Magnetic field strength H of a rectangular “thin” solenoid along its axis | |
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$$ H(t,x) = \frac{I(t) · N}{L·π}· \left( \text{atan} \frac{(2·x+L)·\sqrt{a^2+b^2+(2·x+L)^2}}{a·b} - \text{atan} \frac{(2·x-L)·\sqrt{a^2+b^2+(2·x-L)^2}}{a·b} \right) $$ | (A/m) |
where: $I(t)$ - current (A) at time $t$ (s), $N$ - total number of turns (unitless), $L$ - length of the solenoid (m), $a \approx A$ and $b \approx B$ - length (m) of sides of the rectangular cross-section (inner and outer, as per the diagram), $x$ - location (m) from the centre of the solenoid (the centre is located at the point x = 0) |
^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}
Notes: This equation is valid only for uniformly wound solenoid, with uniform current distribution in each turn. The instantaneous values of H are directly proportional to the instantaneous values of I. The value of B(μ_{0}) is for vacuum. The same equation can be also used for rectangular permanent magnets if magnetisation is calculated to the units of amperes.