Three charges are fixed to an x, y coordinate system. A charge of +12 µC is on the y axis at y = +3.0 m. A charge of -14 µC is at the origin. Lastly, a charge of +50 µC is on the x axis at x = +3.0 m. Determine the magnitude and direction of the net electrostatic force on the charge at x = +3.0 m. Specify the direction relative to the -x axis.
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To determine the net electrostatic force on the charge at x = +3.0 m below the x-axis, we need to consider the individual forces exerted by each charge and then calculate their vector sum.
Let's start by finding the force exerted by the charge of +12 µC on the charge at x = +3.0 m below the x-axis. The force between two charges can be calculated using Coulomb's Law:
F1 = k * (q1 * q2) / r^2
Here, k is the electrostatic constant (9 × 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
The charge of +12 µC exerts a force on the charge at x = +3.0 m below the x-axis along the y-axis. We can find the y-component of this force as follows:
F1y = F1 * sin(theta)
Here, theta is the angle between the force and the -x axis. Since the charge at x = +3.0 m is below the x-axis, the angle will be 90 degrees.
F1y = F1 * sin(90)
Next, let's find the force exerted by the charge of +50 µC on the charge at x = +3.0 m below the x-axis. Again, we can use Coulomb's Law to calculate the force:
F2 = k * (q1 * q2) / r^2
In this case, the charge of +50 µC exerts a force on the charge at x = +3.0 m below the x-axis along the positive x-axis. The angle between this force and the -x axis will be 180 degrees.
F2x = F2 * cos(theta)
F2x = F2 * cos(180)
Now, we can calculate the force exerted by the charge of -14 µC at the origin on the charge at x = +3.0 m below the x-axis. Using Coulomb's Law, we have:
F3 = k * (q1 * q2) / r^2
In this case, the charge of -14 µC exerts a force on the charge at x = +3.0 m below the x-axis along the negative x-axis. The angle between this force and the -x axis will be 0 degrees.
F3x = F3 * cos(theta)
F3x = F3 * cos(0)
Once we have calculated the individual forces, we can find the net force by adding them together:
Fnet = sqrt((F1y)^2 + (F2x + F3x)^2)
Finally, we can complete the calculations using the given values for the charges and their positions. Substitute the values into the equations and calculate the force to find its magnitude and direction relative to the -x axis.