Refer to the diagram (not to scale), which shows a sphere (of outer radius R) th
ID: 1498960 • Letter: R
Question
Refer to the diagram (not to scale), which shows a sphere (of outer radius R) that is made of solid glass except for a spherical cavity at its center (of radius R/3), which is empty. The sphere carries a total net charge of -Q that is uniformly distributed throughout the glass (but no charge resides in the empty central cavity). In terms of R, Q, K (or E_0), find an expression for the electric field (magnitude and direction)-as a function of distance r from the sphere's center-for the following regions: 0 inequality rExplanation / Answer
a)
consider a gaussian sphere of radius r < R/3
charge inside = Qin = 0
from gauss law
total flux = Qin/eo
total flux = E*A = E*4*pi*r^2
E*4*pi*r^2 = 0/eo
E = 0
++++++++++++
(b)
charge density sigma = Q/(4/3*pi*(R^3-(R/3)^3)
sigma = Q/(4/3*pi*(26/27)*R^3)
consider a gaussian sphere of radius R/3 < r < R
effective volume V = 4/3*pi*(r^3-(R/3)^3)) = (4/3)*pi*(r^3-R^3/27)
charge inside Qin = sigma*V = sigma* (4/3)*pi*(r^3-R^3/27)
from gauss law
total flux = Qin/eo
total flux = E*A = E*4*pi*r^2
E*4*pi*r^2 = sigma*(4/3)*pi*(r^3-R^3/27)/eo
E*r^2 = ((1/3)*(Q/(4/3*pi*(26/27)*R^3))*(r^3-(R^3/27)))/eo
E = Q*(27r^3-R^3)/(4*pi*26*R^3*r^2)
(c)
consider a gaussian sphere of radius r > R
charge inside Qin = Q
from gauss law
total flux = Qin/eo
total flux = E*A = E*4*pi*r^2
E*4*pi*r^2 = Q/eo
E = Q/(4*pi*r^2)