Faraday’s law Faraday: A transient current is induced in a circuit if A steady current flowing in an adjacent circuit is switched off An adjacent circuit with a steady current is moved A permanent magnet is moved into or out of the circuit No current flows unless the current in the adjacent circuit changes or there is relative motion of circuits Faraday related the transient current flow to a changing magnetic flux

Faraday’s law Total or convective derivative: ) t x T(x,t) ∂T(x,t)/∂t dx/dt. ∂T(x,t)/∂x ∂T(x,t)/∂t + dx/dt. ∂T(x,t)/∂x

Faraday’s law Consider two situations: (1) Source of B field contributing to  is moving (2) Surface/enclosing contour on which  is measured is moving Which situation applies depends on observer’s rest frame Situation (1) Rest frame of measured circuit (unprimed frame) B is changing on S because source circuit is moving at v S v

Faraday’s law Situation (2) Rest frame of source circuit (primed frame) B’ is changing because measured circuit is moving at v S’ v

Faraday’s law Situation (2) Rest frame of source circuit (primed frame) B’ is changing because measured circuit is moving at v S’ v

Lenz’s Law Minus sign in Faraday’s law is incorporation of Lenz’s Law which states The direction of any magnetic induction effect is such as to oppose the cause of the effect It ensures that there is no runaway induction (via positive feedback) or non-conservation of energy Consider a magnetic North Pole moving towards/away from a conducting loop SNSN v dSdS B B ind SNSN v dSdS B B.dS < 0 Flux magnitude increases d  /dt < 0 B.dS < 0 Flux magnitude decreases d  /dt > 0

Motional EMF Charges in conductor, moving at constant velocity v perpendicular to B field, experience Lorentz force, F = q v x B. Charges move until field established which balances F/q. No steady current established. B v F = q(vxB) - + Completing a circuit does not produce a steady current either B v F = q(vxB)

Motional EMF emf in rod length L moving through B field, sliding on fixed U shaped wire Charge continues to flow while rod continues to move I F = q(vxB) B v emf induced in circuit equals minus rate of change of magnetic flux through circuit

Faraday’s Law in differential form

Electric vector potential

Inductance Self-Inductance in solenoid Faraday’s Law applied to solenoid with changing magnetic flux implies an emf Area of cross section =  R 2 N loops (turns) per unit length B I L

Inductance Work done by emf in LR series circuit VoVo L R First term is energy stored in inductor B field Second term is heat dissipated by resistor solenoid inductance L =  o N 2  r 2 L solenoid field B =  o N  W = ½ LI 2 = ½  o N 2  r 2 L I 2 = ½ (  o N I ) 2  r 2 L/  o = ½ B 2 volume/  o

elastic exchange of field energy Inductance LCR series circuit driven by sinusoidal emf VoVo L C R elastic exchange of kinetic and potential energy

Displacement current Ampere’s Law Problem! Steady current implies constant charge density so Ampere’s law consistent with the continuity equation for steady currents (only). Ampere’s law inconsistent with the continuity equation (conservation of charge) when charge density is time dependent. Continuity equation

Displacement current Add term to LHS such that taking Div makes LHS also identically equal to zero: The extra term is in the bracket extended Ampere’s Law (Maxwell 1862) Displacement current in vacuum (see later)

Displacement current Relative magnitude of displacement and conduction currents

Maxwell Equations in Vacuum Maxwell equations in vacuum