homework_04_spheres

Spheres

An insulator in the shape of a spherical shell is shown in cross-section above. The insulator is defined by an inner radius [a = 4.0 cm] and an outer radius [b = 6.0 cm] and carries a total charge of [Q = + 9.0 \mu C]. (You may assume that the charge is distributed uniformly throughout the volume of the insulator). What is [E_y], the [y]-component of the electric field at point [P] which is located at [(x,y) = (0 m, -5.0 cm)] as shown in the diagram?

• Let
  • [a = 4.0 cm = 0.04 m]
  • [b = 6.0 cm = 0.06 m]
  • [Q = 9.0 \mu C]
  • [(x, y) = (0 m, -5.0 cm) = (0 m, -0.05 m) = ]
• Given
  • For sphical insulators with a point charge [Q] at the center:
  • [V_{sphere} = \frac{ 4}{ 3} \pi r^3]
  • [A_{sphere} = 4 \pi r^2]
  • Guass' Law of Symmetry
    • [ E \oint\limits_{surface}{ dA} = \frac{ q_enclosed}{ \varepsilon_0}]
• Wrong answers
  • [E = \frac{ Q}{ 4 \pi \varepsilon_0 r^2} = 3.23705E7]
    • It looks like you are treating this distribution as a point charge of [Q = 9.0 \mu C] located at the origin. The charge distribution is a little more complicated that that.
• Help
  • What is [q_{enclosed}], the charge enclosed by the Gausian sphere?
    • [q_{enclosed} = \frac{ Q}{ 4 \pi 100 \left((b)^2 - (a)^2\right)} = 3.23705E7]
      • I have no idea why it is dividing by [100]!
• [E = \frac{ q_{enclosed}}{ A \varepsilon_0} = -1.28798E7]