A charge of 2.4 nC and a charge of 5.0 nC are separated by 33.30 cm. Find the equilibrium position for a -4.1 nC charge.
The distance to the 5.0 nC charge must be sqrt(5.0/2.4)= 1.4434 times the distance to the 2.4 nC charge. That will make the q/r^2 terms equal for both charges. The charge of the object at the equilibrium position does not matter.
33.30 cm, divided in a 1:1.4434 ratio, would put the equilbrium position at 13.63 cm from the smaller charge, and 19.67 cm from the larger charge.
To find the equilibrium position for a -4.1 nC charge, we need to analyze the forces acting on it due to the other two charges. The equilibrium position is the point where the net force on the -4.1 nC charge is zero.
The force between two charges is given by Coulomb's Law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them:
F = k * (|q1| * |q2|) / r^2
Where F is the force, k is the electrostatic constant (k = 9 * 10^9 Nm^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between them. The direction of the force depends on the sign of the charges.
Let's calculate the forces between the charges and the -4.1 nC charge.
1. Force due to the 2.4 nC charge:
F1 = k * (|-4.1nC| * |2.4nC|) / (0.3330m)^2
2. Force due to the 5.0 nC charge:
F2 = k * (|-4.1nC| * |5.0nC|) / (0.3330m)^2
To find the net force, we need to consider the direction of the forces. Since the 2.4 nC charge is positive, it will repel the -4.1 nC charge. Since the 5.0 nC charge is positive, it will attract the -4.1 nC charge.
F_net = F2 - F1
Now, set the net force equal to zero, since we want to find the equilibrium position:
0 = F_net
Solve the equation to find the equilibrium position. Rearranging the equation, we have:
F2 = F1
k * (|-4.1nC| * |5.0nC|) / (0.3330m)^2 = k * (|-4.1nC| * |2.4nC|) / (0.3330m)^2
Simplifying the equation, we have:
|-4.1nC| * |5.0nC| = |-4.1nC| * |2.4nC|
Now, you can solve the equation algebraically to find the equilibrium position.
Plugging in the values:
(4.1nC) * (5.0nC) = (4.1nC) * (2.4nC)
20.5 nC^2 = 9.84 nC^2
This equation is not valid, which means that the -4.1 nC charge does not have an equilibrium position between the 2.4 nC and 5.0 nC charges.