Resistance upper bound from persistent-current decay ($R\le -(L/T)\ln (1-ϵ)$)

Suppose a calibrated measurement bounds the fractional loss of a nonzero superconducting persistent current by $ϵ$, with $0<ϵ<1$, over a duration $T>0$. Take $I_0>0$ by choosing the circulation direction. Under the constant-$L$, constant-$R$, undriven RL circuit model,The inequality follows because the logarithm is increasing. The final approximation requires $ϵ\ll 1$ and slightly understates the exact upper bound. If $L$ is uncertain, an independently established upper bound on $L$ gives the conservative electrical resistance bound. The uncertainty in the initial reading, the final reading, background subtraction and drift must all be included in $ϵ$; a display resolution alone does not provide it.

For an original illustrative example, let $L=1\mu \mathrm{H}$ and suppose the corrected electric current ratio after $T=3\mathrm{h}=10800\mathrm{s}$ is at least $0.999$. ThenThese are conditional bounds for stipulated parameters, not values attributed to the 1914 apparatus. With the same $L$ but $R=10^{-11}\Omega$, the model instead gives $\tau =10^5\mathrm{s}$ and $I(24\mathrm{h})/I_0=e^{-0.864}\simeq 0.42147$: a readily visible loss on the chosen scale.

A resolved decay can be checked using its logarithm:A straight line supports this constant-parameter model; curvature or jumps demand a different account. If an effective electrical resistance varies with time while $L$ remains fixed and the loop remains undriven, only its time average is constrained:This does not bound every instantaneous value of $R(t)$. Changing self-inductance or external linked magnetic flux adds terms to the circuit equation, so an uncorrected magnetic signal cannot simply be substituted for $I(t)$. Nonobservation of decay leaves arbitrarily small positive constant electrical resistance compatible with a finite observation: it is not a logical proof that $R=0$.