Chapter 28 – Sources of Magnetic Field- Magnetic Field of a Moving Charge- Magnetic Field of a Current Element- Magnetic Field of a Straight Current-Carrying Conductor- Force Between Parallel Conductors- Magnetic Field of a Circular Current Loop- Ampere’s Law- Applications of Ampere’s Law- Magnetic Materials1. Magnetic Field of a Moving Charge- A charge creates a magnetic field only when the charge is moving.Source point: location of the moving charge.Field point: point P where we want to find the field. 20sin4rvqBϕπµ=Magnetic field from a point charge moving with constant speedµ0= 4π·10-7Wb/A·m = N s2/C2= N/A2= T m/A (permeability of vacuum)c = (1/µ0ε0)1/2 speed of lightMagnetic field of a point charge moving with constant velocity20ˆ4rrvqB×=πµ= vector from source to field point rrr /ˆ=Moving Charge: Magnetic Field Linesdirection of v. Your fingers curl around the charge in direction of magnetic field lines.- The magnetic field lines are circles centered on the line of v and lying in planes perpendicular to that line.- Direction of field line: right hand rule for + charge point right thumb in2. Magnetic Field of a Current Element- The total magnetic field caused by several moving charges is the vectorsum of the fields caused by the individual charges. Current Element: Vector Magnetic FieldnqAdldQ=20200sin4sin4sin42rIdlrAdlvqnrvdQdBddϕπµϕπµϕπµ===∫×=20ˆ4rrlIdBπµLaw of Biot and Savart(total moving charge in volume element dl A)Moving charges in current element are equivalent to dQ moving with drift velocity.(I = n q vdA)Current Element: Magnetic Field Lines - Field vectors (dB) and magnetic field lines of a current element (dl) are likethose generated by a + charge dQ moving in direction of vdrift.- Field lines are circles in planes ┴ to dl and centered on line of dl.3. Magnetic Field of a Straight Current-Carrying Conductor22)sin(sin1ˆyxxdldldlrld+⋅=−⋅=⋅⋅=×ϕπϕ2202/322024)(4axxaIyxdyxIBaa+=+⋅=∫−πµπµxIaxaIB⋅=⋅⋅=πµπµ24)2(00rIB⋅=πµ20∫×=20ˆ4rrlIdBπµIf conductors length 2a >> xB direction: into the plane of the figure, perpendicular to x-y planeField near a long, straight current-carrying conductor- Electric field lines radiate outward from + line charge distribution. They begin and end at electric charges.- Magnetic field lines encircle the current that acts as their source. They form closed loops and never have end points.-The total magnetic flux through any closed surface is zero there are noisolated magnetic charges (or magnetic monopoles) any magnetic fieldline that enters a closed surface must also emerge from that surface.4. Force Between Parallel Conductors- Two conductors with current in same direction. Each conductor lies in B set-up by the other conductor.B generated by lower conductor at the position of upper conductor:rIB⋅=πµ20BLIF×='rILILBIF⋅==πµ2''0- Parallel conductors carrying currents in same direction attract each other. If I has contrary direction they repel each other.rIILF⋅⋅=πµ2'0Two long parallel current-carrying conductorsForce on upper conductor is downward.Magnetic Forces and Defining the Ampere- One Ampere is the unvarying current that, if present in each of the two parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of exactly 2 x 10-7N per meter of length.5. Magnetic Field of a Circular Current Loop( )2204 axdlIdB+=πµ204rIdlBπµ=( )( )2/1222204sinaxxaxdlIdBdBy++==πµθ∫×=20ˆ4rrlIdBπµ( )( )2/1222204cosaxaaxdlIdBdBx++==πµθ- Rotational symmetry about x axis no B component perpendicular to x.For dl on opposite sides of loop, dBxare equal in magnitude and in same direction, dByhave same magnitude but opposite direction (cancel).( )2/322202 axIaBx+=µ( ) ( ) ( ))2(4442/32202/32202/3220aaxaIdlaxaIaxadlIBxππµπµπµ+=+=+=∫∫(on the axis of a circular loop)( )2/322202 axNIaBx+=µMagnetic Field on the Axis of CoilaNIBx20µ=2/3220)(2 axBx+=πµµ(on the axis of N circular loops)(at the center, x=0, of N circular loops)(on the axis of any number of circular
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