Question 23.20: Flux and conducting shells. A charged particle is held at th......
Flux and conducting shells. A charged particle is held at the center of two concentric conducting spherical shells. Figure 23-39a shows a cross section. Figure 23-39b gives the net flux Φ through a Gaussian sphere centered on the particle, as a function of the radius r of the sphere. The scale of the vertical axis is set by Φ_s=5.0 10^5 N⋅m²/C. What are (a) the charge of the central particle and the net charges of (b) shell A and (c) shell B?
Equation 23-6 (Gauss’ law) gives \varepsilon_{ o } \Phi=q_{ enc } .
\varepsilon_0 \Phi=q_{ enc } \quad \text { (Gauss' law), } (23-6)
(a) The value \Phi=-9.0 \times 10^5\, N \cdot m ^2 / C for small r leads to q_{\text {central }}=-7.97 \times 10^{-6} C or roughly – 8.0 μC.
(b) The next (nonzero) value that Φ takes is \Phi=+4.0 \times 10^5 \,N \cdot m ^2 / C , which implies q_{ enc }=3.54 \times 10^{-6} \,C . But we have already accounted for some of that charge in part (a), so the result is
q_A=q_{\text {enc }}-q_{\text {central }}=11.5 \times 10^{-6}\, C \approx 12 \,\mu C .
(c) Finally, the large r value for Φ is \Phi=-2.0 \times 10^5 \,N \cdot m ^2 / C , which implies q_{\text {total enc }}=-1.77 \times 10^{-6} \,C . Considering what we have already found, then the result is
q_{\text {total enc }}-q_A-q_{\text {central }}=-5.3 \,\mu C \text {. }