Field of a finite solenoid

Problem

Measurements of the field $B(x)$ on the axis of a solenoid with $L = 20$ cm, $n = 1000$ turns/m, $I = 5$ A. (a) Verify the value predicted at the centre using the ideal solenoid formula. (b) Explain why at $x = 10$ cm (the end) the field is about half that at the centre.

$x$ (cm) from centre	0	5	10	15	20
$B$ (mT)	6,28	6,20	3,14	0,40	0,05

Solution

(a) At the centre of an ideal solenoid: $B=\mu_0 n I=4\pi\cdot10^{-7}\cdot1000\cdot5=6{,}28\cdot10^{-3}\;\text{T}=6{,}28\;\text{mT}$

in perfect agreement with the value measured at $x=0$.

$\ev{B_\text{centre} \approx 6{,}3\;\text{mT}}$

(b) At the end of the solenoid, half of the winding is missing: the turns are found only on one side of the point considered, whereas at the centre they contribute from both sides. The contribution is therefore roughly halved: $B_\text{end}\approx\frac{B_\text{centre}}{2}\approx3{,}14\;\text{mT}$

consistent with the value measured at $x=10$ cm.

$\ev{B_\text{end} \approx B_\text{centre}/2}$

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