  The running of the coupling is studied in SU(4) gauge theory using the
Schr\"odinger functional technique. Up to energies of the order of the square
root of the string tension $\sigma$, the running is found to agree with the
two-loop perturbative formula. Relating the perturbative to the
non-perturbative regime of the running and converting to the $\msbar$ scheme
allows one to extract the ratio $\Lambda_{\msbar}/\sqrt{\sigma}$. The result is
then used in combination with similar calculations present in the literature
for SU(2) and SU(3) to extract $\Lambda_{\msbar}/\sqrt{\sigma}$ in the large
$N$ limit. Our results for $N=3,4$ agree with a recent study of the same
quantity performed using the Parisi mean field scheme as an intermediate
scheme, while $\Lambda_{\msbar}/\sqrt{\sigma}$ in SU(2) turns out to differ by
2.5%. Possible explanations of this discrepancy are discussed.
