Three charges lie along the x-axis. The positive charge, q1 = 15 µC, is at x= 2.0 m, and The positive charge, q2 = 6.0 μC, is at the origin. At what point on the x-axis must a negative charge, q3 , be placed so that the net force acting on it is zero?
To find the point on the x-axis where a negative charge, q3, must be placed so that the net force acting on it is zero, we need to calculate the electric forces exerted on q3 by q1 and q2 and then determine the position where the net force is zero.
Let's first determine the direction of the net force that q1 and q2 exert on q3. Since both q1 and q2 are positive charges, they will repel q3, causing the net force to be in the positive x-direction.
Now, let's calculate the electric force exerted on q3 by q1 and q2 separately using Coulomb's Law:
The electric force exerted by q1 on q3 is given by:
F1 = k * (q1 * q3) / (r1^2)
The electric force exerted by q2 on q3 is given by:
F2 = k * (q2 * q3) / (r2^2)
where:
k = 8.99 × 10^9 N m^2/C^2 (Coulomb's constant)
q1 = 15 µC = 15 × 10^-6 C (charge of q1)
q2 = 6.0 μC = 6.0 × 10^-6 C (charge of q2)
q3 = unknown charge of q3 (negative charge)
r1 = distance between q1 and q3 (distance from q3 to x=2.0m)
r2 = distance between q2 and q3 (distance from q3 to x=0m)
Since we want the net force on q3 to be zero, F1 and F2 must have equal magnitudes. So, we can set up the equation:
F1 = F2
(k * (q1 * q3) / (r1^2)) = (k * (q2 * q3) / (r2^2))
Now, we can substitute the given values into the equation and solve for r1:
(k * (15 × 10^-6 C * q3) / (r1^2)) = (k * (6.0 × 10^-6 C * q3) / (0^2))
Simplifying the equation:
(r1^2 / r2^2) = (6.0 × 10^-6 C / 15 × 10^-6 C)
(r1^2 / 2^2) = (6.0 / 15)
r1^2 = (4/15) * 2^2
r1^2 = (4/15) * 4
r1^2 = (16/15)
Taking the square root of both sides to solve for r1:
r1 = √(16/15)
r1 ≈ 1.155 m
Therefore, the negative charge, q3, must be placed at x ≈ 1.155 m on the x-axis so that the net force acting on it is zero.
To find the position on the x-axis where the net force acting on the negative charge q3 is zero, we need to calculate the force exerted on q3 by both q1 and q2.
The force between two charges is given by Coulomb's Law:
F = k * (q1 * q3) / r1^2, where k is the electrostatic constant, q1 and q3 are the magnitudes of the respective charges, and r1 is the distance between them.
Similarly, the force between q2 and q3 is:
F = k * (q2 * q3) / r2^2, where r2 is the distance between q2 and q3.
Since the net force acting on q3 must be zero, the forces exerted by q1 and q2 must be equal in magnitude and opposite in direction:
k * (q1 * q3) / r1^2 = k * (q2 * q3) / r2^2
Cancelling out the constants and rearranging the equation, we get:
(q1 * q3) / r1^2 = (q2 * q3) / r2^2
Now, let's substitute the given values:
(15 µC * q3) / (2.0 m)^2 = (6.0 µC * q3) / r2^2
Simplifying the equation further:
(15 * 10^-6 C * q3) / (4.0 m^2) = (6.0 * 10^-6 C * q3) / r2^2
Now, we can cross-multiply and solve for r2^2:
(15 * 10^-6 C * q3 * r2^2) = (6.0 * 10^-6 C * q3 * (4.0 m)^2)
Dividing both sides by (15 * 10^-6 C * q3), we get:
r2^2 = (6.0 * 10^-6 C * (4.0 m)^2) / (15 * 10^-6 C)
Simplifying further:
r2^2 = (6.0 * 10^-6 C * 16.0 m^2) / (15 * 10^-6 C)
Canceling out the C terms gives:
r2^2 = 9.6 m^2
Finally, taking the square root of both sides, we find:
r2 = √(9.6 m^2)
Therefore, the distance r2 between q2 and q3 is approximately 3.10 m.
6 d^2 = 15 (20-d)^2 = 15 (400 - 40 d + d^2)
9 d^2 - 600 d + 6000 = 0
d is the distance in decimeters