Consider three charges arranged in a triangle as shown. + - + 0.266 m 0.599 m −3.4 nC 8.1 nC 4.8 nC y x What is the net electric force on the charge at the origin? The value of the Coulomb constant is 8.99 × 109 N · m2 /C 2 . Answer in units of N. 010 (part 2 of 4) 10.0 points What is the direction of this force (as an angle between −180◦ and +180◦ measured from the positive x-axis, with counterclockwise positive)? Answer in units of my work: 3.49913223E-6/9.74164509E-7=3.591931545
tan^-1(3.59193545)=74.44 degrees (wrong?)
The magnitudes of the forces exerted on the charge at the origin can be calculated like this:
F12 = k*|q1*q2|/r12^2
F13 = k*|q1*q3|/r13^2
where k is the Coulomb constant (8.99 × 10^9 N·m^2/C^2), q1 is the charge at the origin (-3.4 nC), q2 and q3 are the other charges (8.1 nC and 4.8 nC), and r12 and r13 are the distances between charges, respectively.
F12 = (8.99 × 10^9 N·m^2/C^2) * |(-3.4 × 10^(-9) C) * (8.1 × 10^(-9) C)| / (0.266 m)^2
F12 ≈ 3.20 × 10^(-8) N
F13 = (8.99 × 10^9 N·m^2/C^2) * |(-3.4 × 10^(-9) C) * (4.8 × 10^(-9) C)| / (0.599 m)^2
F13 ≈ 2.15 × 10^(-9) N
Now, we need to find the components of those forces in the x and y direction. Let's denote the angle between F12 and the x-axis as theta_12, and the angle between F13 and the x-axis as theta_13. From the geometry, we have:
theta_12 = atan(0.599 / 0.266)
theta_13 = 0
Therefore, the x and y components of the forces are:
F12x = F12 * cos(theta_12)
F12y = F12 * sin(theta_12)
F13x = F13 * cos(theta_13)
F13y = F13 * sin(theta_13)
Calculating those:
F12x ≈ 2.549 × 10^(-8) N
F12y ≈ 7.137 × 10^(-9) N
F13x ≈ 2.15 × 10^(-9) N
F13y = 0 N
The net force components can be calculated by adding the components of the two forces:
Fnet_x = F12x - F13x ≈ 2.33 × 10^(-8) N
Fnet_y = F12y = 7.137 × 10^(-9) N
Now we can find the magnitude and direction of the net force:
Fnet = sqrt(Fnet_x^2 + Fnet_y^2) ≈ 2.50 × 10^(-8) N
theta_net = atan(Fnet_y / Fnet_x)= atan(7.137 × 10^(-9) N / 2.33 × 10^(-8) N) = 16.98°
So the magnitude of the net electric force on the charge at the origin is about 2.50 × 10^(-8) N, and its direction is 16.98° counterclockwise from the positive x-axis.
To find the net electric force on the charge at the origin, we need to consider the forces exerted on it by the other charges in the triangle.
First, we need to find the magnitude of the forces exerted on the charge at the origin by the other charges. We can use Coulomb's law to do this. Coulomb's law states that the magnitude of the force between two charges is given by:
F = (k * |q1 * q2|) / r^2
Where F is the force, k is the Coulomb constant (8.99 × 10^9 N · m^2 / C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Let's calculate the magnitude of the forces:
Force between the charges at (0.266m, 0) and (0, 0):
F1 = (8.99 × 10^9 N · m^2 / C^2) * (8.1 nC * 4.8 nC) / (0.266m)^2
Force between the charges at (0, 0.599m) and (0, 0):
F2 = (8.99 × 10^9 N · m^2 / C^2) * (8.1 nC * 3.4 nC) / (0.599m)^2
Now, we can calculate the net force by adding these forces vectorially. Since we have a triangle configuration, the forces will have both magnitude and direction.
To find the direction of the net force, we can use vectors. The x-component of the net force is the sum of the x-components of the individual forces, and the y-component of the net force is the sum of the y-components of the individual forces. Then, we can find the angle (direction) using the arctangent function.
Let's calculate the net force and its direction:
Net force in the x-direction:
Net Fx = F1x + F2x = F1 * cosθ1 + F2 * cosθ2
Net force in the y-direction:
Net Fy = F1y + F2y = F1 * sinθ1 + F2 * sinθ2
where θ1 and θ2 are the angles made by the individual forces with respect to the positive x-axis.
Finally, we can find the magnitude (net F) and direction (angle) of the net force using the following formulas:
net F = sqrt(Net Fx^2 + Net Fy^2)
angle = atan(Net Fy / Net Fx)
By plugging in the calculated values of Net Fx, Net Fy, we can find the magnitude and direction of the net force.