Potential Difference Between Two Points — Question 9

Question 9

A prescribed set of voltage readings must satisfy loop consistency

Three stationary circuit nodes have claimed readings $V_B-V_A=5.00$ V, $V_C-V_B=-2.00$ V and $V_A-V_C=-4.00$ V. A single reading contains an additive error; the first two are trusted.

1. Test the closed-path sum and determine the corrected third reading.

2. If a particle of charge q travels from A to C, find its potential-energy change from either trusted path.

Question 9 - Solution

A prescribed set of voltage readings must satisfy loop consistency

(a) Method and calculation

$$(V_B-V_A)+(V_C-V_B)+(V_A-V_C)=5-2-4=-1.00\,\mathrm V\ne0.$$$$V_C-V_A=5-2=3.00\,\mathrm V\Rightarrow(V_A-V_C)_{\mathrm{correct}}=-3.00\,\mathrm V.$$

(b) Interpretation and further calculation

$$\Delta U_{A\to C}=q(V_C-V_A)=3.00q\,\mathrm V.$$$$q[(V_B-V_A)+(V_C-V_B)]=q(5-2)\,\mathrm V=\Delta U_{A\to C}.$$

Physical check

The consistency test concerns electrostatic node potentials; circuits with appreciable time-varying magnetic flux require a broader treatment.