Electric Current — Question 5

Question 5

A drifting instrument zero biases a finite current measurement

A sensor reports $I_m(t)=I(t)+b_0+kt$ over $0\le t\le T$. Independent calibration gives the offsets $b_0$ and $b_T$ at the endpoints; the offset varies linearly.

1. Express the actual transported charge in terms of the measured integral and calibrations.

2. For a measured integral 120 mC, T=20 s, $b_0=1.0$ mA and $b_T=3.0$ mA, find actual charge and average current.

Question 5 - Solution

A drifting instrument zero biases a finite current measurement

(a) Method and calculation

$$k=(b_T-b_0)/T,\quad Q_m=\int_0^T I_m\,dt.$$$$Q=Q_m-b_0T-kT^2/2=Q_m-\tfrac12(b_0+b_T)T.$$

(b) Interpretation and further calculation

$$Q=120-\tfrac12(1+3)(20)=80.0\,\mathrm{mC}.$$$$\overline I=Q/T=4.00\,\mathrm{mA}.$$

Physical check

The correction is the area under the offset curve, not just the final offset multiplied by time.