The Electric Field — Question 5

Question 5

Two oblique sensor projections reconstruct a planar field

A nonperturbing sensor measures a planar field projection of 40.0 N/C along $+x$ and 70.0 N/C along a direction $60^\circ$ counterclockwise from $+x$.

1. Reconstruct $E_x,E_y$ and the field magnitude.

2. The second direction is changed to $0^\circ$ while the first direction stays fixed. Explain algebraically why the two readings then cannot determine $E_y$.

Question 5 - Solution

Two oblique sensor projections reconstruct a planar field

(a) Method and calculation

$$E_x=40.0,\quad70.0=E_x\cos60^\circ+E_y\sin60^\circ.$$$$E_y=100/\sqrt3\approx57.7350\,\mathrm{N/C},\quad|\mathbf E|=\sqrt{1600+10000/3}\approx70.2377\,\mathrm{N/C}.$$

(b) Interpretation and further calculation

$$E_{\phi}=E_x\cos\phi+E_y\sin\phi\Rightarrow E_y=\frac{E_{\phi}-E_x\cos\phi}{\sin\phi}.$$$$\phi=0:\quad\sin\phi=0,\quad E_\phi=E_x.$$Parallel sensor directions give redundant information. Near-parallel directions also amplify measurement error in the inferred transverse component.

Physical check

The second reading is a projection, not the vertical component.