit area passing through a plane perpendicular to the flow). The magnetostatic curl equation is analogous to the electric field source equation D 19 EEL 3472 Magnetostatics Stokes’ Theorem This theorem will be used to derive Ampere’s circuital law which is similar to Gauss’s law in electrostatics. According to Stokes’ theorem, the circulation of A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L. LA dl A ds S The direction of integration around L is related to the direction ofds by the right-hand rule. 20 EEL 3472 Magnetostatics Determining the sense of dl and dS involved in Stokes’s theorem Stokes’ theorem converts a surface integral of the curl of a vector to a line integral of the vector, and vice versa. (The divergence theorem relates a volume integral of the divergence of a vector to a surface integral of the vector, and vice versa). Adv A ds v 21 S EEL 3472 Magnetostatics 1 Surface S2 (circular) 22 A dl A ds L S EEL 3472 Magnetostatics Ampere’s Circuital Law Choosing any surface S bounded by the border line L and applying Stokes’ theorem to the magnetic field intensity vector H , we have S H ds H dl L Substituting the magnetostatic curl equation we obtain H J I J d s H dl enc S L which is Ampere’s Circuital Law. It states that the circulation of H around a closed path is equal to the current enclosed by the path. 23 EEL 3472 Magnetostatics (Amperian Path) Ampere’s law is very useful in determining H when there is a closed path L around the current I such that the magnitude of H is constant over the path. 24 EEL 3472 Magnetostatics Boundary conditions are the rules that relate fields on opposite sides of a boundary. T= tangential components N= normal components We will make use of Gauss’s law for magnetic fields for 25 BN 2 B ds 0 and Ampere’s circuital law BN1 and for HT1 and HT 2 H dl Ienc EEL 3472 Magnetostatics Boundary condition on the tangential component of H In the limit as L2 0 HT1 HT 2 L 4 0 we have H dl H l HT 2l Ienc T1 L The tangential component of H is continuous across the boundary, while that of B is discontinuous where Ienc is the current on the boundary surface (since the integration path in the limit is infinitely narrow). When the conductivities of both media are finite, Ienc 0 and HT1 HT 2 BT1 / 1 BT2 / 2 26 EEL 3472 Magnetostatics Gaussian surface (cylinder with its plane faces parallel to the boundary) 27 EEL 3472 Magnetostatics Applying the divergence theorem to B and noting that B 0 (magnetic field is solenoidal) we have B ds B S 0 V In the limit as h 0 , the surface integral over the curved surface of the cylinder vanishes. Thus we have BN 2A BN1A 0 BN1 BN 2 28 (the normal component of B is continuous at the boundary) 1HN1 2HN 2 (the normal component of H is discontinuous at the boundary) EEL 3472