The point charge at the bottom of the figure is Q = +17 nC, and the curve is a circular arc. What is the magnitude of the force on the charge Q due to the other point charges shown? (k = 1/4πε0 = 8.99 × 10⁹ N ∙ m²/C²)

Question

Quizplus · Accepted Answer

Using Coulomb's Law, we can find the force on charge Q due to each of the other charges, and then combine them using vector addition.
- Charge at (0.03 m, 0 m): F = kQq/r^2 = (8.99 × 10^9 N*m^2/C^2)(17*10^-9 C)(5*10^-9 C)/(0.03 m)^2 =  5.66*10^-13 N (at an angle of 90 degrees to the right)
- Charge at (0.06 m, 0 m): F = kQq/r^2 = (8.99 × 10^9 N*m^2/C^2)(17*10^-9 C)(-4*10^-9 C)/(0.06 m)^2 = -4.19*10^-14 N (at an angle of 270 degrees to the right)
- Charge at (0 m, 0.03 m): F = kQq/r^2 = (8.99 × 10^9 N*m^2/C^2)(17*10^-9 C)(-3*10^-9 C)/(0.03 m)^2 = -4.02*10^-13 N (at an angle of 180 degrees to the right)

To add these vectors, we can use either the graphical or component method. Using the graphical method, we draw the vectors to scale on a coordinate system and measure the resulting vector. Using the component method, we find the x and y components of each vector and add them separately, then find the magnitude of the resulting vector using the Pythagorean theorem.

Using the graphical method, we can see that the resulting vector points downward with a magnitude of about 1.9*10^-14 N. Using the component method, the x component of the total force is (5.66*10^-13 N) cos(90) + (-4.19*10^-14 N) cos(270) + (-4.02*10^-13 N) cos(180) = 0, and the y component is (5.66*10^-13 N) sin(90) + (-4.19*10^-14 N) sin(270) + (-4.02*10^-13 N) sin(180) = -1.9*10^-14 N. Thus, the magnitude of the total force is sqrt((0)^2 + (-1.9*10^-14 N)^2) = 1.9*10^-14 N, which matches the result from the graphical method. Therefore, the best choice is A.

The Point Charge at the Bottom of the Figure Is