The Electric Field — Question 9

Question 9

Different local fields may produce the same force magnitude

At P the field is $(30,40)$ N/C; at R it is $(-40,30)$ N/C. A $+2.00$ nC probe can be placed at either point without changing the source. A sensor records only force magnitude.

1. Find the force magnitudes and the angle between the two field vectors.

2. Find the component-sensitive force change when moving the probe from P to R. State what additional measurement would distinguish the locations.

Question 9 - Solution

Different local fields may produce the same force magnitude

(a) Method and calculation

$$|\mathbf E_P|=|\mathbf E_R|=50\,\mathrm{N/C}\Rightarrow|\mathbf F_P|=|\mathbf F_R|=100\,\mathrm{nN}.$$$$\mathbf E_P\cdot\mathbf E_R=30(-40)+40(30)=0\Rightarrow\theta=90^\circ.$$

(b) Interpretation and further calculation

$$\Delta\mathbf E=(-70,-10)\,\mathrm{N/C},\quad\Delta\mathbf F=(-140,-20)\,\mathrm{nN}.$$$$|\Delta\mathbf F|=100\sqrt2\,\mathrm{nN}.$$One signed directional component, such as $F_x$, distinguishes the locations even though a magnitude-only sensor does not.

Physical check

The change of a vector magnitude is not generally the magnitude of the vector change.