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 $(\mathcal{E})$:
  $\mathcal{E}=-{\displaystyle \frac{d\mathrm{\Phi }}{dt}}$

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

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: $\oint Edl=-{\displaystyle \frac{d\mathrm{\Phi }}{dt}}$

Eddy Current

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

Self-Induction

• $\mathrm{\Phi }=Li$

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

Inductors

Self-Inductance of a Long Solenoid

• $L={\mu }_0{n}^{2}Al$

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

Growth and Decay of Current in an LR Circuit

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

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

Energy Stored in an Inductor

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

Energy Density in a Magnetic Field

• $B={\mu }_{0}ni$

• $u={\displaystyle \frac{B^2}{2{\mu }_0}}$

Mutual Induction

• Mutual Inductance $(M)$:
  $M={\displaystyle \frac{{\mu }_0{N}_1{N}_{2}A}{l}}$

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