Gauss' Law — Question 4

Question 4

A missing face flux can be inferred with a hard uncertainty bound

A closed six-face surface encloses exact charge $Q=60\epsilon_0$ in SI units, so its total flux is 60 N m$^2$/C. Five outward face fluxes are reported as 10, $-20$, 30, $-40$ and 50 N m$^2$/C, each with an independent hard error bound of $\pm2$ N m$^2$/C.

1. Find the nominal flux through the sixth face.

2. Find a guaranteed interval for that flux without assuming random errors. If the sixth face has area $.500$ m$^2$, bound its average outward normal field.

Question 4 - Solution

A missing face flux can be inferred with a hard uncertainty bound

(a) Method and calculation

$$\sum_{i=1}^{5}\Phi_i=10-20+30-40+50=30\,\mathrm{N\,m^2/C}.$$$$\Phi_6=Q/\epsilon_0-\sum_{i=1}^{5}\Phi_i=30\,\mathrm{N\,m^2/C}.$$

(b) Interpretation and further calculation

$$\left|\delta\sum_{i=1}^{5}\Phi_i\right|\le\sum_{i=1}^{5}|\delta\Phi_i|\le10\Rightarrow20\le\Phi_6\le40.$$$$\langle E_n\rangle_6=\Phi_6/A_6\Rightarrow40\le\langle E_n\rangle_6\le80\,\mathrm{N/C}.$$Quadrature of errors would not provide this guaranteed bound.

Physical check

The inferred mean normal field need not equal the field at the face center.