homework_03_electric_flux_and_field_from_lines_of_charge

Electric Flux and Field from Lines of Charge

An infinite line of charge with charge density [\lambda_1 = -2.0 \mu C/cm] is aligned with the [y]-axis as shown.

1)

What is [E_x( P)], the value of the [x]-component of the electric field produced by by the line of charge at point [P] which is located at [(x,y) = (a,0)], where [a = 9.5 cm]?

• [E_x( P) = 2 k \frac{ \lambda}{ r} = -3.78526 \times 10^7]
  • [\lambda_1 = -2.0 \mu C/cm = -200 \mu C/m]
  • [a = 0.095 m]

2)

What is [E_y( P)], the value of the [y]-component of the electric field produced by by the line of charge at point P which is located at [(x,y) = (a,0)], where [a = 9.5 cm]?

• [E_y( P) = 0]

3)

A cylinder of radius [a = 0.095 m] and height [h = .086 m] is aligned with its axis along the [y]-axis as shown. What is the total flux [\Phi] that passes through the cylindrical surface? Enter a positive number if the net flux leaves the cylinder and a negative number if the net flux enters the cylnder.

• Let
  • [\lambda_1 = -2.0 \mu C/cm = -200 \mu C/m]
  • [a = 0.095 m]
  • [h = 0.086 m ]
• Given
  • [\oint\limits_{surface}{ \vec E \cdot d \vec A = \frac{ q_{enclosed}}{ \varepsilon_0}]
  • [\varepsilon_0 = 8.85 \times 10^{-12} \frac{ C^2}{N m^2}]
• [\Phi = h \left( \frac{ \lambda_1}{ \varepsilon_0} \right)= -1.9435 \times 10^6]

4)

Another infinite line of charge with charge density [\lambda_2 = 6.0 \mu C/cm] parallel to the [y]-axis is now added at [x = 4.75 cm] as shown.

What is the new value for [E_x( P)], the [x]-component of the electric field at point [P]?

• Let
  • [\lambda_1 = -2.0 \mu C/cm = -200 \mu C/m]
  • [\lambda_2 = 6.0 \mu C/cm = 600 \mu C/m]
  • [a = 0.095 m]
  • [h = 0.086 m ]
  • [\frac{ a}{ 2} = x = 0.0475 m]
• Given
  • [E_x( P) = 2 k \frac{ \lambda}{ r}]
• [E_x( P) = 2k \left( \frac{ \lambda_2}{ \frac{ a}{ 2}}
  • \frac{ \lambda_1}{ a} \right) = 1.89263 \times 10^8 ]

5)

What is the total flux [\Phi] that now passes through the cylindrical surface? Enter a positive number if the net flux leaves the cylinder and a negative number if the net flux enters the cylnder.

• Let
  • [\lambda_1 = -2.0 \mu C/cm = -200 \mu C/m]
  • [\lambda_2 = 6.0 \mu C/cm = 600 \mu C/m]
  • [a = 0.095 m]
  • [h = 0.086 m ]
  • [\frac{ a}{ 2} = x = 0.0475 m]
• Given
  • [\oint\limits_{surface}{ \vec E \cdot d \vec A = \frac{ q_{enclosed}}{ \varepsilon_0}]
  • [\varepsilon_0 = 8.85 \times 10^{-12} \frac{ C^2}{N m^2}]
• [\Phi = h \left( \frac{ \lambda_1}{ \varepsilon_0}
  • \frac{ \lambda_2}{ \varepsilon_0} \right) = 3.88701 \times 10^6]

6)

The initial infinitie line of charge is now moved so that it is parallel to the [y]-axis at [x = -4.75cm].

What is the new value for [E_x( P)], the [x]-component of the electric field at point [P]?

• Let
  • [\lambda_1 = -2.0 \mu C/cm = -200 \mu C/m]
  • [\lambda_2 = 6.0 \mu C/cm = 600 \mu C/m]
  • [a = 0.095 m]
  • [h = 0.086 m ]
  • [\frac{ a}{ 2} = x = 0.0475 m]
• Given
  • [E_x( P) = 2 k \frac{ \lambda}{ r}]
• [E_x( P) = 2k \left( \frac{ \lambda_2}{ \frac{ a}{ 2}}
  • \frac{ \lambda_1}{ \frac{3 a}{ 2}} \right) = 1.89263 \times 10^8 ]

7)

What is the total flux [\Phi] that now passes through the cylindrical surface? Enter a positive number if the net flux leaves the cylinder and a negative number if the net flux enters the cylnder.

• Let
  • [\lambda_1 = -2.0 \mu C/cm = -200 \mu C/m]
  • [\lambda_2 = 6.0 \mu C/cm = 600 \mu C/m]
  • [a = 0.095 m]
  • [h = 0.086 m ]
  • [\frac{ a}{ 2} = x = 0.0475 m]
• Given
  • [\oint\limits_{surface}{ \vec E \cdot d \vec A = \frac{ q_{enclosed}}{ \varepsilon_0}]
  • [\varepsilon_0 = 8.85 \times 10^{-12} \frac{ C^2}{N m^2}]
• [\Phi = h \left( \frac{ \lambda_1}{ \varepsilon_0}
  • \frac{ \lambda_2}{ \varepsilon_0} \right) = 3.88701 \times 10^6]