Chapter 22--Examples

Problem A 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 in Region 1: r<a Region 2: a<= r <=b Region 3: r> b +q a b -s +s

Step 1: Pick your shape I choose spherical! So

Region 1: r<a +q a b -s +s qenclosed=q

Region 2: a<=r<=b +q a b -s +s qenclosed=q+(4pa2)*(-s)

Region 3: r>b +q a b -s +s qenclosed=q+(4pa2)*(-s)+ (4pb2)*(s)

Problem An electric filed given by E=4i-3(y2+2)j pierces the Gaussian cube shown below. (E is in newtons/coulomb and y is meters). What net charge is enclosed by the Gaussian cube? X=1.0 m X=3.0 m x y z

First, let’s get a sense of direction 4 y Planes y-z: Normal to +x x-z: Normal to –y y-z: Normal to –x x-z: Normal to +y x-y: Normal to +z x-y: Normal to -z 6 1 3 x 2 z 5 X=3.0 m X=1.0 m

Need to find qenclosed

Integrating each side (start with surface 1) Region 3, in which the normal vector points in the opposite direction, will have a value of -16

The rest of the sides Since E is perpendicular to sides 5 & 6, the result is zero.

Problem The 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>a Prove that E=0 for r>b Prove that, for a<r<b, a b -l l

First, I choose a shape I choose cylindrical! So

For r<a qenclosed =0 a b -l l

For a<r<b qenclosed =lL a b -l l

For r>b qenclosed =lL-lL=0 This is the principle of a coaxial cable

Problem A 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. R b a

First, let’s do a solid cylinder of radius, R

Now what if we have an off-axis cylinder We learned in Phys 250, that we can “translate” coordinates by r’=r-b Where b is the direction and distance of the center of the off-axis cylinder r is a vector from the origin b r’ r

Ehole= Esolid cylinder-Eoff-axis hole All of these are constants and do not depend on r.