Magnetostatics 3rd 1
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Magnetostatic fields originate from steady currents like direct currents in wires. Most equations for electric fields can also describe magnetic fields by substituting analogous quantities. According to Biot-Savart's law, the magnetic field produced by a current element is proportional to the current and inversely proportional to the distance squared. Ampere's circuital law relates the line integral of magnetic field around a closed loop to the current passing through the enclosed surface. Forces can act on moving charges and on current-carrying wires placed in magnetic fields.
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Magnetostatics 3rd 1
- 1. Unit-2Unit-2 MagnetostaticsMagnetostatics Mr. HIMANSHU DIWAKAR Assistant Professor Department of ECE
- 2. 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. MagnetostaticsMagnetostatics
- 3. π = 4 1 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.IN SI units, , so α 2 R sinIdl kdH α = 2 R4 sinIdl kdH π α = MagnetostaticsMagnetostatics Biot – Savart’s Law
- 4. Using the definition of cross product we can represent the previous equation in vector form as The direction of 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 and the position of the source point by where is the unit vector directed from the source point to the field point, and is the distance between these two points. )sin( nABaABBA θ=× 32 44 R RdlI R adlI Hd r ππ × = × = RR = R Rar = r′ rre ′ rr ′− r 32 4 )]([ 4 )( rr rrdlI rr adlI Hd r ′− ′−× = ′− × = ππ Hd MagnetostaticsMagnetostatics
- 5. 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). MagnetostaticsMagnetostatics
- 6. Types of Current Distributions Line current density (current) - occurs for infinitesimally thin filamentary bodies (i.e., wires of negligible diameter). Surface current density (current per unit width) - occurs when body is perfectly conducting. Volume current density (current per unit cross sectional area) - most general. GETGI
- 7. Ampere’s Circuital Law in Integral Form Ampere’s Circuit Law states that the line integral of H around a closed path proportional to the total current through the surface bounding the path GETGI enc C IldH =⋅∫
- 8. Ampere’s Circuital Law in Integral Form (Cont’d) By convention, dS is taken to be in the direction defined by the right-hand rule applied to dl. ∫ ⋅= S encl sdJI Since volume current density is the most general, we can write Iencl in this way. S dl dS
- 9. Applying Stokes’s Theorem to Ampere’s Law GETGI ∫ ∫∫ ⋅== ⋅×∇=⋅ S encl SC sdJI sdHldH ⇒ Because the above must hold for any surface S, we must have JH =×∇ Differential form of Ampere’s Law
- 10. Ampere’s Law in Differential Form Ampere’s law in differential form implies that the B-field is conservative outside of regions where current is flowing. GETGI
- 11. Applications of Ampere’s Law Ampere’s law in integral form is an integral equation for the unknown magnetic flux density resulting from a given current distribution. GETGI encl C IldH =⋅∫ known unknown
- 12. Applications of Ampere’s Law (Cont’d) In general, solutions to integral equations must be obtained using numerical techniques. However, for certain symmetric current distributions closed form solutions to Ampere’s law can be obtained. GETGI
- 13. Applications of Ampere’s Law (Cont’d) Closed form solution to Ampere’s law relies on our ability to construct a suitable family of Amperian paths. An Amperian path is a closed contour to which the magnetic flux density is tangential and over which equal to a constant value. GETGI
- 14. Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law Consider an infinite line current along the z-axis carrying current in the +z-direction: GETGI I
- 15. Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (1) Assume from symmetry and the right-hand rule the form of the field (2) Construct a family of Amperian paths GETGI φφ HaH ˆ= circles of radius ρ where ∞≤≤ ρ0
- 16. Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) GETGI Amperian path IIencl = I ρ x y
- 17. Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (4) For each Amperian path, evaluate the integral GETGI HlldH C =⋅∫ ρπφ 2HldH C =⋅∫ magnitude of H on Amperian path. length of Amperian path.
- 18. Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (5) Solve for B on each Amperian path GETGI l I H encl = ρπ φ 2 ˆ I aH =
- 19. The magnetic flux density vector is related to the magnetic field intensity by the following equation 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, H/m (henry per meter) The magnetic flux through a given surface S is given by Wb (weber) H ,HB µ= µ 7 o 104 − ⋅π=µ ∫ ⋅=Ψ S ,sdB µ
- 20. 0=•∫s sdB Law of conservation of magnetic flux Law of conservation of magnetic flux or Gauss’s law for magnetostatic field
- 22. There are no magnetic flow sources, and the magnetic flux lines always close upon themselves
- 23. Forces due to magnetic field There are three ways in which force due to magnetic fields can be experienced 1.Due to moving charge particle in a B field 2.On a current element in an external field 3.Between two current elements
- 24. Force on a Moving Charge A moving point charge placed in a magnetic field experiences a force given by GETGI BvQ The force experienced by the point charge is in the direction into the paper. B BuqFm ×=
- 25. BuqF EqF m e ×= = FFF me =+ )( BuEqF ×+= Lorentz’s Force equation Note: Magnetic force is zero for q moving in the direction of the magnetic field (sin0=0) If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by
- 26. One can tell if a magneto-static field is present because it exerts forces on moving charges and on currents. If a current element is located in a magnetic field , it experiences a force This force, acting on current element , is in a direction perpendicular to the current element and also perpendicular to . It is largest when and the wire are perpendicular. Idl B dF=Idl×B B Idl B __ uQlId ⇔ Magnetic Force on a current element
- 27. FORCE ON A CURRENT ELEMENT The total force exerted on a circuit C carrying current I that is immersed in a magnetic flux density B is given by GETGI ∫ ×= C BldIF
- 28. Force between two current elements Experimental facts: Two parallel wires carrying current in the same direction attract. Two parallel wires carrying current in the opposite directions repel. GETGI × × I1 I2 F12F21 × • I1 I2 F12F21
- 29. Force between two current elements Experimental facts: A short current-carrying wire oriented perpendicular to a long current-carrying wire experiences no force. GETGI × I1 F12 = 0 I2
- 30. Force between two current elements Experimental facts: The magnitude of the force is inversely proportional to the distance squared. The magnitude of the force is proportional to the product of the currents carried by the two wires. GETGI