Three point charges are located at the corners of an equilateral triangle as in the figure below. Find the magnitude and direction of the net electric force on the 0.70 µC charge. (A = 0.70 µC, B = 6.50 µC, and C = -3.90 µC.)
To find the net electric force on the 0.70 µC charge, we need to calculate the individual electric forces exerted by each of the three charges and then find their vector sum.
The electric force between two charges is given by Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
where F is the force, k is the electrostatic constant (8.99 x 10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between them.
Let's label the charges as q1 = 0.70 µC, q2 = 6.50 µC, and q3 = -3.90 µC.
1. Calculate the electric force between q1 and q2:
F12 = k * (|q1| * |q2|) / r^2
The charges q1 and q2 are at the corners of an equilateral triangle, and the distance between them is the length of one side of the triangle.
Using the Pythagorean theorem, we can find that the distance between q1 and q2 is given by:
r = √(a^2 + b^2)
where a is the side length of the triangle.
Since the triangle is equilateral, all sides are equal. Let's denote the side length as 's'.
Using trigonometry, we can find that a = s and b = s * √3 / 2.
Substituting these values, we get:
r = √((s)^2 + (s * √3 / 2)^2)
r = √(s^2 + 3s^2/4)
r = √(7s^2/4)
r = (s/2) √7
Now we can substitute the values into Coulomb's Law to find F12:
F12 = k * (|q1| * |q2|) / ((s/2) √7)^2
2. Calculate the electric force between q1 and q3:
The charges q1 and q3 are also at the corners of an equilateral triangle, so the distance between them is the same as before: r = (s/2) √7.
Using Coulomb's Law, we can find F13:
F13 = k * (|q1| * |q3|) / ((s/2) √7)^2
3. Calculate the electric force between q2 and q3:
The charges q2 and q3 are not in an equilateral triangle configuration, but rather a linear configuration. The electric force will act along the line connecting q2 and q3.
The distance between q2 and q3 can be found using the Pythagorean theorem:
r = √(a^2 + b^2)
where a is the distance between q1 and q3 (which we found to be (s/2) √7) and b is the distance between q1 and q2 (which is also (s/2) √7).
Substituting the values, we get:
r = √(((s/2) √7)^2 + ((s/2) √7)^2)
r = √(7s^2/4 + 7s^2/4)
r = √(7s^2/2)
r = (√7/√2) s
Using Coulomb's Law, we can find F23:
F23 = k * (|q2| * |q3|) / ((√7/√2) s)^2
Finally, we can calculate the magnitudes and directions of the net electric force on the 0.70 µC charge by summing the forces vectorially.