Chapter 28Sources of the Magnetic FieldBiot-Savart Law – IntroductionBiot and Savart conducted experiments on the force exerted by an electric current on a nearby magnetThey arrived at a mathematical expression that gives the magnetic field at some point in space due to a currentBiot-Savart Law – Set-UpThe magnetic field is at some point PThe length element is The wire is carrying a steady current of IPlease replace with fig. 30.1dBrdsrBiot-Savart Law –ObservationsThe vector is perpendicular to both and to the unit vector directed from toward PThe magnitude of is inversely proportional to r2, where r is the distance from to PdBrrˆdBrdsrdsrdsrBiot-Savart Law –Observations, contThe magnitude of is proportional to the current and to the magnitude ds of the length element The magnitude of is proportional to sin θ, where θis the angle between the vectors and dsrrˆdsrdBrdBrThe observations are summarized in the mathematical equation called the Biot-Savart law:The magnetic field described by the law is the field due to the current-carrying conductorDon’t confuse this field with a field external to the conductorBiot-Savart Law – Equation24oµddπ r×=s rBrrˆIPermeability of Free SpaceThe constant µois called the permeability of free spaceµo= 4π x 10-7T.m / ATotal Magnetic Fieldis the field created by the current in the length segment dsTo find the total field, sum up the contributions from all the current elements IThe integral is over the entire current distributiondBr24oµdπ r×=∫s rBrrˆIdsrBiot-Savart Law – Final NotesThe law is also valid for a current consisting of charges flowing through spacerepresents the length of a small segment of space in which the charges flowFor example, this could apply to the electron beam in a TV setdsrCompared to Distance The magnitude of the magnetic field varies as the inverse square of the distance from the sourceThe electric field due to a point charge also varies as the inverse square of the distance from the chargeBrErCompared to , 2DirectionThe electric field created by a point charge is radial in directionThe magnetic field created by a current element is perpendicular to both the length element and the unit vectorrˆdsrBrErCompared to , 3SourceAn electric field is established by an isolated electric chargeThe current element that produces a magnetic field must be part of an extended current distributionTherefore you must integrate over the entire current distributionBrErfor a Long, Straight ConductorThe thin, straight wire is carrying a constant currentIntegrating over all the current elements gives ( )211 2 44θoθoµBθ dθπaµθ θπa= −= −∫IcosIsin sin() sin d dxθ× =s r krˆˆBrfor a Long, Straight Conductor, Special CaseIf the conductor is an infinitely long, straight wire, θ1= π/2 and θ2= -π/2The field becomes 2IoµBπa=Brfor a Long, Straight Conductor, DirectionThe magnetic field lines are circles concentric with the wireThe field lines lie in planes perpendicular to to wireThe magnitude of the field is constant on any circle of radius aThe right-hand rule for determining the direction of the field is shownBrfor a Curved Wire SegmentFind the field at point Odue to the wire segmentI and R are constantsθwill be in radians4IoµBθπR=Brfor a Circular Loop of WireConsider the previous result, with a full circleθ= 2πThis is the field at the center of the loop24 4 2o o oµ µ µB θ ππa πa a= = =I I IBrfor a Circular Current LoopThe loop has a radius of R and carries a steady current of IFind the field at point P( )232 222oxµ aBa x=+IBrComparison of LoopsConsider the field at the center of the current loopAt this special point, x = 0 Then, This is exactly the same result as from the curved wire( )232 2222o oxµ a µBaa x= =+I IMagnetic Field Lines for a LoopFigure (a) shows the magnetic field lines surrounding a current loopFigure (b) shows the field lines in the iron filingsFigure (c) compares the field lines to that of a bar magnetMagnetic Force Between Two Parallel ConductorsTwo parallel wires each carry a steady currentThe field due to the current in wire 2 exerts a force on wire 1 of F1= I1ℓB22BrPLAYACTIVE FIGUREMagnetic Force Between Two Parallel Conductors, cont.Substituting the equation for givesParallel conductors carrying currents in the same direction attract each otherParallel conductors carrying current in opposite directions repel each other1 212I IoµFπa=l2BrMagnetic Force Between Two Parallel Conductors, finalThe result is often expressed as the magnetic force between the two wires, FBThis can also be given as the force per unit length:1 22I IB oF µπa=lDefinition of the AmpereThe force between two parallel wires can be used to define the ampereWhen the magnitude of the force per unit length between two long, parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7N/m, the current in each wire is defined to be 1 ADefinition of the CoulombThe SI unit of charge, the coulomb, is defined in terms of the ampereWhen a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 CAndre-Marie Ampère1775 – 1836French physicistCreated with the discovery of electromagnetismThe relationship between electric current and magnetic fieldsAlso worked in mathematicsMagnetic Field of a WireA compass can be used to detect the magnetic fieldWhen there is no current in the wire, there is no field due to the currentThe compass needles all point toward the Earth’s north pole Due to the Earth’s magnetic fieldMagnetic Field of a Wire, 2 Here the wire carries a strong current The compass needles deflect in a direction tangent to the circle This shows the direction of the magnetic field produced by the wire Use the active figure to vary the currentPLAYACTIVE FIGUREMagnetic Field of a Wire, 3The circular magnetic field around the wire is shown by the iron filingsAmpere’s LawThe product of can be evaluated for small length elements on the circular path defined by the compass needles for the long straight wireAmpere’s law states that the line integral ofaround any closed path equals µoIwhere I is the