Potential Due to a Group of Point Charges — Question 5

Question 5

Two voltage measurements recover two unknown source charges

Point charges Q1 and Q2 are fixed a distance d apart. At P their distances are d and 2d; at R the distances are 2d and d. Measurements give $V_P=5kq_0/d$ and $V_R=4kq_0/d$, with known $q_0>0$.

1. Solve for Q1 and Q2.

2. Find the potential at the midpoint and discuss uniqueness of this two-equation inversion.

Question 5 - Solution

Two voltage measurements recover two unknown source charges

(a) Method and calculation

$$Q_1+Q_2/2=5q_0,\quad Q_1/2+Q_2=4q_0.$$$$2Q_1+Q_2=10q_0,\quad Q_1+2Q_2=8q_0\Rightarrow Q_1=4q_0,\ Q_2=2q_0.$$

(b) Interpretation and further calculation

$$V_M=\frac{2k}{d}(Q_1+Q_2)=12kq_0/d.$$$$\det\begin{pmatrix}1&1/2\\1/2&1\end{pmatrix}=3/4\ne0,$$so the fixed-reference measurements determine a unique pair.

Physical check

The two measurement points lie on opposite exterior extensions of the source segment, consistent with the specified distances.