Gauss' Law — Question 2

Question 2

Nested flux measurements resolve the charge in separate layers

Use $\epsilon_0=8.854\times10^{-12}\,\mathrm{C^2/(N\,m^2)}$. Three nested closed surfaces have measured outward fluxes $\Phi_1=80.0$, $\Phi_2=-40.0$ and $\Phi_3=120$ N m$^2$/C. Each lies entirely inside the next, with no charge on any boundary.

1. Find the charge inside the smallest surface and in each intervening layer.

2. Find the combined charge in the two layers, and check that all three regions sum to the charge inside the largest surface.

Question 2 - Solution

Nested flux measurements resolve the charge in separate layers

(a) Method and calculation

$$Q_1=\epsilon_0\Phi_1=708.32\,\mathrm{pC}.$$$$Q_{12}=\epsilon_0(\Phi_2-\Phi_1)=-1062.48\,\mathrm{pC},\quad Q_{23}=\epsilon_0(\Phi_3-\Phi_2)=1416.64\,\mathrm{pC}.$$

(b) Interpretation and further calculation

$$Q_{12}+Q_{23}=\epsilon_0(\Phi_3-\Phi_1)=354.16\,\mathrm{pC}.$$$$Q_1+Q_{12}+Q_{23}=1062.48\,\mathrm{pC}=\epsilon_0\Phi_3.$$

Physical check

A layer charge is determined by a difference of fluxes; the larger surface reading alone includes all deeper regions.