Given the arrangement of charged particles in the figure below, find the net electrostatic force on the q2 = 12.33-µC charged particle. (Assume q1 = 5.15 µC and q3 = −15.18 µC. Express your answer in vector form.) The figure is a xy coordinate measure in cm. q1 at the point (-2cm.0), q2 at the point (1cm,1cm and q3 at the point (0,-1cm).
To find the net electrostatic force on q2, we need to consider the forces exerted by q1 and q3 on q2.
We can use Coulomb's Law to calculate the force between charged particles:
F = (k * |q1| * |q2|) / r^2
Where:
- F is the force between the particles
- k is the Coulomb constant, approximately equal to 9 x 10^9 Nm^2/C^2
- |q1| and |q2| are the magnitudes of the charges
- r is the distance between the particles
Let's calculate the force exerted on q2 by q1:
|q1| = 5.15 µC = 5.15 x 10^-6 C
|q2| = 12.33 µC = 12.33 x 10^-6 C
r1 = distance between q2 and q1 = [(1 cm - (-2 cm))^2 + (1 cm - 0 cm)^2]^0.5 = (9 + 1)^0.5 = 10^0.5 cm
F1 = (k * |q1| * |q2|) / r1^2
Now, let's calculate the force exerted on q2 by q3:
|q3| = 15.18 µC = 15.18 x 10^-6 C
r2 = distance between q2 and q3 = [(1 cm - 0 cm)^2 + (1 cm - (-1 cm))^2]^0.5 = (1 + 4)^0.5 = 5^0.5 cm
F2 = (k * |q2| * |q3|) / r2^2
Finally, the net force on q2 is the vector sum of F1 and F2:
Net force = F1 + F2
To express the answer in vector form, we need to calculate the x and y components of the forces.
The x-component of F1 can be found using cosine:
F1x = F1 * cos(theta1)
where theta1 is the angle between the x-axis and the line connecting q1 and q2.
Similarly, the y-component of F1 can be found using sine:
F1y = F1 * sin(theta1)
The x-component of F2 can be found using cosine:
F2x = F2 * cos(theta2)
where theta2 is the angle between the x-axis and the line connecting q2 and q3.
Similarly, the y-component of F2 can be found using sine:
F2y = F2 * sin(theta2)
Finally, the net force on q2 in vector form is:
Net Force = (F1x + F2x) i + (F1y + F2y) j
Let's calculate the values.
To find the net electrostatic force on the charged particle q2, we need to calculate the individual forces between q2 and q1 as well as q2 and q3.
The electrostatic force between two charged particles, q1 and q2, is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
where F is the force between the charges, k is Coulomb's constant (approximately 9 x 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 first calculate the force between q1 and q2.
Distance (r) between q1 and q2 can be found using the distance formula:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Given that q1 is at (-2 cm, 0) and q2 is at (1 cm, 1 cm), we can substitute these values into the distance formula:
r = sqrt((1 cm - (-2 cm))^2 + (1 cm - 0)^2)
= sqrt(3^2 + 1^2)
= sqrt(9 + 1)
= sqrt(10) cm
Now we can calculate the force between q1 and q2:
F1-2 = (k * q1 * q2) / r^2
= (9 x 10^9 N m^2/C^2) * (5.15 x 10^-6 C) * (12.33 x 10^-6 C) / (sqrt(10) cm)^2
Calculating this expression will give us the magnitude of the force between q1 and q2.
Next, let's calculate the force between q2 and q3.
Using the same distance formula, we can find the distance (r) between q2 and q3.
r = sqrt((x2 - x3)^2 + (y2 - y3)^2)
= sqrt((1 cm - 0 cm)^2 + (1 cm - (-1 cm))^2)
= sqrt(1^2 + 2^2)
= sqrt(1 + 4)
= sqrt(5) cm
Now, we can calculate the force between q2 and q3:
F2-3 = (k * q2 * q3) / r^2
= (9 x 10^9 N m^2/C^2) * (12.33 x 10^-6 C) * (-15.18 x 10^-6 C) / (sqrt(5) cm)^2
Calculating this expression will give us the magnitude of the force between q2 and q3.
Finally, to find the net electrostatic force on q2, we need to calculate the vector sum of the two forces, F1-2 and F2-3.
Since the charges q1 and q3 have opposite signs, their forces will have opposite directions. We can define the positive x-axis as pointing towards the right and the positive y-axis as pointing upwards. The force F1-2 on q2 will have some x and y-components, and the force F2-3 will have some x and y-components as well. The net force can be found by summing these components separately.
So, the net force on q2 can be expressed in vector form as:
F_net = F1-2_x + F2-3_x î + F1-2_y + F2-3_y ĵ
where F1-2_x and F1-2_y are the x and y-components of the force F1-2, and F2-3_x and F2-3_y are the x and y-components of the force F2-3.