  Many estimates for the magnetic relaxation time of magnetic nanoparticle
systems neglect the effect of the applied field strength. This is despite many
applications of magnetic nanoparticles involving relaxation dynamics under the
influence of applied fields. Here, an analytic approximation for the
field-dependent Brownian relaxation time of single-domain, spherical magnetic
nanoparticles in an external applied field is developed mathematically. This
expression is validated by comparison with existing empirically-derived
expressions and by comparison to particle-level simulations that allow particle
rotations. Our approximation works particularly well for larger particles. We
then use the developed expression to analytically calculate the total magnetic
relaxation time when both Brownian and N\'eel relaxation mechanisms are at
play. Again, we show that the results match those found using particle-level
simulations, this time with both particle rotations and internal magnetization
dynamics allowed. However, for some particle parameters and for large field
strengths, our simulations reveal that the Brownian and N\'eel relaxation
mechanisms are decoupled and it is not appropriate to combine these to
calculate a total relaxation time.
