Magnetism

Magnetic field

• A magnetic field is a vector field $\vec{B}$
  • The SI unit of magnetic field is the tesla (T) which is defined as $1\text{T}=1\text{N}/(\text{A}\cdot \text{m})$.
  • $\vec{B}=\frac{{\mu }_0}{4\pi }\frac{q\vec{v}\times \hat{r}}{r^2}$ is the magnetic field (in $\mathsf{T}$) at distance $r$ from a point charge $q$ moving with velocity $\vec{v}$ in the direction $\hat{r}$. $\vec{B}$ 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.

Lorentz force

• (Lorentz Force) $\vec{F}=\int I(d\vec{\ell }\times \vec{B})$ is the magnetic force (in $\mathsf{N}$) on a current-carrying conductor in a magnetic field $\vec{B}$.
  • $F=ILB\sin \theta$ (or $\vec{F}=I(\vec{L}\times \vec{B})$) is the magnetic force exerted on a straight wire of length $L$ carrying a current $I$ in a uniform magnetic field $B$ where $\theta$ is the angle between $\vec{L}$ and $\vec{B}$.
  • $F=qvB\sin \theta$ (or $\vec{F}=q(\vec{v}\times \vec{B})$) is the magnetic force exerted on a particle of charge $q$ moving with velocity $\vec{v}$ in a uniform magnetic field $\vec{B}$ where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

Biot-Savartlaw

• (differential form) $d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{I(d\vec{\ell }\times \hat{r})}{r^2}$ is the infinitesimal magnetic field produced by a current element $Id\vec{\ell }$ at a distance $r$ in the direction $\hat{r}$.

• (integrated form) $\vec{B}=\frac{{\mu }_0}{4\pi }\int \frac{Id\vec{l}\times \hat{r}}{r^2}$

• $B={\mu }_{0}nI$ 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

• ${\Phi }_B=\vec{B}\cdot \vec{A}$ is the magnetic flux (in $\mathsf{Wb}$) through a surface area $\vec{A}$ in a magnetic field $\vec{B}$.
  • ${\Phi }_B=\int \vec{B}\cdot d\vec{A}$ is the magnetic flux through a surface area $A$ in a magnetic field $\vec{B}$.

Electromagnetic induction

• (Faraday’s Law of Induction) $\mathcal{E}=-N\frac{d{\Phi }_B}{dt}$ is the induced emf (in $\mathsf{V}$) in a closed loop of wire with $N$ turns due to a changing magnetic flux ${\Phi }_B$ (in $\mathsf{Wb}$) through the loop.

  • (The negative sign is due to Lenz’s Law)
  • (Motional EMF) $\mathcal{E}=B\ell v\sin \theta$ is the induced emf (in $\mathsf{V}$) in a straight conductor of length $\ell$ moving with velocity $v$ in a magnetic field $B$ at an angle $\theta$ to the field.

• The inductance (in henry, $\mathsf{H}$) of a

  • An inductor is a circuit elementthathassignificantinductance.
  • $L=\frac{{\mu }_0{N}^{2}A}{\ell }$
  • $U=\frac{1}{2}L{I}^2$ is the energy stored in an inductor with inductance $L$ carrying current $I$.
  • $\mathcal{E}=-L\frac{dI}{dt}$

Magnetic moment

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

Ampère’s Law

• (Ampère’s circuital Law)
  • ${\oint }_{\mathcal{C}}\vec{B}\cdot d\vec{\ell }={\mu }_0{I}_{\text{enc}}$ where $I_{\text{enc}}$ is the current through the area enclosed by the closed path $\mathcal{C}$ in a magnetic field $\vec{B}$.

Maxwell’s equations

Gauss’s Law for Electricity

• $\nabla \cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}$
  • where $\rho$ is the charge density

Gauss’s Law for Magnetism

• $\nabla \cdot \mathbf{B}=0$
• $\oint \vec{B}\cdot d\vec{A}=0$
  • (the magnetic flux through a closed surface is zero, indicating that there are no magnetic monopoles)

Maxwell–Faraday equation

• $\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$
• $\oint \vec{E}\cdot d\vec{\ell }=-\frac{d{\Phi }_B}{dt}$

Ampère–Maxwell law

• $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\partial \mathbf{E}}{\partial t}$
• $\oint \vec{B}\cdot d\vec{\ell }={\mu }_{0}I+{\mu }_0{ϵ}_0\frac{d{\Phi }_E}{dt}$