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 = 0.115 m). The charges on the circle are -4.70 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°).
To solve this problem, we can use Coulomb's law, which states that 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.
Given:
Charge at the center, q1 = -3.00 µC
Charge on the circle at north, q2 = -4.70 µC
Charge on the circle at east, q3 = +5.00 µC
Radius of the circle, r = 0.115 m
To find the net electrostatic force on the charge at the center, we need to calculate the individual forces due to charges q2 and q3 and then combine them vectorially.
Step 1: Calculate the force between q1 and q2:
Use Coulomb's law:
F2 = k * |q1 * q2| / r^2
Where k is the electrostatic constant ( k = 9 × 10^9 N m^2 / C^2) and r is the distance between the charges at the center and north.
Plugging in the values:
F2 = (9 × 10^9 N m^2 / C^2) * |(-3.00 × 10^-6 C) * (-4.70 × 10^-6 C)| / (0.115 m)^2
Step 2: Calculate the force between q1 and q3:
Use Coulomb's law:
F3 = k * |q1 * q3| / r^2
where r is the distance between the charges at the center and east.
Plugging in the values:
F3 = (9 × 10^9 N m^2 / C^2) * |(-3.00 × 10^-6 C) * (5.00 × 10^-6 C)| / (0.115 m)^2
Step 3: Find the net force:
To find the net force, we need to combine the forces F2 and F3 vectorially. Since the forces are at 90 degrees to each other, we can use the Pythagorean theorem to find the magnitude of the net force.
Net force (F_net) = √(F2^2 + F3^2)
Step 4: Find the direction:
The direction of the net force can be found by calculating the angle between F_net and the positive x-axis (east direction). We will use trigonometry for this.
angle = tan^(-1)(F3 / F2)
So the solution would provide the magnitude of the net electrostatic force (F_net) and the direction relative to the east direction with 0°.
You can now substitute the values into the formulae and perform the calculations to find the final answers.