Ohm's Law — Question 5

Question 5

A saturating transport law sets a quantitative low-field regime

A prescribed current-density law is $J(E)=\sigma_0E/\sqrt{1+(E/E_s)^2}$, with $\sigma_0,E_s>0$.

1. Find the limiting current density as $E\to+\infty$ and the low-field expansion.

2. Find the positive field for which J is exactly 95 percent of the ohmic prediction $\sigma_0E$.

Question 5 - Solution

A saturating transport law sets a quantitative low-field regime

(a) Method and calculation

$$\lim_{E\to\infty}J=\sigma_0E_s.$$$$J=\sigma_0E\left[1-\tfrac12(E/E_s)^2+O((E/E_s)^4)\right].$$

(b) Interpretation and further calculation

$$J/(\sigma_0E)=\frac1{\sqrt{1+(E/E_s)^2}}=.95.$$$$E/E_s=\sqrt{.95^{-2}-1}=.328684.$$

Physical check

The linear law is a limiting constitutive approximation with a calculable range of accuracy.