A)When point charges q1 = +1.5 ìC and q2 = +5.7 ìC are brought near each other, each experiences a repulsive force of magnitude 0.77 N. Determine the distance between the charges.
B)Two point charges are fixed on the y axis: a negative point charge q1 = -29 ìC at y1 = +0.21 m and a positive point charge q2 at y2 = +0.35 m. A third point charge q = +9.1 ìC is fixed at the origin. The net electrostatic force exerted on the charge q by the other two charges has a magnitude of 21 N and points in the +y direction. Determine the magnitude of q2.
A) To determine the distance between the charges, we can use Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
where F is the force between the charges, 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 the charges.
Since both charges experience a repulsive force of magnitude 0.77 N, we can equate the magnitudes of the forces:
0.77 N = k * (|q1| * |q2|) / r^2
Substituting the given values for the charges, q1 = +1.5 μC and q2 = +5.7 μC, and solving for r:
0.77 N = (8.99 x 10^9 N*m^2/C^2) * ((1.5 x 10^-6 C) * (5.7 x 10^-6 C)) / r^2
Simplifying the equation:
r^2 = ((1.5 x 10^-6 C) * (5.7 x 10^-6 C)) / (0.77 N / (8.99 x 10^9 N*m^2/C^2))
r^2 = (8.55 x 10^-12 C^2) / (8.56 x 10^-2 m^2/C^2)
r^2 = 9.99 x 10^-11 m^2
Taking the square root of both sides:
r = √(9.99 x 10^-11 m^2)
r ≈ 3.16 x 10^-6 m
Therefore, the distance between the charges is approximately 3.16 x 10^-6 meters.
B) To determine the magnitude of q2, we can apply the principle of superposition to calculate the net force on q due to q1 and q2 separately, and then find the magnitude of q2.
The net electrostatic force between point charges q and q1 or q2 is given by Coulomb's Law:
F = k * (|q1| * |q|) / r1^2 (force due to q1)
F = k * (|q2| * |q|) / r2^2 (force due to q2)
where F is the force between the charges, k is the electrostatic constant (8.99 x 10^9 N*m^2/C^2), |q1|, |q2|, and |q| are the magnitudes of the charges, and r1 and r2 are the distances between the charges.
Given that the net force on q due to q1 and q2 has a magnitude of 21 N and points in the +y direction, we can write the equation for the net force:
√(Fq1^2 + Fq2^2) = 21 N
Plugging in the values, |q1| = 29 μC, |q| = 9.1 μC, r1 = 0.21 m, r2 = 0.35 m:
√((8.99 x 10^9 N*m^2/C^2 * (29 x 10^-6 C) * (9.1 x 10^-6 C))^2 + (8.99 x 10^9 N*m^2/C^2 * (q^2) * (9.1 x 10^-6 C))^2) = 21 N
Simplifying the equation:
√(2.721 * 10^-3 + ((8.99 x 10^9 N*m^2/C^2 * q^2 * 9.1 x 10^-6 C))^2) = 21 N
√(2.721 * 10^-3 + (8.783 x 10^-3 * q^2)^2) = 21 N
Simplifying further:
2.721 * 10^-3 + (8.783 x 10^-3 * q^2)^2 = 441 N^2
(8.783 x 10^-3 * q^2)^2 = 441 - 2.721 * 10^-3
(8.783 x 10^-3 * q^2)^2 ≈ 440.997
Taking the square root of both sides:
8.783 x 10^-3 * q^2 ≈ √440.997
q^2 ≈ (√440.997) / (8.783 x 10^-3)
q^2 ≈ 15 C
Taking the square root of both sides:
q ≈ √15 C
Therefore, the magnitude of q2 is approximately √15 C.