Crossed Fields: The Hall Effect — Question 7

Question 7

Finite voltmeter resistance suppresses a Hall voltage and inflates the inferred density

Treat the transverse Hall contacts as an ideal open-circuit voltage VH0 in series with resistance rH. A voltmeter of resistance RV is connected between them. Assume it does not otherwise change the imposed longitudinal current.

1. Find the measured voltage and the density inferred by ignoring the meter loading.

2. Find the minimum RV/rH needed to keep the inferred density no more than one percent above its true value.

Question 7 - Solution

Finite voltmeter resistance suppresses a Hall voltage and inflates the inferred density

(a) Method and calculation

$$V_m=V_{H0}\frac{R_V}{r_H+R_V}.$$$$\frac{n_{\mathrm{inferred}}}n=\frac{|V_{H0}|}{|V_m|}=1+\frac{r_H}{R_V}.$$

(b) Interpretation and further calculation

$$\frac{n_{\mathrm{inferred}}-n}{n}=r_H/R_V\le.0100.$$$$R_V/r_H\ge100.$$

Physical check

The density error is the inverse of the voltage attenuation, so its exact expression differs from the fractional voltage error.