Two point charges, q1 and q2, of 4.00 uc each, are placed -6.0 cm and 6.0 cm away from the origin on the x-axis. A charge q3 of -1.00 uC is placed 8.0 cm away from the origin on the y-axis.

Question

a. Find the distance from q3 to q1 and from q3 to q2.
b, Find the magnitude and the direction of the force F13 exerted by q1 on q3.
c. Find the magnitude and the direction of the force F23 exerted by q2 on q3.
d. Find the magnitude and the direction of the force F12 exerted by q1 on q2.
e. In the space below, sketch the vectors representing forcesF13 and F23.
f Find he angle between the q1-q3 line and the x-axis.

Answer 1

a. The distances from q3 to q1 and q3 to q2 can be found using the Pythagorean theorem:

For q3 to q1: The x-coordinate is (-6.0 cm) and the y-coordinate is (8.0 cm).
r13 = sqrt((-6.0 cm)^2 + (8.0 cm)^2) = 10.0 cm

For q3 to q2: The x-coordinate is (6.0 cm) and the y-coordinate is still (8.0 cm).
r23 = sqrt((6.0 cm)^2 + (8.0 cm)^2) = 10.0 cm

b. The magnitude of force F13 can be found using Coulomb's law as:
F13 = k * |q1 * q3| / r13^2, where k is the Coulomb's constant (k ≈ 8.9875 x 10^9 N * m^2 / C^2)
F13 = (8.9875 x 10^9 N * m^2 / C^2) * (4.00 x 10^-6 C * 1.00 x 10^-6 C) / (10.0 x 10^-2 m)^2
F13 ≈ 0.3595 N

The direction can be found by determining the angle between the x-axis and the q1-q3 line:
tan(θ13) = (8.0 cm) / (-6.0 cm)
θ13 = arctan(-4/3)
θ13 ≈ -53.13 degrees

c. The magnitude of force F23 can be found in the same way as F13:
F23 = k * |q2 * q3| / r23^2
F23 = (8.9875 x 10^9 N * m^2 / C^2) * (4.00 x 10^-6 C * 1.00 x 10^-6 C) / (10.0 x 10^-2 m)^2
F23 ≈ 0.3595 N

The direction can also be found in the same way:
tan(θ23) = (8.0 cm) / (6.0 cm)
θ23 = arctan(4/3)
θ23 ≈ 53.13 degrees

d. The magnitude of force F12 can be found as:
F12 = k * |q1 * q2| / r12^2, where r12 is the distance between q1 and q2 which equals 12.0 cm
F12 = (8.9875 x 10^9 N * m^2 / C^2) * (4.00 x 10^-6 C * 4.00 x 10^-6 C) / (12.0 x 10^-2 m)^2
F12 ≈ 0.1997 N

The direction is along the x-axis, so it's either 0 degrees or 180 degrees. Since q1 and q2 are both positive, they repel each other, so the direction is 180 degrees (opposite directions along the x-axis).

e. For F13, the vector starts at q3 and points towards q1, making an angle of -53.13 degrees with the x-axis. For F23, the vector also starts at q3 and points towards q2, making an angle of 53.13 degrees with the x-axis.

f. The angle between the q1-q3 line and the x-axis is the same as the direction angle for F13 in part b, which is -53.13 degrees.