1. In a region of two-dimensional space, there are three fixed charges: +1 mC at (0, 0), −2 mC at (18 mm, -3 mm), and +3 mC at (-8 mm, 12 mm). What is the net force on the −2-mC charge?
Find the magnitude and direction (counterclockwise from the x-axis).
To find the net force on the -2 mC charge, we need to consider the forces exerted on it by the other charges.
The force between two charges can be calculated using Coulomb's Law, which states that the force (F) between two charges (q1 and q2) is given by the equation:
F = k * (|q1| * |q2|) / r^2
where k is the electrostatic constant (k ≈ 9 × 10^9 N m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
To find the magnitude of the force between two charges, we can plug in the given values into Coulomb's Law. Let's first calculate the force between the -2 mC charge and the +1 mC charge at (0, 0).
F1 = k * ((|-2 mC| * |+ 1 mC|) / r^2
Since the distance between the two charges is not given, we can assume it is negligible (i.e., both charges are assumed to be very close to each other). Therefore, r ≈ 0, and the force between them can be considered infinite, pointing directly towards the +1 mC charge.
Next, let's calculate the force between the -2 mC charge and the +3 mC charge at (-8 mm, 12 mm). To do this, we need to calculate the distance between the charges.
Using the distance formula between two points (x1, y1) and (x2, y2):
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((-8 mm - 18 mm)^2 + (12 mm - -3 mm)^2)
= sqrt((-26 mm)^2 + (15 mm)^2)
≈ sqrt(676 mm^2 + 225 mm^2)
≈ sqrt(901 mm^2)
≈ 30.03 mm
Now we can calculate the force between the -2 mC charge and the +3 mC charge.
F2 = k * ((|-2 mC| * |+ 3 mC|) / r^2
F2 = (9 × 10^9 N m^2/C^2) * ((2 × 10^-6 C) * (3 × 10^-6 C) / (30.03 mm)^2)
F2 ≈ 18 N
The magnitude of the net force on the -2 mC charge is equal to the sum of F1 and F2:
F_net = |F1| + |F2|
= ∞ + 18 N
= ∞
Here, ∞ represents an infinite force due to the negligible distance between the -2 mC and +1 mC charges. This infinite net force means that the -2 mC charge will experience a significant force towards the +1 mC charge.
Regarding the direction of the net force, it points directly towards the +1 mC charge, which is in the positive x-direction. Therefore, the direction of the net force is counterclockwise from the x-axis.
the answer:
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