Ampere's Circuital Law And The Solenoid

NCERT Class 12, Chapter 4 (Moving Charges and Magnetism) derives, by applying Ampere's circuital law to a rectangular loop straddling the solenoid wall, that the field inside a long solenoid is uniform and given by $B={\mu }_{0}nI$, where $n$ is the number of turns per unit length. Only the segment of the Amperian loop lying inside the solenoid contributes, since the field outside an ideal long solenoid is essentially zero, and inside it is directed uniformly along the axis. Substituting ${\mu }_0=4\pi \times 10^{-7}$ T·m/A, $n=500$ turns/m, and $I=3$ A: $B=(4\pi \times 10^{-7})(500)(3)=6\pi \times 10^{-4}\approx 1.88\times 10^{-3}$ T. The value $1.88\times 10^{-4}$ T is ten times too small from a unit slip in $n$. The value $6.0\times 10^{-3}$ T drops the factor $\pi$ from ${\mu }_0$. The value $9.4\times 10^{-4}$ T is half the answer from mistakenly using $n=250$. Plausibility check: the field is independent of the solenoid radius, and multiplying the scales $10^{-7}\times 10^2\times 10^0$ gives order $10^{-3}$ T, a typical solenoid interior field, so the magnitude is sensible.