Induction

Electromagnetic Induction

Faraday’s Law

• Whenever the flux of magnetic field through the area bounded by a closed conducting loop changes, and emf in produced in the loop.

• EMF induced $(\mathcal{E})$:
  $\mathcal{E} = -\dfrac{d\Phi}{dt}$

• Flux ($\Phi$):
  $\Phi = \int{\vec{B}\cdot \vec{dS}} = BA \cos \theta$

Lenz’s Law

• The direction of induced current is such theat it opposes the change that has induced it.

Motional EMF

• EMF in a conductor moving with velocity $v$ in magnetic field $B$:
  $\mathcal{E} = vBl$

Induced Electric Field

• Induced electric field $E$ around a loop: $\oint E \, dl = -\dfrac{d\Phi}{dt}$

Eddy Current

• Electromagnetic damping.
• Circular currents induced in conductors due to changing magnetic flux.
• $i \propto \left|\dfrac{d\Phi}{dt}\right|$.

Self-Induction

• $\Phi = Li$

• Induced EMF $(\mathcal{E})$ in coil due to its own current $I$:
  $\mathcal{E} = -L \dfrac{di}{dt}$

Inductors

Self-Inductance of a Long Solenoid

• $L = \mu_0\:n^2Al$

• $n$: Turns per unit length,
  $A$: Cross-sectional area,
  $l$: Length of solenoid.

Growth and Decay of Current in an LR Circuit

1. Growth:
   $i = i_0 \biggr(1 - e^{-t/\tau} \biggr)$
2. Decay:
   $i = i_0\: e^{-t/\tau}$
3. Time constant ($\tau$):
   $\tau = \dfrac{L}{R}$

At $t = \tau$,
Growth: $i = i_0 (1-\dfrac{1}{e}) = 0.63 i_0$
Decay: $i = i_0 \dfrac{1}{e} = 0.37 i_0$

Energy Stored in an Inductor

• Energy $(U)$: $U = \dfrac{1}{2} L i^2$

Energy Density in a Magnetic Field

• $B = \mu_0\:ni$

• $u = \dfrac{B^2}{2 \mu_0}$

Mutual Induction

• Mutual Inductance $(M)$:
  $M = \dfrac{\mu_0 N_1 N_2 A}{l}$

• Induced EMF $\mathcal{E}_2$ in $\text{coil}_2$ due to change in current $i_1$ in $\text{coil}_1$:
  $\mathcal{E}_2 = -M \dfrac{di_1}{dt}$