The Electric Field — Question 4

Question 4

A compensating source must cancel unwanted components as well as set magnitude

Two independent fixed sources produce $\mathbf E_A=(30,-10,20)$ N/C and $\mathbf E_B=(-5,25,-8)$ N/C at P. A third adjustable source must make the total field point along $+z$ with magnitude 50 N/C. Component triples refer to Cartesian axes.

1. Find the required third-source field.

2. If that source operates at 90 percent of its required strength with direction unchanged, find the actual field and the residual transverse magnitude.

Question 4 - Solution

A compensating source must cancel unwanted components as well as set magnitude

(a) Method and calculation

$$\mathbf E_A+\mathbf E_B=(25,15,12)\,\mathrm{N/C}.$$$$\mathbf E_C=(0,0,50)-(25,15,12)=(-25,-15,38)\,\mathrm{N/C}.$$

(b) Interpretation and further calculation

$$\mathbf E_{\mathrm{actual}}=(25,15,12)+.9(-25,-15,38)=(2.5,1.5,46.2)\,\mathrm{N/C}.$$$$E_\perp=\sqrt{2.5^2+1.5^2}=\sqrt{8.5}\approx2.91548\,\mathrm{N/C}.$$A strength error in the compensator introduces direction error as well as longitudinal error.

Physical check

Vector superposition applies component by component; adding field magnitudes would not solve the compensation problem.