A charge q1 of -5.00*10^-9C and a charge q2 of -2.00*10^-9C are separated by a distance of 40.0cm. Find the equilibrium position for a third charge of 15.0*10^-9C
To find the equilibrium position for the third charge, we need to consider the forces acting on it due to the other two charges.
The force between two charges can be calculated using Coulomb's Law:
F = (k * |q1| * |q2|) / r^2
Where:
- F is the electrostatic force between the charges
- k is the Coulomb's constant, approximately 8.99 * 10^9 Nm^2/C^2
- |q1| and |q2| are the magnitudes of the charges
- r is the distance between the charges
Now, since we want to find the equilibrium position for the third charge, the forces acting on it from the other two charges should cancel out. This means that the magnitudes of the forces should be equal, but opposite in direction.
Let's consider the force acting on the third charge due to q1:
F1 = (k * |q1| * |q3|) / r1^2
And the force acting on the third charge due to q2:
F2 = (k * |q2| * |q3|) / r2^2
For equilibrium, F1 = -F2, since the forces should be opposite in direction. Therefore:
(k * |q1| * |q3|) / r1^2 = -(k * |q2| * |q3|) / r2^2
Now we can substitute the given values into the equation:
(8.99 * 10^9 Nm^2/C^2 * 5.00 * 10^-9C * 15.0 * 10^-9C) / r1^2 = - (8.99 * 10^9 Nm^2/C^2 * 2.00 * 10^-9C * 15.0 * 10^-9C) / r2^2
Simplifying the equation:
5.00 * 15.0 * r2^2 = 2.00 * 15.0 * r1^2
Dividing both sides by 15.0:
r2^2 = (2.00 / 5.00) * r1^2
Taking the square root of both sides:
r2 = sqrt((2.00 / 5.00) * r1^2)
Now we can substitute the given distance (40.0 cm) for r1:
r2 = sqrt((2.00 / 5.00) * (0.40 m)^2)
Evaluating the expression:
r2 = sqrt((2.00 / 5.00) * 0.16 m^2)
r2 = sqrt(0.064 m^2)
r2 ≈ 0.25 m
Therefore, the equilibrium position for the third charge is approximately 0.25 meters away from the other two charges in the direction away from q1.