homework_04_line_charge_and_charged_cylindrical_shell

Line Charge and Charged Cylindrical Shell

An infinite line of charge with linear density [\lambda_1 = 6.9 \mu \frac{ C}{ m}] is positioned along the axis of a thick insulating shell of inner radius [a = 2.5 cm] and outer radius [b = 4.8 cm]. The insulating shell is uniformly charged with a volume density of [\rho = -665.0 \mu \frac{ C}{ m^3}].

1)

What is [\lambda_2], the linear charge density of the insulating shell?

• Given
  • [E = \frac{ \lambda}{ 2 \pi \varepsilon_0 r}]
    • where [\lambda \Rightarrow] liniear charge density
  • [\rho = \frac{ Q}{ 4 \pi r^2}]
• Let

  • [a = 2.5 cm = 0.025 m]

    • inner radius of the insulating shell

  • [b = 4.8 cm = 0.048 m]

    • outer radius of the insulating shell

  • [\rho = -665.0 \mu \frac{ C}{ m^3}]

    • insulating shell

  • [\lambda_2 = \rho * A_{cylindrical_shell} = \rho * \pi * (b^2 - a^2)]

2)

What is [E_x( P)], the value of the [x]-component of the electric field at point [P], located a distance [8.2 cm] along the [y]-axis from the line of charge?

[E_x( P) = 0]

3)

What is [E_y( P)], the value of the [x]-component of the electric field at point [P], located a distance [8.2 cm] along the [y]-axis from the line of charge?

4)

What is [E_x( R)], the value of the [x]-component of the electric field at point [R], located a distance [1.25 cm] along a line that makes an angle of [30^\circ] with the [x]-axis?

[E_y( P) = 2 k \frac{ \lambda_1}{ r} = 8.59527E6]