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* a
** 26
*** B from x_{dist} On axis of a current loop of radius a
$B = \frac{\mu_0 I a^2}{2(x_{dist}_{}^2 + a^2)^{3/2}}$


*** B on axis from magnetic dipole
$B = \frac{\mu_0}{2 \pi} \frac{\mu}{x^3}$


*** Net Torque on closed loop with area A at orientation \theta
$\tau = I A B \text{sin}(\theta)$


*** Field outside, inside any current distribution with line symmetry
$B = \frac{\mu_0 I}{2 \pi r}$

$B = \frac{\mu_0 I r_{inside}}{2 \pi R_{outside}^2}$

*** Sheet with uniform current density J
*** Solenoid with turns n per unit length

** 27
*** Flux through solenoid with n turns per unit length
$\phi_B = BA = \mu_0 n I \pi R^2$


*** Flux through rectangular loop with $l$ parallel to wire at distance $a$

$\phi_B = \int B dA = \int_{a}^{a+w} \frac{\mu_0 I}{2 \pi r} l dr = \frac{\mu_0 I l}{2 \pi} \text{ln}(\frac{a+w}{a})$


*** Induced current through circuit with bars at distance $l$ and moving bar velocity $v$
$I = \frac{Blv} {r}$

*** Flux through coil with $N$ turns turning at frequency $f$ in field $B$
$\phi_B = N B \pi r^2 \text{cos}(2 \pi f t)$

$E = - \frac{d \phi_B}{dt}$

*** Inductance of a solenoid
$L = \frac{\phi_B}{I} = \mu_0 n^2 A l$

*** Electric field of a solenoid of radius $R$ at loop radius $r$ with $B = bt$
$E = \frac{R^2 b}{2r}$