Chapter 3a Magnetostatic ECT1026 Field Theory Chapter 3a Magnetostatic By Dr Mardeni Roslee mardeni.roslee@mmu.edu.my 0383125481

Contents 3.1 Introduction 3.2 Magnetic Force 3.3 The Biot-Savart Law 3.4 Ampère’s Law

Basic Concepts and Quantities ECT1026 Field Theory Lecture 3-1 2009/2010 3.1 Introduction Basic Concepts and Quantities of Magnetostatics Current Carrying Conductor Magnetic Field or Magnetic Flux Lines Right-Hand Screw Rule Magnetic Flux Density (Wb/m2) Magnetic Field Intensity (A/m) Current & Current Density Magnetic Dipoles & Current Loops

Magnetostatics steady motion of electric charges ECT1026 Field Theory 3.1 Introduction Magnetostatics phenomenon associated with the steady motion of electric charges 1st Observed Magnetic Phenomena Natural stone - ancient city of Magnesia now known as magnetite (Fe3O4) examples of permanent magnets

Permanent Magnets & Compass Needles Interaction ECT1026 Field Theory 3.1 Introduction Permanent Magnets & Compass Needles Interaction described in terms of Magnetic Poles Point north  North-pole (N) – filed lines emerge Point south  South-pole (S) – field lines enter S N Magnetic Poles always exist in Pairs

Opposite Poles  Attract ECT1026 Field Theory 3.1 Introduction S N Like Poles  Repel N S Opposite Poles  Attract

Chapter 3 Magnetostatics ECT1026 Field Theory 3.1 Introduction Chapter 2 Electrostatics electric charges  electric field Chapter 3 Magnetostatics bar magnet  magnetic filed Magnetic-Field Lines Magnetic Flux Density B

Hans Christian Oersted in 1819 Electric Current  Magnetic Field ECT1026 Field Theory 3.1 Introduction Hans Christian Oersted in 1819 Electric Current  Magnetic Field compass needle was deflected by a current in a wire The needle always turned in the direction perpendicular to the current-carrying wire and to the radial line connecting the wire to the needle

ECT1026 Field Theory 3.1 Introduction Current-carrying wire induces a magnetic field that formed closed circular loops around the wire

Common Effects of Magnetic Field (on Matter) ECT1026 Field Theory 3.1 Introduction Common Effects of Magnetic Field (on Matter) When a current-carrying conductor is placed near to a magnetic needle the needle will deflect A moving charge particle experiences magnetic force Any current carrying conductor also experiences a force Two parallel conductors carrying same direction current are attracted towards each other, and vice versa.

Magnetic Field or Magnetic Flux Lines ECT1026 Field Theory 3.1 Introduction Characteristics of Magnetic Field or Magnetic Flux Lines The distribution and density of magnetic field is visualized as lines of magnetic flux. Magnetic flux is the basis used to explain magnetic effects and magnitudes Direction of magnetic field/lines – direction of the north-seeking pole of a compass needle placed in the field Each line of magnetic flux forms a closed-loop

Magnetic Field or Magnetic Flux Lines ECT1026 Field Theory 3.1 Introduction Characteristics of Magnetic Field or Magnetic Flux Lines Lines of magnetic flux never intersect Lines of magnetic flux always trying to shorten themselves causing unlike poles to attract each other Lines of magnetic flux (parallel) are in the same direction and repel each other. They exerted a lateral pressure on one another A piece of soft iron can be magnetized thru’ magnetic induction

ECT1026 Field Theory 3.1 Introduction Magnetic Flux, B lines of force The magnetic field consists of lines of force, which form complete circles around the conductor These circles centered around the center of the current carrying conductor and the circular planes are perpendicular to it

direction of the circling magnetic flux around the conductor ECT1026 Field Theory 3.1 Introduction Right-Hand Screw Rule Used to relate the direction of current flowing in a conductor bar, & direction of magnetic field circling it B I Thumb direction of the current flowing in the conductor  Directed OUT of the paper  Directed INTO the page Fingers direction of the circling magnetic flux around the conductor

Parallel and Anti-Parallel Linear Currents ECT1026 Field Theory 3.1 Introduction Parallel and Anti-Parallel Linear Currents Two current carrying conductors are placed in each other parallel  each will possess a magnetic field of its own, &  each will exert a magnetic field on the other conductor Parallel Linear Current Anti-Parallel Linear Current A sign “CROSS” indicates the current is flowing IN along the conductor A sign “DOT” indicates the current is flowing OUT along the conductor

ECT1026 Field Theory 3.1 Introduction Magnetic Flux Density B (Wb/m2) Magnetic Field Intensity H (A/m) B = mH = mrmo H m= mrmo mo : Permeability of free space (= 4p 10-7 H/m) mr : relative permeability of the material 1 Wb/m2 = 1 T [Wb = weber; T = tesla] Relative Permeability mr of some common materials Diamagnetic Gold, Silver, Copper, Water mr ~ 1 Paramagnetic Air, Aluminium, Tungsten, platinum ~ 1 Ferromagmetic Cobalt ~ 250, Nickel ~ 600: Iron ~ 4,000-5,000: Mild Steel ~ 2600

 • B = 0 B • ds = 0 H • dl = I   H = J ECT1026 Field Theory 3.1 Introduction The two fundamental postulates of magnetostatics that specify the divergence and the curl of B in free space are: Integral Form: Differential Form: B • ds = 0  • B = 0 H • dl = I   H = J

Attributes of Electrostatics & Magnetostatics ECT1026 Field Theory 3.1 Introduction Attributes of Electrostatics & Magnetostatics

uppercase letter I = steady current ECT1026 Field Theory 3.1 Introduction Steady Current & Current Density in a Conductor Current = (A) the rate at which the charge is transported past a given point in a conducing medium 1 A = transportation of one coulomb of charge in one second = 1 C/s uppercase letter I = steady current

J Δq´=vΔv= vΔlΔs´ Current Density (A/m2) 3.1 Introduction ECT1026 Field Theory 3.1 Introduction Current Density (A/m2) J Fig. 3-1 (Cos 0 =1) (4) The charges are moving with a mean velocity u along the axis of the tube. Over a period Δt, the chares move a distance Δl = uΔt The amount of charge that crosses the tube’s cross-sectional surface in time is therefore: (3) Distance = velocity x time Δv= ΔlΔs´ Δq´=vΔv= vΔlΔs´ (2) (1)

More General Case (surface normal is not parallel to u) ECT1026 Field Theory 3.1 Introduction (Cos  = 0) More General Case (surface normal is not parallel to u) Amount of charge Δq flowing through Δs: OR Δq=vu•ΔsΔt Δq=vuΔsΔtcos = vu•Δs ΔI = Δt Δq Corresponding current ΔI = J • Δs J = v u  (A/m2)

I dl = Js ds = Jdv 3.1 Introduction ECT1026 Field Theory 3.1 Introduction Current sources specified in terms of Js over a surface S, or J over a volume V are related by the following equation: I dl = Js ds = Jdv Surface Current Density, Js Volume Current Density, J The total current flowing across the surface of the conductor is The total current crossing the cross section S of the cylinder is Js dl l I = I = J• ds s l dl ds S

Magnetic Dipoles and Current Loops ECT1026 Field Theory 3.1 Introduction Magnetic Dipoles and Current Loops A current loop with dimensions much smaller that the distance between the loop and the observation point is called a magnetic dipole. Magnetic Field Pattern is similar to that of a permanent magnetic and a pattern of the electric field of the electric dipole (c)=(a)