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.
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.
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.
To solve this problem, let's break it down step-by-step:
a. To find the distance from q3 to q1 and from q3 to q2, we can use the Pythagorean theorem.
For q3 to q1:
Distance = sqrt((distance along x-axis)^2 + (distance along y-axis)^2)
Distance = sqrt((-6.0 cm)^2 + (8.0 cm)^2)
Distance = sqrt(36.0 cm^2 + 64.0 cm^2)
Distance = sqrt(100.0 cm^2)
Distance = 10.0 cm
For q3 to q2:
Distance = sqrt((distance along x-axis)^2 + (distance along y-axis)^2)
Distance = sqrt((6.0 cm)^2 + (8.0 cm)^2)
Distance = sqrt(36.0 cm^2 + 64.0 cm^2)
Distance = sqrt(100.0 cm^2)
Distance = 10.0 cm
b. To find the magnitude and direction of the force F13 exerted by q1 on q3, we can use Coulomb's law.
Magnitdue of the force F13 = (k * |q1| * |q3|) / distance^2
k is the Coulomb's constant = 8.99 * 10^9 N m^2 / C^2
|q1| and |q3| are the magnitudes of the charges = 4.00 uC and 1.00 uC respectively
distance is the distance between q1 and q3 = 10.0 cm (from part a)
Magnitude of F13 = (8.99 * 10^9 N m^2 / C^2) * (4.00 * 10^-6 C) * (1.00 * 10^-6 C) / (10.0 cm)^2
Magnitude of F13 = 3.59 * 10^-2 N
The direction of the force F13 can be found using the right-hand rule or by considering the signs of the charges. Since q1 is positive and q3 is negative, the force will be attractive, pulling q3 towards q1.
c. To find the magnitude and direction of the force F23 exerted by q2 on q3, we use the same equations as in part b, but with the charges and distances adjusted accordingly.
Magnitude of F23 = (8.99 * 10^9 N m^2 / C^2) * (4.00 * 10^-6 C) * (1.00 * 10^-6 C) / (10.0 cm)^2
Magnitude of F23 = 3.59 * 10^-2 N
Since q2 is also positive and q3 is negative, the force F23 will also be attractive, pulling q3 towards q2.
d. To find the magnitude and direction of the force F12 exerted by q1 on q2, we can use Coulomb's law again.
Magnitude of F12 = (k * |q1| * |q2|) / distance^2
k is the Coulomb's constant = 8.99 * 10^9 N m^2 / C^2
|q1| and |q2| are the magnitudes of the charges = 4.00 uC each
distance is the distance between q1 and q2 = 12.0 cm
Magnitude of F12 = (8.99 * 10^9 N m^2 / C^2) * (4.00 * 10^-6 C) * (4.00 * 10^-6 C) / (12.0 cm)^2
Magnitude of F12 = 2.77 * 10^-2 N
Since both q1 and q2 have positive charges, the force F12 will be repulsive, pushing q1 away from q2.
e. Sketch the vectors representing forces F13 and F23.
(Unfortunately, as a text-based AI, I am unable to create visual sketches. However, you can imagine two vectors originating at q3 - one pointing towards q1 representing F13 and one pointing towards q2 representing F23.)
f. To find the angle between the q1-q3 line and the x-axis, we can use trigonometry.
tan(angle) = (y-component of q1-q3 line) / (x-component of q1-q3 line)
The x-component of the q1-q3 line is -6.0 cm (negative because it's on the left side of the origin)
The y-component of the q1-q3 line is 8.0 cm
angle = arctan(8.0 cm / -6.0 cm)
angle = -53.13 degrees
Therefore, the angle between the q1-q3 line and the x-axis is approximately -53.13 degrees.
To solve this problem, we can use Coulomb's Law, which states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
a. To find the distance from q3 to q1 and q2, we can use the Pythagorean theorem:
Distance from q3 to q1 (d13) = sqrt((distance from origin to q1)^2 + (distance from origin to q3)^2)
= sqrt((-6.0 cm)^2 + (8.0 cm)^2)
Distance from q3 to q2 (d23) = sqrt((distance from origin to q2)^2 + (distance from origin to q3)^2)
= sqrt((6.0 cm)^2 + (8.0 cm)^2)
b. To find the magnitude and direction of the force F13 exerted by q1 on q3, we can use Coulomb's Law:
Force F13 = (k * |q1| * |q3|) / (d13^2)
= (9.0 x 10^9 Nm^2/C^2 * (4.00 x 10^-6 C) * (1.00 x 10^-6 C)) / (d13^2)
where k is the electrostatic constant (9.0 x 10^9 Nm^2/C^2).
To find the direction of the force, we can consider that like charges repel each other, so the force will be directed away from q1 towards q3.
c. To find the magnitude and direction of the force F23 exerted by q2 on q3, we can follow the same steps as in part b, using the distance d23:
Force F23 = (k * |q2| * |q3|) / (d23^2)
Again, the force will be directed away from q2 towards q3.
d. To find the magnitude and direction of the force F12 exerted by q1 on q2, we can use Coulomb's Law:
Force F12 = (k * |q1| * |q2|) / (distance between q1 and q2)^2
= (9.0 x 10^9 Nm^2/C^2 * (4.00 x 10^-6 C) * (4.00 x 10^-6 C)) / ((6.0 cm + 6.0 cm)^2)
Once again, the force will be repulsive, pushing q2 away from q1.
e. In the space below, sketch the vectors representing forces F13 and F23.
[Description: Draw two vectors representing the forces F13 and F23, pointing away from q1 and q2 respectively, towards q3.]
f. To find the angle between the q1-q3 line and the x-axis, we can use trigonometry. Let's denote this angle as θ.
tan(θ) = opposite/adjacent = (distance from origin to q3) / (distance from origin to q1)
By calculating the arctan of this ratio, we can find θ.