homework_01_point_charges_in_one_dimension

Point Charges in One Dimension

A point chare [q_1 = -3.4 \mu C ] is located at the origin of a coordinate system. Another point charge [q_2 = 6.6 \mu C] is located along the [x]-axis at a distance [x_2 = 9.5 cm] from [q_1].

1)

What is [F_{12,x}], the value of the [x] component of the force that [q_1] exerts on [q_2]?

• [F_{ij} = \kappa \frac{ q_i q_j}{ r^2} \hat r]
  • [ \kappa = \frac{ 1}{ 4 \pi \epsilon_0} = 8.99 \times 10^9 \frac{ Nm^2}{ C^2} ]
  • [ \mu = 10^{-6}]
• Let:
  • [q_1 = q_i = -3.4 \mu C]
  • [q_2 = q_j = 6.7 \mu C ]
  • [r = 9.5 cm = .095 m]
• [F_{ij,x} = \kappa \frac{ q_i q_j}{ r^2} = -22.6917 N]

2)

Charge [q_2] is now displaced a distance [y_2 = 2.9 cm] in the positive [y]-direction. What is the new value for the [x]-component of the force that [q_1] exerts on [q_2]?

• [F_{ij} = \kappa \frac{ q_i q_j}{ r^2} \hat r]
  • [ \kappa = \frac{ 1}{ 4 \pi \epsilon_0} = 8.99 \times 10^9 \frac{ Nm^2}{ C^2} ]
  • [ \mu = 10^{-6}]
• Let:
  • [q_1 = q_i = -3.4 \mu C]
  • [q_2 = q_j = 6.7 \mu C ]
  • [x_2 = x = 9.5 cm = .095 m]
  • [y_2 = y = 2.9 cm = .029 m]
  • [r = \sqrt{ x^2 + y^2} = .09932774 m]
  • [\theta_{xr} = \arctan{ \left( \frac{ x}{ y} \right)} = 16.9755^\circ]
• [F_{ij,x} = \kappa \frac{ q_i q_j}{ r^2} \cos{ \left( \theta_{xr} \right)} = -19.853 N]

3)

A third point charge [q_3] is now positioned halfway between [q_1] and [q_2]. The net force on [q_2] now has a magnitude of [F_{2,net} = 6.9 N] and points away from [q_1] and [q_3]. What is the value (sign and magnitude) of the charge [q_3]?

• [F_{ij} = \kappa \frac{ q_i q_j}{ r^2} \hat r]

  • [ \kappa = \frac{ 1}{ 4 \pi \epsilon_0} = 8.99 \times 10^9 \frac{ Nm^2}{ C^2} ]
  • [ \mu = 10^{-6}]

• Let:

  • [q_1 = q_i = -3.4 \mu C]
  • [q_2 = q_j = 6.7 \mu C ]
  • [x_2 = x = 9.5 cm = .095 m]
  • [y_2 = y = 2.9 cm = .029 m]
  • [r = \sqrt{ x^2 + y^2} = .09932774 m]
  • [\theta_{xr} = \theta = \arctan{ \left( \frac{ x}{ y} \right)} = 16.9755^\circ]

• [F_{ij,x} = \kappa \frac{ q_i q_j}{ r^2} \cos{ \left( \theta \right)} = -19.853 N]

• [F_{ij} = \kappa \frac{ q_i q_j}{ r^2} = -20.754 N]

• Let:

  • [q_3 = q_k = ? C ]
  • [F_{2,net} = F_{j,net} = 6.9 N ]

• [F_{kj} = \left| F_{j,net} - F_{ij} \right| = 27.6574 N ]

  • We subtract since [F_{ij}] points toward [q_i] and [F_{j,net}] points away from [q_i] and [q_j].

• [F_{kj} = \kappa \frac{ q_k q_j}{ \left( \frac{ r}{ 2} \right)^2} = 1.13255]

  • Solve for [q_k]

4)

How would you change [q_1] (keeping [q_2] and [q_3] fixed) in order to make the net force on [q_2] equal to zero?

• Increase its magnitude and change its sign
• Decrease its magnitude and change its sign
• Increase its magnitude and keep its sign the same
  • It need to conterbalance [F_{kj}] which is [F_{kj}> F_{ij]
• Decrease its magnitude and keep its sign the same
• There is no change you can make to q1 that will result in the fet force on q2 being equal to zero.

4)

How would you change [q_3] (keeping [q_1] and [q_2] fixed) in order to make the net force on [q_2] equal to zero?

• Increase its magnitude and change its sign
• Decrease its magnitude and change its sign
• Increase its magnitude and keep its sign the same
• Decrease its magnitude and keep its sign the same
  • It need to balance [F_{ij}] which is [F_{ij}< F_{kj]
• There is no change you can make to [q_3] that will result in the fet force on [q_2] being equal to zero.