A charge of -3.00 µC is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass (radius = 0.130 m). The charges on the circle are -4.00 µC at the position due north and +5.00 µC at the position due east. What is the magnitude and direction of the net electrostatic force acting on the charge at the center? Specify the direction relative to due east (0°).
Magnitude
Direction
Use Coulomb's law to compute the force at the center due to each particle. The -3 uC will exert a repulsion force towards the "South" direction and the +5 uC will exert an attraction force in the East direction. Perform a vector addition of those two forces to get the magnitude and direction.
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To find the magnitude and direction of the net electrostatic force acting on the charge at the center, we can use Coulomb's Law. Coulomb's Law 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.
First, let's find the individual forces acting on the charge at the center due to the charges on the circle. Since the charge at the center is negative and the charges on the circle are positive, the forces will be attractive.
The force due to the -4.00 µC charge at the position due north can be calculated using Coulomb's Law:
F1 = (k * q1 * q2) / r^2
where k is the electrostatic constant (9 x 10^9 N•m^2/C^2), q1 is the charge at the center (-3.00 µC), q2 is the charge on the circle (-4.00 µC), and r is the distance between them (0.130 m).
F1 = (9 x 10^9 N•m^2/C^2) * (-3.00 x 10^-6 C) * (-4.00 x 10^-6 C) / (0.130 m)^2
Similarly, the force due to the +5.00 µC charge at the position due east can be calculated using Coulomb's Law:
F2 = (k * q1 * q2) / r^2
F2 = (9 x 10^9 N•m^2/C^2) * (-3.00 x 10^-6 C) * (5.00 x 10^-6 C) / (0.130 m)^2
Now, we can calculate the magnitude and direction of the net force using vector addition. Since the forces are at right angles to each other (due north and due east), we can use the Pythagorean theorem and trigonometry to find the resultant force.
Magnitude:
The magnitude of the net force, F_net, is given by:
F_net = √(F1^2 + F2^2)
Direction:
The direction of the net force relative to due east is given by the angle, θ, between the net force and the due east direction.
θ = atan(F2 / F1)
Now, we can substitute the known values and calculate the magnitude and direction of the net electrostatic force acting on the charge at the center.