Question 23.12: Figure 23-36 shows two nonconducting spherical shells fixed ......
Figure 23-36 shows two nonconducting spherical shells fixed in place. Shell 1 has uniform surface charge density +6.0 μC/m² on its outer surface and radius 3.0 cm; shell 2 has uniform surface charge density +4.0 μC/m² on its outer surface and radius 2.0 cm; the shell centers are separated by L = 10 cm. In unit-vector notation, what is the net electric field at x = 2.0 cm?
We note that only the smaller shell contributes a (nonzero) field at the designated point, since the point is inside the radius of the large sphere (and E = 0 inside of a spherical charge), and the field points toward the -x direction. Thus, with R = 0.020 m (the radius of the smaller shell), L = 0.10 m and x = 0.020 m, we obtain
\begin{aligned} \vec{E} & =E(-\hat{ j })=-\frac{q}{4 \pi \varepsilon_0 r^2} \hat{ j }=-\frac{4 \pi R^2 \sigma_2}{4 \pi \varepsilon_0(L-x)^2} \hat{ j }=-\frac{R^2 \sigma_2}{\varepsilon_0(L-x)^2} \hat{ j } \\ & =-\frac{(0.020 \,m )^2\left(4.0 \times 10^{-6} \,C / m ^2\right)}{\left(8.85 \times 10^{-12} \, C ^2 / N \cdot m ^2\right)(0.10 \,m -0.020 \, m )^2} \hat{ j }=\left(-2.8 \times 10^4 \,N / C \right) \hat{ j } . \end{aligned}