Time-dependent fields

Static fields decoupled .D = r  x E = 0 .B = 0  x H = J D = eE B = mH Electric fields have zero curl Caused by Static Charges Magnetic fields have zero divergence Caused by Static Currents Asymmetry in E and B field properties

Maxwell’s equations for Electromagnetism Dynamic fields coupled .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t D = eE B = mH Maxwell’s equations for Electromagnetism Field equations more symmetric (fields resemble each other)

Deriving Circuit Theory ! .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t Kirchhoff’s Law V=LdI/dt

Gauss’ Law for electrostatics (Flux prop. to enclosed charge) The 4 Maxwell equations .D = r Gauss’ Law for electrostatics (Flux prop. to enclosed charge)

Gauss’ Law for magnetostatics (There is no magnetic charge) The 4 Maxwell equations .B = 0 Gauss’ Law for magnetostatics (There is no magnetic charge)

Faraday’s law of induction (Changing magnetic flux The 4 Maxwell equations  x E = - ∂B/∂t Faraday’s law of induction (Changing magnetic flux creates voltage)

(Changing electric flux creates magnetic field) The 4 Maxwell equations  x H = J + ∂D/∂t Ampere’s law (Changing electric flux creates magnetic field)

The 4 Maxwell’s equations .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t Not just E,B but D,H and also inputs r, J = sE B = mH D = eE Constitutive equations (Maxwell’s eqns don’t give e, m, s, r Need quantum mechanics/solid state/statistical physics for this!) We treat them as external inputs

The 4 Maxwell’s equations .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t Most important consequence: Electromagnetic Waves (Chapter 7) Here we’ll learn the two new equations (Faraday’s Law and Ampere’s Law)

Changing magnetic flux Faraday’s Law  x E = - ∂B/∂t Since  x E ≠ 0 , can’t have E = -U Changing magnetic flux creates voltage

Faraday’s Law  x E .dA = - ∂B.dA/∂t Integrate both sides

Faraday’s Law E .dl = - ∂FB/∂t Stokes Theorem

Changing magnetic flux Faraday’s Law E .dl = - ∂FB/∂t Changing magnetic flux creates voltage

Faraday’s Law Vemf = - ∂FB/∂t Definition: FB/I = L Vemf = - LdI/dt (Solenoid) Vemf = - LdI/dt

Changing magnetic flux Faraday’s Law  x E = - ∂B/∂t Changing magnetic flux creates voltage (and thus current)

How to change magnetic flux? http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

Lenz’s Law Induced current opposes any change in flux (Nature Prefers Inertia) opposing induced flux Increasing flux

Lenz’s Law Induced current opposes any change in flux (Nature Prefers Inertia) opposing induced flux Decreasing flux

Force argument for Lenz’s Law Lorentz Force -e(v x B) downwards on el Wire motion Current flows upward in slider Current flows upward in slider Current flows upward in slider Decreasing B Flux towards you  induced B also towards you Increasing B Flux towards you  induced B away from you

No matter which way you see it, the current in the slider flows the same way !!

http://micro.magnet.fsu.edu/electromag/java/faraday/ /faraday2 /lenzlaw/ /transformer /detector