Flux of an Electric Field — Question 8

Question 8

Angular uncertainty gives asymmetric flux bounds near a grazing orientation

A uniform field of magnitude E crosses an area A. The angle between field and chosen normal is measured as $80^\circ$ with a hard bound of $\pm5^\circ$; E and A are exact.

1. Find exact lower and upper bounds on $\Phi/(EA)$ and their deviations from the nominal result.

2. Compare the exact deviations with the first-order angular estimate. State why degree values must be converted in the differential formula.

Question 8 - Solution

Angular uncertainty gives asymmetric flux bounds near a grazing orientation

(a) Method and calculation

$$75^\circ\le\theta\le85^\circ\Rightarrow\cos85^\circ\le\Phi/(EA)\le\cos75^\circ.$$$$.0871557\le\Phi/(EA)\le.258819;\quad (\Phi/EA)_0=\cos80^\circ=.173648.$$

(b) Interpretation and further calculation

$$\Delta_-\approx-.0864924,\quad\Delta_+\approx+.0851709.$$$$\delta(\Phi/EA)\simeq-\sin80^\circ\,\delta\theta,\quad|\delta\theta|=\pi/36\Rightarrow|\delta(\Phi/EA)|\approx.0859407.$$The derivative of cosine has this form only when angle increments are in radians.

Physical check

These are hard geometric bounds, not a statistical confidence interval.