Faraday’s and Maxwell-Ampere Laws • A changing magnetic flux produces a curly electric field: d Edl Bnˆ dA dt • A changing electric flux produces a (curly) magnetic field: d Bdl 0 I encl 0 Enˆ dA dt A moving “slab” of E and B fields • Thin region in which there are uniform electric and magnetic fields. – – – – No charges or currents in the vicinity of the slab. Outside the slab, both fields are zero. Inside the slab, E and B are perpendicular to each other. Slab moves in a direction perpendicular to both fields. E E B v v E B front view v B top view Does Slab obey Faraday’s Law? E E B v v B front view d Edl Bnˆ dA dt Does Slab obey Maxwell-Ampere Law? top view E B v E v B d Bdl 0 I encl 0 Enˆ dA dt Constraints imposed by Faraday’s and Maxwell-Ampere Laws E Bv B 0 0 Ev c 0 8.85 10 12 C2 Nm2 0 4 10 c 3.00 108 m s 7 Ns 2 C2 1 0 0 Properties of EM Waves • • • • 1 Move at the speed of light (in vacuum): c 0 0 Magnitudes of fields related by E Bc. E and B are perpendicular to each other. Wave moves in direction of E B , i.e. perpendicular to both fields. E B v Dipole Antenna • To create electromagnetic waves, charges must be accelerating. • Recall that in the steady state, the fields are constant. Figure 31.6b t=0 Later time Figure 31.7 Plane waves • Far from the antenna, the EM waves are well described as plane waves – the fields are uniform within planes perpendicular to the direction of propagation. • Sinusoidal waves – fields are sinusoidal functions of position and time. – E.g. wave moving along x axis: 2 x 2 t ˆ E E0 sin( ) j E0 sin(kx t )ˆj T 2 x 2 t ˆ B B0 sin( )k B0 sin(kx t )kˆ T Figure 31.9 2 x 2 t ˆ E E0 sin( ) j E0 sin(kx t )ˆj T 2 x 2 t ˆ B B0 sin( )k B0 sin(kx t )kˆ T c T f Energy density in EM Waves Electric Field Energy density uE 12 0 E 2 Magnetic Field Energy density uB 1 2 B2 0 E Bc c 1 0 0 Half of the energy in an EM wave is in the electric field and half is in the magnetic field.