homework_04_point_charge_and_charged_sphere

Point Charge and Charged Sphere

A point charge [q_1 = -8.5 \mu C] is located at the center of a thick conducting shell of inner radius [a = 2.8 cm] and outer radius [b = 4.1 cm], The conducting shell has a net charge of [q_2 = 2.1 \mu C].

1)

What is [E_x( P)], the value of the [x]-component of the electric field at point [P], located a distance [7.3 cm] along the [x]-axis from [q_1]?

• Given
  • [E = k \frac{ Q}{ r^2}]
    • Electric field of a point charge
• Let
  • [q_1 = -8.5 \mu C]
    • point charge at the center
  • [q_2 = 2.1 \mu C]
    • net charge of the conducting shell
  • [a = 2.8 cm = 0.028]
    • inner radius of the conducting shell
  • [b = 4.1 cm = 0.041 m]
    • outer radius of the conducting shell
  • [r = 7.3 cm = 0.073 m]
• [E_x( P) = k \left(\frac{ q_1 + q_2}{ r^2}\right) = -1.07968 \times 10^7 \frac{ N}{ C}]

2)

What is [E_y( P)], the value of the [y]-component of the electric field at point [P], located a distance [7.3 cm] along the [x]-axis from [q_1]?

• [E_y( P)\ = 0]

3)

What is [E_x( R)], the value of the [x]-component of the electric field at point [R], located a distance [1.4 cm] along the [y]-axis from [q_1]?

• [E_x( R) = 0]

4)

What is [E_y( R)], the value of the [y]-component of the electric field at point [R], located a distance [1.4 cm] along the [y]-axis from [q_1]?

• Given
  • [E = k \frac{ Q}{ r^2}]
    • Electric field of a point charge
• Let
  • [q_1 = -8.5 \mu C]
    • point charge at the center
  • [q_2 = 2.1 \mu C]
    • net charge of the conducting shell
  • [a = 2.8 cm = 0.028]
    • inner radius of the conducting shell
  • [b = 4.1 cm = 0.041 m]
    • outer radius of the conducting shell
  • [r = 1.4 cm = 0.014 m]
• [E_x( P) = k \left(\frac{ q_1}{ r^2} \right) = -3.89872 \times 10^8 \frac{ N}{ C}]

5)

What is [\sigma_b], the surface charge density at the outer edge of the shell?

• Given
  • [\sigma = \frac{ q_{enclosed}}{ 4 \pi r^2}]
• Let
  • [q_1 = -8.5 \mu C]
    • point charge at the center
  • [q_2 = 2.1 \mu C]
    • net charge of the conducting shell
  • [a = 2.8 cm = 0.028]
    • inner radius of the conducting shell
  • [b = 4.1 cm = 0.041 m]
    • outer radius of the conducting shell
  • [r = b]
• [\sigma_b = \frac{ q_1 + q_2}{ 4 \pi b^2} = -.000303 \frac{ C}{ m^2}]

6)

What is [\sigma_a], the surface charge density at the inner edge of the shell?

• Given
  • [\sigma = \frac{ q_{enclosed}}{ 4 \pi r^2}]
• Let
  • [q_1 = -8.5 \mu C]
    • point charge at the center
  • [q_2 = 2.1 \mu C]
    • net charge of the conducting shell
  • [a = 2.8 cm = 0.028]
    • inner radius of the conducting shell
  • [b = 4.1 cm = 0.041 m]
    • outer radius of the conducting shell
  • [r = b]
• [ \sigma_a = -\left(\frac{ q_1}{ 4 \pi a^2}\right) = .000402 \frac{ C}{ m^2}]

7)

For how many values of [ x: (4.1 cm < x < \infty)] is it true that [E_x = 0]?

• none
  • The field is treated as if it is a single point charge outside the conducting wall and there after extends to infinity diminishing by a rate of [r^2].

8)

Define [E_2] to be equal to the magnitude of the electic field at [r = 1.4 cm] when the charge on the outer shell [q_2] is equal to [2.1 \mu C]. Define [E_0] to be equal to the magnitude of the electric field at [r = 1.4 cm] if the charge on the outer shell [q_2] were changed to [0]. Compare [E_2] and [E_0].

• [E_2 = E_0]
  • The fields are equal as the charge on the outer shell does not effect the field on within the shell.