2ProblemA charge, +q, is surrounded by a thin, spherical shell of radius, a, which has a charge density of –s on its surface. This shell is, in turn, surrounded by another thin shell of radius, b, which has a surface charge of +s. Find the electric fields inRegion 1: r<aRegion 2: a<= r <=bRegion 3: r> b+qab-s+s
10Integrating each side (start with surface 1) Region 3, in which the normal vector points in the opposite direction, will have a value of -16
11The rest of the sidesSince E is perpendicular to sides 5 & 6, the result is zero.
12ProblemThe figure below shows a cross-section of two thin concentric cylinders with radii of a and b where b>a. The cylinders equal and opposite charges per unit length of l.Prove that E = 0 for r>aProve that E=0 for r>bProve that, for a<r<b,ab-ll
16For r>b qenclosed =lL-lL=0 This is the principle of a coaxial cable
17ProblemA very long, solid insulating cylinder with radius R has a cylindrical hole with radius, a, bored along its entire length. The axis of the hole is a distance b from the axis of the cylinder, where a<b<R. The solid material of the cylinder has a uniform charge density, p.Find the magnitude and direction of the electric field inside the hole and show that E is uniform over the entire hole.Rba
19Now what if we have an off-axis cylinder We learned in Phys 250, that we can “translate” coordinates by r’=r-bWhereb is the direction and distance of the center of the off-axis cylinderr is a vector from the originbr’r
20Ehole= Esolid cylinder-Eoff-axis hole All of these are constants and do not depend on r.