Three point charges are arranged along the x-axis. Charge q1 = +2.75 µC is at the origin, and charge q2 = -5.25 µC is at x = 0.220 m. Charge q3 = -6.50 µC. Where is q3 located if the net force on q1 is 7.50 N in the −x-direction?
To determine the location of q3, we can use Coulomb's law and the principle of superposition. Coulomb's law states that the force between two point charges is given by:
F = (k * |q1 * q2|) / r^2
where,
F is the force between the charges,
k is the electrostatic constant (k = 8.99 * 10^9 Nm^2/C^2),
q1 and q2 are the magnitudes of the charges,
r is the distance between the charges.
Applying the principle of superposition, the net force on q1 is the vector sum of the forces exerted by q2 and q3:
Fnet = F1+2 + F1+3
where,
Fnet is the net force on q1,
F1+2 is the force between q1 and q2,
F1+3 is the force between q1 and q3.
Since the net force on q1 is given as 7.50 N in the -x direction, we can assume that the force between q1 and q2 (F1+2) is in the +x direction. Therefore, the force between q1 and q3 (F1+3) must be in the -x direction and have a magnitude of 7.50 N.
Now let's calculate the forces:
F1+2 = (k * |q1 * q2|) / r1+2^2
F1+3 = (k * |q1 * q3|) / r1+3^2
Substituting the values:
F1+2 = (8.99 * 10^9 Nm^2/C^2 * (2.75 * 10^-6 C) * (-5.25 * 10^-6 C)) / (0.220 m)^2
F1+3 = 7.50 N
Solving for the distances:
From F1+3 = (8.99 * 10^9 Nm^2/C^2 * (2.75 * 10^-6 C) * (-6.50 * 10^-6 C)) / r1+3^2
we can find r1+3.
Now, we rearrange the equation to solve for r1+3:
r1+3^2 = (8.99 * 10^9 Nm^2/C^2 * (2.75 * 10^-6 C) * (-6.50 * 10^-6 C)) / F1+3
r1+3 = sqrt((8.99 * 10^9 Nm^2/C^2 * (2.75 * 10^-6 C) * (-6.50 * 10^-6 C)) / F1+3)
Calculating this expression will give you the distance r1+3, which represents the location of q3.