Magnetism

Magnetic field

• A magnetic field is a vector field $\vec{\mathbf{B}}$
  • The SI unit of magnetic field is the tesla ($\mathrm{T}$) which is defined as $1\text{T}=1\text{N}/(\text{A}\cdot \text{m})$.
• $\vec{\mathbf{H}}=\frac{\vec{\mathbf{B}}}{{\mu }_0}-\vec{\mathbf{M}}$ is the magnetic field strength (in $A/m$) where ${\mu }_0$ is the permeability of free space and $\vec{\mathbf{M}}$ is the magnetization of the material.

Lorentz force

• $\vec{\mathbf{F}}=q(\vec{\mathbf{E}}+\vec{\mathbf{v}}\times \vec{\mathbf{B}})$ is the force (in $\mathrm{N}$) on a particle of charge $q$ moving with velocity $\vec{\mathbf{v}}$ in an electric field $\vec{\mathbf{E}}$ and a magnetic field $\vec{\mathbf{B}}$.
  • $\vec{\mathbf{F}}={\int }_{\mathcal{C}}I(d\vec{\mathbf{\ell }}\times \vec{\mathbf{B}})$ is the magnetic force (in $\mathrm{N}$) on a current-carrying conductor in a magnetic field $\vec{\mathbf{B}}$.
  • $\vec{\mathbf{F}}=I(\vec{\mathbf{\ell }}\times \vec{\mathbf{B}})$ is the magnetic force exerted on a straight wire of length $\ell$ carrying a current $I$ in a uniform magnetic field $B$ where $\theta$ is the angle between $\vec{\mathbf{\ell }}$ and $\vec{\mathbf{B}}$.

Biot-Savartlaw

• $\vec{\mathbf{B}}=\frac{{\mu }_0}{4\pi }\frac{q\vec{v}\times \hat{r}}{r^2}$ is the magnetic field (in $\mathrm{T}$) at distance $r$ from a point charge $q$ moving with velocity $\vec{v}$ in the direction $\hat{r}$. $\vec{\mathbf{B}}$ is perpendicular to both $\vec{v}$ and $\hat{r}$.
  • (superposition) The total magnetic field caused by multiple point charges is the vector sum of the magnetic fields caused by each individual charge.
• $\vec{\mathbf{B}}=\frac{{\mu }_0}{4\pi }{\oint }_{\mathcal{C}}\frac{Id\vec{\mathbf{\ell }}\times \hat{\mathbf{r}}}{r^2}$
• $B={\mu }_{0}nI$ is the magnetic field inside a long solenoid with $n$ turns per unit length ($n=\frac{N}{L}$) carrying current $I$.
• $B=\frac{{\mu }_{0}I}{2\pi r}$ is the magnitude of magnetic field at distance $r$ from a long straight wire carrying current $I$.
• $B=\frac{{\mu }_{0}I}{2R}$ is the magnitude of magnetic field at the center of a circular loop of radius $R$ carrying current $I$.
• $B={\mu }_{0}nI$ is the magnetic field inside of a long solenoid with $n$ turns per unit length ($n=\frac{N}{L}$) carrying current $I$.

Magnetic flux

• ${\Phi }_B={\iint }_S\vec{\mathbf{B}}\cdot d\vec{\mathbf{S}}$ is the magnetic flux (in $\mathrm{Wb}$) through a surface $S$ in a magnetic field $\vec{\mathbf{B}}$.
  • ${\Phi }_B=\vec{\mathbf{B}}\cdot \vec{\mathbf{A}}$ is the magnetic flux through a flat surface with a vector area $\vec{\mathbf{A}}$ in a uniform magnetic field $\vec{\mathbf{B}}$.
  • (See Gauss’s Law for magnetism)

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

Magnetic moment

• $\vec{\mathbf{\mu }}=I\vec{\mathbf{A}}$ is the magnetic (dipole) moment (in $\mathrm{A}\cdot {\mathrm{m}}^2$) of a current loop with area $\vec{\mathbf{A}}$ carrying current $I$.
• $\vec{\mathbf{\tau }}=\vec{\mathbf{\mu }}\times \vec{\mathbf{B}}$ is the torque on a magnetic dipole $\vec{\mathbf{\mu }}$ in a magnetic field $\vec{\mathbf{B}}$.
• $U=-\vec{\mathbf{\mu }}\cdot \vec{\mathbf{B}}$ is the potential energy of a magnetic dipole $\vec{\mathbf{\mu }}$ in a magnetic field $\vec{\mathbf{B}}$.

Ampère’s circuital Law

	Differential Form	Integral Form
Ampère’s (circuital) Law	$\nabla \times \vec{\mathbf{B}}={\mu }_0\vec{\mathbf{J}}$	${\oint }_{\mathcal{C}}\vec{\mathbf{B}}\cdot d\vec{\mathbf{\ell }}={\mu }_0{I}_{\text{enc}}$ where $I_{\text{enc}}$ is the current through the area enclosed by the closed path $\mathcal{C}$

Ampère’s circuital Law is a special case Ampère–Maxwell law when there is no changing electric field (steady current).

Maxwell’s equations

	Differential Form	Integral Form
Gauss’s Law for Electricity	$\nabla \cdot \vec{\mathbf{E}}=\frac{\rho }{{\epsilon }_0}$ $\rho$ is the charge density	${\iint }_S\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}=\frac{Q_{\text{enc}}}{{\epsilon }_0}$ where $S$ is any closed surface
Gauss’s Law for Magnetism	$\nabla \cdot \vec{\mathbf{B}}=0$	${\iint }_S\vec{\mathbf{B}}\cdot d\vec{\mathbf{A}}=0$ where $S$ is any closed surface
Maxwell–Faraday equation	$\nabla \times \vec{\mathbf{E}}=-\frac{\partial \vec{\mathbf{B}}}{\partial t}$	${\oint }_{\partial S}\vec{\mathbf{E}}\cdot d\vec{\mathbf{\ell }}=-\frac{d}{dt}{\iint }_S\vec{\mathbf{B}}\cdot d\vec{\mathbf{A}}$ where $\partial S$ is the boundary curve of a surface $S$
Ampère–Maxwell law	$\nabla \times \vec{\mathbf{B}}={\mu }_0\left(\vec{\mathbf{J}}+{\epsilon }_0\frac{\partial \vec{\mathbf{E}}}{\partial t}\right)$	${\oint }_{\partial S}\vec{\mathbf{B}}\cdot d\vec{\mathbf{\ell }}={\mu }_0\left(I_{\text{enc}}+{\epsilon }_0\frac{d}{dt}{\iint }_S\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\right)$ where $\partial S$ is the boundary curve of a surface $S$