# 4 Concentric Spheres (Homework 5-310)

The system consists of four thin concentric spherical shells with radii $R$, $2R$, $3R$, and $4R$, respectively. The surface charge densities on these shells are $4sigma$, $-3sigma$, $2sigma$ and$-sigma$, respectively. Determine the electric field in all regions.

The charge on each spherical shell is in order

$Q_1=4sigma*S_1=4sigma*4pi R^2=16pisigma R^2$

$Q_2=-3sigma*S_2=-3sigma*4pi*(2R)^2=-48pisigma R^2$

$Q_3=2sigma*S_3=2sigma*4pi(3R)^2=72pisigma R^2$

$Q_4=-sigma*S_4=-sigma*4pi(4R)^2=-64pisigma R^2$

The arrangement has spherical symmetry which means that finally $E=E(r)$ and does not depend on angles. There are 5 regions in space (see the figure).

Take 5 spherical Gaussian surfaces of radius $r$ in each of the 5 regions of space.

General Gauss law is

$E(r)*S(r)=frac{Q_{inside}}{epsilon$}$

For $0≤r<R$ we have

$E(r)*4pi r^2=0$ that is $E_1(r)=0$

For $R≤ r< 2R$ we have

$E(r)*4pi r^2=frac{Q_1}{epsilon}=frac{16pisigma R^2}{epsilon}$ that is $E_2 (r)={4 pisigma}{epsilon}*frac{R}{r^2}$

For $2R≤r<3R$ we have

$E(r)*4 pi r^2= frac{Q_1+Q_2}{epsilon} (=-frac{32pisigma R^2}{epsilon}$ that is $E_3 (r)=-frac{32pisigma}{epsilon}*frac{R}{r^2}$

For $3R≤r<4R$ we have

$E(r)*4pi r^2=frac{Q_1+Q_2+Q_3}{epsilon} (=frac{40pisigma R^2}{epsilon}$ that is $E_4 (r)=frac{40pisigma}{epsilon}*frac{R}{r^2}$

For $4R≤r<infty$ we have

$E(r)*4pi r^2=frac{Q_1+Q_2+Q_3+Q_4}{epsilon} (=-frac{24pisigma R^2}{epsilon}$ that is $E_5 (r)=-frac{24pisigma}{epsilon}*frac{R}{r^2}$