Electric Charge — Question 10

Question 10

A signed calibration separates charge-dependent force from a background

A fixed-source force sensor obeys the supplied model $F_x(q)=Aq+B$, with $q$ in nC and $F_x$ in mN. The constant $B$ represents a charge-independent background. Measurements give $F_x(+2)=7$ and $F_x(-2)=-1$. Source geometry and background stay fixed.

1. Find $A$ and $B$ by adding and subtracting the two measurements.

2. Infer an unknown charge from measured force 1.0 mN. Find the charge giving zero total force, and explain why zero force need not mean zero charge.

Question 10 - Solution

A signed calibration separates charge-dependent force from a background

(a) Method and calculation

$$7=2A+B,\quad-1=-2A+B.$$$$A=\frac{7-(-1)}4=2.0\,\mathrm{mN/nC},\quad B=\frac{7+(-1)}2=3.0\,\mathrm{mN}.$$

(b) Interpretation and further calculation

$$q=\frac{F_x-B}{A}=\frac{1-3}{2}=-1.0\,\mathrm{nC}.$$$$F_x=0:\quad q=-B/A=-1.5\,\mathrm{nC}.$$The charge-dependent force cancels the background at a nonzero charge. A neutral object instead gives the background reading $B$.

Physical check

The inferred charge is meaningful only within the stated linear calibration and unchanged experimental conditions.