ENE 325 Electromagnetic Fields and Waves Lecture 5 Ampére’s law, Scalar and Vector Magnetic Potentials, Magnetic Force, Torque, and Magnetic Material

Review (1) Ampere’s circuital law - the integration of around any closed path is equal to the net current enclosed by that path. ‘Curl’ is employed to find the point form of Ampère’s circuital law. Curl of or is the maximum circulation of per unit area as the area shrinks to zero

Review (2) Magnetic flux density is related to the magnetic field intensity in the free space by Weber/m2 or Tesla (T) where 0 is the free space permeability, given in units of henrys per meter, or 0 = 410-7 H/m. Magnetic flux  (units of Webers) passing through a surface is found by

Outline Curl and point form of Ampére’s law Magnetic flux density Scalar and vector magnetic potentials Magnetic force and torque Magnetic material and permeability

Curl and the point form of Ampére’s circuital law (1) ‘Curl’ is employed to find the point form Ampère’s circuital law, analogous to ‘Divergence’ to find the point form of Gauss’s law. Curl of or is the maximum circulation of per unit area as the area shrinks to zero.

Curl and the point form of Ampére’s circuital law (2) ‘Curl´ operator perform a derivative of vector and returns a vector quantity. For Cartesian coordinates, can be written as

Physical view of curl Field lines indicating divergence A simple way to see the Field lines indicating curl direction of curl using right hand rule

Stokes’s Theorem Stokes’s Theorem relates a closed line integral into a surface integral

Magnetic flux density, B Magnetic flux density is related to the magnetic field intensity in the free space by Magnetic flux  (units of Webers) passing through a surface is found by Weber/m2 or Tesla (T) 1 Tesla = 10,000 Gauss. where 0 is the free space permeability, given in units of henrys per meter, or 0 = 410-7 H/m.

Gauss’s law for magnetic fields

EX1 A solid conductor of circular cross section is made of a homogeneous nonmagnetic material. If the radius a = 1 mm, the conductor axis lies on the z axis, and the total current in the direction is 20 A, find a) H at  = 0.5 mm b) B at  = 0.8 mm c) The total magnetic flux per unit length inside the conductor

Maxwell’s equations for static fields Integral form Differential form

The scalar and vector magnetic potentials (1) Scalar magnetic potential (Vm) is the simple practical concept to determine the electric field. Similarly, the scalar magnetic potential, Vm, is defined to relate to the magnetic field but there is no physical interpretation. Assume To make the above statement true, J = 0.

The scalar and vector magnetic potentials (2) From Laplace’s equation This equation’s solution to determine the potential field requires that the potential on the boundaries is known.

The scalar and vector magnetic potentials (3) The difference between V (electric potential) and Vm (scalar magnetic potential) is that the electric potential is a function of the positions while there can be many Vm values for the same position.

The scalar and vector magnetic potentials (4) While for the electrostatic case does not depend on path.

The scalar and vector magnetic potentials (5) Vector magnetic potential (A) is useful to find a magnetic filed for antenna and waveguide. From Let assume so and

The scalar and vector magnetic potentials (6) It is simpler to use the vector magnetic potential to determine the magnetic field. By transforming from Bio-savart law, we can write The differential form

Ex2 Determine the magnetic field from the infinite length line of current using the vector magnetic potential Find at point P(, , z) then

Vector magnetic potential for other current distributions For current sheet For current volume

Magnetic force Force on a moving charge Force on a differential current element N

Hall effect Hall effect is the voltage exerted from the separation of electrons and positive ions influenced by the magnetic force in the conductor. This Hall voltage is perpendicular to both magnetic field and the charge velocity. N

Magnetic force on the current carrying conductor (1) For the current carrying conductor, consider the magnetic force on the whole conductor not on the charges. From and then dQ = vdv

Magnetic force on the current carrying conductor (2) From we can write then For a straight conductor in a uniform magnetic field (still maintains the closed circuit), Force between differential current elements determine the force on the conductor influenced by the other nearby. or F = ILBsin

Ex3 Determine the force action on circuit 2 by circuit 1.

Force and torque on a closed circuit (1) Nm where = torque (Nm) = distance from the origin (m) = Force (N) If the current is uniform,

Force and torque on a closed circuit (2) For a current loop, we can express torque as If is constant or uniform, we can express torque as Define magnetic dipole moment where m = magnetic dipole moment (Am2). Therefore, torque can be shown as

Ex4 To illustrate some force and torque calculations, consider the rectangular loop shown. Calculate the total force and torque contribution on each side. Let the current I flow in the loop lied in the uniform magnetic field tesla.

Ex5 A 2.5 m length conductor is located at z = 0, x =4m and has a uniform current of 12 A in the direction . Determine in this area if the force acting on the conductor is 1.210-2 N in the direction .

The nature of magnetic materials Combine our knowledge of the action of a magnetic field on a current loop with a simple model of an atom and obtain some appreciation of the difference in behavior of various types of materials in magnetic fields. The magnetic properties of the materials depend on ‘magnetic moment’. Three types of magnetic moment are 1. The circular orbiting of electrons around the positive nucleus results in the current and then the magnetic field m = IdS. 2. Electron spinning around its own axis and thus generates a magnetic dipole moment. 3. Nuclear spin, this factor provides a negligible effect on the overall magnetic properties of materials.

Types of magnetic material (1) diamagnetic The small magnetic filed produced by the motion of the electrons in their orbits and those produced by the electron spin combine to produce a net field of zero or we can say the permanent magnetic moment m0 = 0. The external field would produce an internal magnetic field. Some examples of materials that has diamagnetic effect are Metallic bismuth, hydrogen, helium, the other ‘inert’ gases, sodium chloride, copper, gold silicon, germanium, graphite, and sulfur.

Types of magnetic material (2) paramagnetic The net magnetic moment of each atom is not zero but the average over the volume is, due to random orientation of the atoms. The material shows no magnetic effects in the absence of the external field. Whenever there is an external field and the alignment of magnetic moments acts to increase the value of , the material is called ‘paramagnetic’ but if it acts to decrease the value of , it is still called diamagnetic. For example, Potassium, Oxygen, Tungsten, and some rare earth elements.

Types of magnetic material (3) Ferromagnetic each atom has a relatively large dipole moment due to uncompensated electron spin moments. These moments are forced to line up in parallel fashion over region containing a large number of atoms, these regions are called ‘domains’. The domain moments vary in direction from domain to domain. The overall effect is therefore one of cancellation, and the material as a whole has no magnetic moment. When the external field is applied, those domains which are in the direction of the applied field increase their size at the expense of their neighbors, and the internal field increases greatly over that of the external field alone. When the external field is removed, a completely random domain alignment is not usually attained, and a residual dipole field remains in the macroscopic structure.

Types of magnetic material (4) The magnetic state of material is a function of its magnetic history or ‘hysteresis’. For example, Iron, Nickel, and Cobalt. Antiferromagnetic The forces between adjacent atoms cause the atomic moments to line up in anti parallel fashion. The net magnetic moment is zero. The antiferromagnetic materials are affected slightly by the presence of and external magnetic field. For example, nickel oxide (NiO), ferrous sulfide (FeS), and cobalt chloride (CoCl2).

Types of magnetic material (5) Ferrimagnetic Substances show an antiparallel alignment of adjacent atomic moments, but the moments are not equal. A large response to an external magnetic field therefore occurs. For example, the ferrites, the iron oxide magnetite (Fe3O4), a nickel-zinc ferrite, and a nickel ferrite.

Types of magnetic material (6) Superparamagnetic materials are composed of an assembly of ferromagnetic particles in a nonferromagnetic matrix. The domain walls cannot penetrate the intervening matrix material to the adjacent particles. For example, the magnetic tape.