Physics II - B16
TABLE OF CONTENTS
Motional emf
\[\displaystyle \sum F = 0 \to qE - qvb = 0 \to E = vB\]
\[\Delta V = El = vBl\] \[I = \frac{\Delta V}{R} = \frac{El}{R} = \frac{vBl}{R}\] \[F = IlB = \frac{vBl}{R}lB\] \[P = Fv = \frac{(vBl)^2}{R} = \frac{(\Delta V)^2}{R} = I^2R\]
Faraday’s Laws of Induction
Faraday
\[\displaystyle \Phi_m = Blx\]
\[\displaystyle \xi = -\frac{d\Phi_B}{dt} = \frac{-d}{dt}(Blx) = -Bl\frac{dx}{dt} = -Blv\]
Lenz’ Law
\[\displaystyle \xi = \oint \vec E .d\vec s = -\frac{d\Phi_B}{dt} = -\frac{d}{dt} \int\vec B.d\vec A\] \[\Phi_B \uparrow \, : \vec B_{ind} \uparrow \downarrow \vec B\] \[\Phi_B \downarrow \, : \vec B_{ind} \uparrow \uparrow \vec B\]
Remind: $\displaystyle \oint \vec E. d\vec s$ over a closed loop is zero for conservative electric field.
Four E&M equations
AC Generator
\[\displaystyle \Phi_B = BAcos\theta = BAcos(\omega t)\]
\[\xi = -\frac{d}{dt}(BAcos(\omega t)) = \omega BAsin(\omega t)\]
Eddy Currents (Foucault’s currents)
\[\displaystyle B_1 = \mu_0\frac{N}{l}I_1\] \[\Phi_2 = B_1 \pi r^2 = \mu_0\frac{N}{l}I_1 \pi r^2\] \[\displaystyle \xi_2 = -\frac{d\Phi_2}{dt} = -M_{12}\frac{dI_1}{dt}\] \[M_{12} = \mu_0\frac{N}{l} \pi r^2\, (mutual \, inductance)\]