Review - Induced Electric Field • Definition dΦB r r εind = − = ∫ Eind • dl dt circuit • Question: Magnetic field strength inside a solenoid is (0.5T/A)I where I is the solenoid current. The current is increasing at a rate of 2 A/s. What is the strength of the induced electric field 0.1 m from the axis of the solenoid? Eddy Currents - Another Induced emf Effect • A solid conductor placed in a changing magnetic field will have many “loops” of induced current, called “eddy currents”. • Can also be a conductor moving in or out of a magnetic field • Example: Pendulum with metal sheet • Lenz’s law says response will be such as to oppose change-- pendulum will be slowed down by magnet “Eddies” of current • Eddy current converts KE of pendulum into heat • Other examples • Can reduce eddy current effects by restricting paths for current flow:

Will have much less Eddy current, even for same amount of metal

• Use of laminated cores in transformer • Question-- Displacement Current r dB r • We have seen that ⇒ E dt r dE r • What about ⇒ B dt • Yes! dE/dt acts like a current -- displacement current • Can derive this by considering a charging capacitor Curve C I r v • Ampere’s Law for curve C: ∫ B • dl = µ0I C r r dEcapacitor ∫ B • dl ≠ 0 ⇒ Id ∝ S 2 dt Displacement Current - Definition • Displacement current through a surface is defined in terms of the change in electric flux through surface: dΦ I ≡ ε E Displacement Current d 0 dt • Thus we see two kinds of current as capacitor charges

I I (Conduction current)

Id Displacement current; no Φ motion of charge, just d E/dt • Check how Id agrees with I: E Plates of area A I

σ -σ r  σ  Q Q ΦE = ∫ E • nˆdA = EA =  A = ( )A =  ε0  ε0 A ε0 •Thus dΦE d  Q  dQ Id = ε0 = ε0   = = I dt dt  ε0  dt • Checks!! Ampere-Maxwell Law • Include displacement current in Ampere’s Lax to get Ampere-Maxwell Law: r r ∫ B • dl = µ0()I + Id through C where the current is taken through the surface bounded by curve C • Example (Q): Parallel-plate capacitor with plates of area 0.1 m2 is being charged by 0.2A current. What is the rate of change of electric flux between the plates? Φ ε ε 10 • We found d E/dt = Id/ 0 = 0.2A/ 0 = 2.3 x 10 Vm/s. We can also find dE/dt between the capacitor plates; Φ 11 dE/dt = (1/A) d E/dt = 2.3 x 10 V/m-s • We now have the full set of Maxwell Equations that describe the “divergence” and circulation of E and B: r Q ∫ E • nˆdA = in closed..S ε0 r ∫ B • nˆdA = 0 closed..S r r r d r dB E • dl = − B • nˆdA = − • nˆdA ∫ dt ∫ ∫ dt C S r S r r  r dE  ∫ B • dl = µ0 ∫ j + ε0  • nˆdA C s  dt 