Metals — Question 2

Question 2

A field step reveals the time required to establish current

A metal initially carries no current. At $t=0$ a constant field $E_0$ is applied. Use $\dot J+J/\tau=\sigma_0E_0/\tau$, where $\sigma_0$ is the dc conductivity.

1. Solve for J(t) and its initial slope.

2. Find the time to reach 95 percent of the final current and the charge per area transported up to that time.

Question 2 - Solution

A field step reveals the time required to establish current

(a) Method and calculation

$$\frac{d}{dt}(Je^{t/\tau})=\frac{\sigma_0E_0}{\tau}e^{t/\tau}\Rightarrow\boxed{J=\sigma_0E_0(1-e^{-t/\tau})}.$$$$\boxed{\dot J(0)=\sigma_0E_0/\tau}.$$

(b) Interpretation and further calculation

$$e^{-t_{95}/\tau}=.05\Rightarrow\boxed{t_{95}=\tau\ln20=2.996\tau}.$$$$\boxed{Q/A=\int_0^{t_{95}}Jdt=\sigma_0E_0\tau(\ln20-.95)}.$$

Physical check

The initial response is inertial. At long times the relaxation term balances the field, leaving a finite steady current.