Electric Charge — Question 9

Question 9

Attraction observations can reveal an inconsistent point-charge model

Three objects are asserted to be nonzero point charges. Separate tests report that every pair attracts. Define $s_i=\operatorname{sgn}(q_i)$, so $s_i=\pm1$, and use attraction if and only if $s_is_j=-1$.

1. Show algebraically that the three observations cannot all hold in this model.

2. If the A–B and B–C attraction reports are reliable, predict the A–C interaction. Give two physically distinct ways the original experimental description might fail.

Question 9 - Solution

Attraction observations can reveal an inconsistent point-charge model

(a) Method and calculation

$$(s_As_B)(s_Bs_C)(s_Cs_A)=s_A^2s_B^2s_C^2=+1.$$$$(-1)(-1)(-1)=-1\ne+1.$$Thus no assignment of three nonzero point-charge signs satisfies all three attraction reports.

(b) Interpretation and further calculation

$$s_As_C=(s_As_B)(s_Bs_C)=(-1)(-1)=+1.$$$$A\text{ and }C:\quad\text{repulsion}.$$A report could be wrong, or the objects could be extended polarizable bodies rather than the asserted point charges. Additional non-electrostatic forces are another possible model failure.

Physical check

An inconsistent inference identifies failure of the combined observations and assumptions; it does not identify which individual report is wrong.