Flux of an Electric Field — Question 10

Question 10

A growing aperture reconstructs a nonuniform normal field

A family of rectangular apertures lies in $z=0$, spanning $0\le x\le a$ and $0\le y\le W$. The normal field depends only on x. Measurements give $\Phi(a)=C a^2+D a^3$ for every $0\le a\le L$, with positive-z orientation.

1. Infer $E_z(x)$ by differentiating the measured flux with respect to a.

2. Compare the field at the far edge of an aperture with its area-averaged field, and determine when they agree for $a>0$.

Question 10 - Solution

A growing aperture reconstructs a nonuniform normal field

(a) Method and calculation

$$\Phi(a)=W\int_0^aE_z(x)\,dx,\quad\frac{d\Phi}{da}=WE_z(a).$$$$E_z(x)=\frac{2Cx+3Dx^2}{W}.$$

(b) Interpretation and further calculation

$$\langle E_z\rangle=\frac{\Phi(a)}{Wa}=\frac{Ca+Da^2}{W}.$$$$E_z(a)-\langle E_z\rangle=\frac{Ca+2Da^2}{W}=0\Rightarrow C+2Da=0.$$For nonzero C and D, a positive solution exists only when their signs are opposite and $a=-C/(2D)$ lies within L.

Physical check

The field reconstruction uses a continuous family of aperture measurements; a single integrated reading cannot determine an arbitrary profile.