Magnetic field and force

Magnetic field of a moving charge

\begin{equation} \vec{B} = \frac{\mu_0}{4 \pi} \frac{q \, \vec{v} \times \hat{r}}{r^2} \qquad \textnormal{(B field of moving charge)} \end{equation}

Magnetic field of current-carrying conductor

\begin{equation} d\vec{B} = \frac{\mu_0}{4 \pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2} \qquad \textnormal{(B field of current carrying conductor)} \end{equation}

Biot-Savart Law

\begin{equation} \vec{B} = \frac{\mu_0 \, I}{4 \pi} \int \frac{ d\vec{l} \times \hat{r}}{r^2} \qquad \textnormal{(Biot-Savart Law)} \end{equation}

Field of long and straight conductor

Outside conductor, r > R \begin{equation} B = \frac{\mu_0 \, I}{2 \pi r} \qquad \textnormal{(Field of long, straight conductor)} \end{equation} Inside conductor, r < R \begin{equation} B = \frac{\mu_0 \, I}{2 \pi} \frac{r}{R^2} \qquad \textnormal{(Field of long, straight conductor)} \end{equation}

Solenoid

Inside solenoid, near center, n = turns / length \begin{equation} B = \mu_0 \, n \, I \qquad \textnormal{(B field in a solenoid)} \end{equation} Outside solenoid \begin{equation} B \approx 0 \qquad \textnormal{(B field outside solenoid)} \end{equation}

Toroid

Within the windings, distance from the axis of symmetry, N = turns \begin{equation} B = \frac{\mu_0 \, I \, N}{2 \, \pi \, r} \qquad \textnormal{(B field in a toroid)} \end{equation} Outside the enclosure of the windings \begin{equation} B \approx 0 \qquad \textnormal{(Outside windings of a toroid)} \end{equation}

Magnetic force

\begin{equation} \vec{F} = q \, \vec{v} \times \vec{B} \qquad \textnormal{(Magnetic force)} \end{equation}

Magnetic force between current carrying conductors

\begin{equation} \frac{F}{L} = \frac{\mu_0 \, I \, I'}{2 \pi r} \qquad \textnormal{(Magnetic force per length between conductors)} \end{equation}

Magnetic field of a current loop

\begin{equation} B_x = \frac{ \mu_0 \, I \, p^2}{2 (x^2 + p^2)^{3/2}} \qquad \textnormal{(Circular loop)} \end{equation} \begin{equation} B_x = \frac{ \mu_0 \, N \, I}{2 \, p} \qquad \textnormal{(Center of n circular loops)} \end{equation}

Magnetic field and flux

\begin{equation} \phi_B = \int \vec{B} \cdot d\vec{A} \end{equation} \begin{equation} = \int B_{\perp} \, dA = \int B \textnormal{ cos } \chi \, dA \qquad \textnormal{(Magnetic flux)} \end{equation} \begin{equation} \phi_B = \oint_S \vec{B} \cdot d\vec{A} = 0 \end{equation}

Movement in a magnetic field

\begin{equation} R = \frac{mv}{|q| B} \qquad \textnormal{(Radius of motion)} \end{equation}

Magnetic force on a conductor

\begin{equation} \vec{F} = I \vec{l} \times \vec{B} \qquad \textnormal{(Radius of motion)} \end{equation} \begin{equation} d\vec{F} = I d\vec{l} \times \vec{B} \end{equation}

Torque in a magnetic field

\begin{equation} \vec{\mu} = I\vec{A} \qquad \textnormal{(Magnetic moment)} \end{equation} \begin{equation} \vec{\tau} = \vec{\mu} \times \vec{B} \qquad \textnormal{(Torque due to magnetic moment and B)} \end{equation} \begin{equation} \tau = I B A \textnormal{ sin } \lambda \end{equation} \begin{equation} U = - \vec{\mu} \cdot \vec{B} = - \mu \, B \textnormal{ cos } \beta \qquad \textnormal{(Potential energy of magnetic moment in a B field)} \end{equation}

Ampere's law (no correction)

\begin{equation} \oint_{loop} B \cdot ds = \mu_0 \, I_{enc} \end{equation}