The Electric Field Due to a Point Charge — Question 8

Question 8

Two symmetry-point readings separate the sum and difference of source charges

Unknown charges $q_L,q_R$ are fixed at $(-a,0)$ and $(a,0)$. At the origin, $E_x=kQ_0/a^2$. At $(0,a)$, the measured y component is $kQ_0/(\sqrt2a^2)$, where $Q_0>0$ is known. Other sources are absent.

1. Derive equations for the charge sum and difference.

2. Find both charges and predict the x component of the field at $(0,a)$.

Question 8 - Solution

Two symmetry-point readings separate the sum and difference of source charges

(a) Method and calculation

$$E_x(0)=\frac{k(q_L-q_R)}{a^2}\Rightarrow q_L-q_R=Q_0.$$$$E_y(0,a)=\frac{k(q_L+q_R)}{2\sqrt2a^2}\Rightarrow q_L+q_R=2Q_0.$$

(b) Interpretation and further calculation

$$q_L=\tfrac32Q_0,\quad q_R=\tfrac12Q_0.$$$$E_x(0,a)=\frac{k(q_L-q_R)}{2\sqrt2a^2}=\frac{kQ_0}{2\sqrt2a^2}.$$Measurements on different symmetry locations isolate different linear combinations of the unknown charges.

Physical check

The x component at the elevated point need not vanish when the source charges differ.