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Transcript
Today’s lecture V. Magnetostatics (in vacuum) • • • • • Magnetic field Lorentz force Current and current densities Continuity equation The Law of Biot-Savart Magnetic field B and Lorentz Force In electrostatics, we studied forces and fields of stationary charges. • Stationary charge is surrounded by electric field E What happens if the charges are moving? • • Moving charge is surrounded by E and magnetic field B The Lorentz Force describes the force on a charged particle Q in an electric and magnetic field. It has been found experimentally. F = Q(E + v × B ) Felec Fmag The force acting on the component v⊥ perpendicular to the magnetic field is a central force and forces the charge on a circle. The component v|| parallel to the field is not affected.⇒ Trajectory becomes a helix. No work done by magnetic force For a charge Q moved along a displacement the work done is: dWmag dl = v ⋅ dt = Fmag ⋅ dl = Q ⋅ (v × B )⋅ v ⋅ dt = 0 0 Magnetic forces do no work! Magnetic forces can only change the direction (of the Velocity) of charged particles. They cannot change the magnitude (of their velocity). Example: J.J. Thomson’s experiment (1897) for q/m 1. Determine v by measuring E/B: 2. Determine q/m by measuring R with E turned off. Electric currents - line charge Current is charge per unit time passing a given point. Convention: positive current points in the direction the positive charges are flowing. Currents only relate to the moving charges. It is measured in Ampere: 1 A=1 C/s. A line charge traveling in a wire with speed v constitutes a current of a line charge: I = λ ⋅v The force on a wire with current I in a magnetic field B is: Fmag = ∫ (v × B )⋅ dq = ∫ (v × B )⋅ λ ⋅ dl = ∫ (I × B )⋅ dl = ∫ I dl × B = I ∫ dl × B “magnetostatic case” I const in wire ( ( ) ) Surface and volume current densities Charge flowing over a surface is described by the surface current density: dI K := =σ ⋅v dl⊥ σ is the mobile surface charge density. K is current per unit width perpendicular to the flow. K will change in general over the surface due to changes in σ or v. The magnetic force on the surface current in a magnetic field is Fmag = ∫∫ (v × B )⋅ σ ⋅ da = ∫∫ (K × B )⋅ da The flow of charge through a volume is described by means of the volume current density: dI J := = ρ ⋅v da⊥ ρ is the mobile volume charge density. The magnetic force on the volume current in a magnetic field is F = v × B ⋅ ρ ⋅ dτ mag ∫∫∫ ( ) = ∫∫∫ (J × B )⋅ dτ Continuity equation Current crossing a surface S I = ∫∫ J ⋅ da⊥ = ∫∫ J ⋅ da S S The total charge per unit time leaving a volume V is ∫∫ J ⋅ da = ∫∫∫ ∇ ⋅ J ⋅ dτ S V ( ) Due to the conservation of charge d δρ ∫∫∫ ∇ ⋅ J ⋅ dτ = − dt ∫∫∫ ρ ⋅ dτ = − ∫∫∫ δ t ⋅ dτ V V V ( ) This holds for any volume V and therefore the result is the mathematical expression for local charge conservation, the Continuity equation: δρ ∇⋅ J = − δt Steady currents and the law of Biot-Savart Stationary charges ⇒constant electric fields ⇒electrostatics Steady currents⇒constant magnetic fields⇒magnetostatics A steady current means that charge is continuously flowing And not piling up anywhere, which means: δρ =0 δt In magnetostatics the continuity equation becomes ∇⋅ J = 0 The magnetic field of a steady line current is given by the Law of Biot-Savart: µ0 I × ur −r ' µ 0 dl '×ur − r ' B(r ) = I∫ 2 2 dl ' = ∫ 4π (r − r ') 4π (r − r ') • Steady current (I=const) produces a time-independent magnetic field B, • Integration along flow of the current, • Superposition principle applies, • µ0 =4π×10-7 N/A2 is permeability of free space Biot-Savart Law for cont. current distributions The Biot-Savart Law for line currents: µ0 I (r ' ) × ur −r ' B(r ) = 2 dl ' ∫ 4π (r − r ') The Biot-Savart Law for surface currents: µ0 K (r ' ) × ur − r ' B(r ) = 2 da ' ∫∫ 4π S (r − r ') The Biot-Savart Law for volume currents: µ0 J (r ' ) × ur −r ' B(r ) = 2 dτ ' ∫∫∫ 4π V (r − r ') Note: Due to our definition of source point and field point, the integral is over primed Coordinates. Div and curl Are taken over unprimed coordinates! field point r = (x, y , z ) d = r − r' dτ ' source point Examples Example: (5.9 a and b) Find the magnetic field at point P for each of the steady current configurations.