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Chapter 22--Examples

Chapter 22--Examples. + s. - s. b. +q. a. Problem.

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Chapter 22--Examples

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  1. Chapter 22--Examples

  2. +s -s b +q a 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

  3. Step 1: Pick your shape • I choose spherical! • So

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

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

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

  7. y x z X=3.0 m X=1.0 m 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?

  8. 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

  9. Need to find qenclosed

  10. 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

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

  12. a b 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, -l l

  13. First, I choose a shape • I choose cylindrical! • So

  14. a b For r<a • qenclosed =0 -l l

  15. a b For a<r<b • qenclosed =lL -l l

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

  17. 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

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

  19. 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

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

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