# Encyclopedia Magnetica™

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calculator:solenoid_rectangular

#### Calculator of H along axis of rectangular "thin" solenoid

Magnetic field strength H of a rectangular “thin” solenoid along its axis
$$H(t,x) = \frac{I(t) · N}{L·π}· \left( \text{atan} \frac{(2·x+L)·\sqrt{a^2+b^2+(2·x+L)^2}}{a·b} - \text{atan} \frac{(2·x-L)·\sqrt{a^2+b^2+(2·x-L)^2}}{a·b} \right)$$ (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)
“Rectangular” and “thin” solenoid ($a \approx A, b \approx B$)

Current I =

Side a = A =

Side b = B =

Length L =

Number of turns N = (unitless)

Position on axis x =

H =      B0) =

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(μ0) is for vacuum. The same equation can be also used for rectangular permanent magnets if magnetisation is calculated to the units of amperes.