Magnetic Flow Density

From point dipole

m=mez\bm m = m\bm e_z

:

B(r)=μ0m4πr3(2cos⁡θer+sin⁡θeθ)\bm B(\bm r) = \frac{\mu_0m}{4\pi r^3}(2\cos\theta\bm e_r + \sin\theta\bm e_\theta)

From current density

Jtot(r′)\bm J_{tot}(\bm r')

:

B(r)=μ04π∫Jtot(r′)×eRR2 dv′\bm B(\bm r) = \frac{\mu_0}{4\pi}\int\frac{\bm J_{tot}(\bm r')\times \bm e_R}{R^2}\: dv'

where

Jtot=J+Jm\bm J_{tot} = \bm J + \bm J_m

. From current line:

B(r)=μ04π∫I dl′×eRR2\bm B(\bm r) = \frac{\mu_0}{4\pi}\int\frac{I\:d\bm l'\times\bm e_R}{R^2}

From circular thread loop:

B(x=0,y=0,z)=μ0I2b2(b2+z2)3/2ez\bm B(x = 0, y = 0, z) = \frac{\mu_0 I}2\frac{b^2}{(b^2 + z^2)^{3/2}}\bm e_z

From coil:

B=μ0NIℓcos⁡(α2)−cos⁡(α1)2ez\bm B = \frac{\mu_0NI}\ell \frac{\cos(\alpha_2) - \cos(\alpha_1)}2\bm e_z

From long straight current path:

B(r)=μ0I2πrceφ\bm B(\bm r) = \frac{\mu_0 I}{2\pi r_c}\bm e_\varphi