Thick Solenoid Axial Field

An air-cored multilayer coil of finite length and finite winding depth. The plate evaluates the exact closed-form axial field (not the infinite-solenoid approximation) and shows how the field sags towards the ends.

Parameters

Total turns

N

Current

I

Coil length

L

Inner radius

R_1

Outer radius

R_2

Probe position on axis

z

Fig. 7Axial section. The axis is tinted by local field strength; the red probe marks the point reported as B(z).

Axial field

Centre field

B_0

17.2 mT

At probe

B(z)

17.2 mT

At coil end

B(L/2)

9.86 mT

End / centre57.5 %

Infinite-solenoid

\mu_0 NI/L

20.9 mT

Field strength

H_0

13.6 kA/m

In gauss171.5 G

Ampere-turns

NI

1 kA·t

☞ Field within 1 % of centre value over 10.5 mm (17.5 % of coil length).

Flux density along the axis

How it works

Theory · assumptions · formulas

From a loop to a thick coil

A single circular turn of radius $r$ produces on its axis $B = \mu_0 I r^2 / \bigl(2 (r^2 + z^2)^{3/2}\bigr)$ . Spreading $N$ turns uniformly over a rectangle of length $L$ and radii $R_1 \ldots R_2$ and integrating gives the classical closed form:

B(z) = \frac{\mu_0 N I}{2L\,(R_2 - R_1)} \Bigl[ F\bigl(z + \tfrac{L}{2}\bigr) - F\bigl(z - \tfrac{L}{2}\bigr) \Bigr]

F(u) = u\,\ln\frac{R_2 + \sqrt{R_2^2 + u^2}}{R_1 + \sqrt{R_1^2 + u^2}}

Compared with the long solenoid

For a coil much longer than its radius the centre field approaches $\mu_0 N I / L$ , and at the very ends it falls to half of that, since each end "sees" only one semi-infinite coil. Short, fat field coils fall well below the long-solenoid figure, which is why iron is used to concentrate flux in practical magnets.

AssumptionsUniform current density through the winding cross-section, no magnetic material nearby, and points on the axis only. Off-axis fields require elliptic integrals.