Electromagnetic induction

• $\mathcal{E}={\oint }_{\mathcal{C}}(\vec{\mathbf{E}}+\vec{\mathbf{v}}\times \vec{\mathbf{B}})\cdot d\vec{\mathbf{\ell }}$ is the induced emf (in $\mathrm{V}$) in a moving conductor in a magnetic field $\vec{\mathbf{B}}$ where $\vec{\mathbf{E}}$ is the electric field and $\vec{\mathbf{v}}$ is the velocity of the conductor.

  • (constant flux, $\frac{d{\Phi }_B}{dt}=0$) $\mathcal{E}={\oint }_{\mathcal{C}}(\vec{\mathbf{v}}\times \vec{\mathbf{B}})\cdot d\vec{\mathbf{\ell }}$ (Motional EMF)
    • (faraday disk) $\mathcal{E}=\frac{1}{2}\omega B{R}^2$
  • (stationary conductor $\vec{\mathbf{v}}=0$) $\mathcal{E}={\oint }_{\mathcal{C}}\vec{\mathbf{E}}\cdot d\vec{\mathbf{\ell }}$
    • (changing flux, $\frac{d{\Phi }_B}{dt}\ne 0$) $\mathcal{E}=-\frac{d{\Phi }_B}{dt}$ (Faraday’s law of induction)
      • (The negative sign is due to Lenz’s law)
      • The mutual inductance (in $\mathrm{H}$) between two circuits is $M=\frac{{\Phi }_{21}}{I_1}=\frac{{\Phi }_{12}}{I_2}$ where ${\Phi }_{21}$ is the magnetic flux through circuit 2 due to current $I_1$ in circuit 1, (and vice versa).
        • ${\mathcal{E}}_2=-M\frac{d{I}_1}{dt}$ is the emf induced in circuit 2 due to a changing current $I_1$ in circuit 1.
        • ${\mathcal{E}}_1=-M\frac{d{I}_2}{dt}$ is the emf induced in circuit 1 due to a changing current $I_2$ in circuit 2.
        • The henry ($\mathrm{H}$) can be defined as $1\mathrm{H}=1Wb/A$.
      • The self-inductance (or inductance) (in $\mathrm{H}$) of a circuit is $L=\frac{{\Phi }_B}{I}$ where ${\Phi }_B$ is the magnetic flux through the circuit due to current $I$ in this circuit.
        • $\mathcal{E}=-L\frac{dI}{dt}$ is the emf induced in a circuit due to a changing current $I$ in the same circuit.

• An inductor (or coil),

• $L=\frac{{\mu }_0{N}^{2}A}{\ell }$

  • $U=\frac{1}{2}L{I}^2$ is the energy stored in an inductor with inductance $L$ carrying current $I$.

Inductor symbols	Inductor symbol	Inductor symbol