Three point charges are arranged along the x axis: q1 = −4.43 nC at x1 = 199 mm, q2 = 2.33 nC at x2 = -302 mm, and a positive charge q3 at the origin. What is the value of q3 if the net force on this point charge has magnitude 4.50 µL?
I have never heard of force measured in µL. Is that supposed to be microliters? Check again.
Then write the equation
F = k q3 (q1/x1^2 - q2/x2^2), making sure you use the negavive value for q1. The forces due to q1 and q2 will be in the same direction. k is the Columb constant. Solve for q1
Make sure the q's are in Coulombs, the distances x are in meters, and the force in Newtons.
To find the value of charge q3, we can use the principle of superposition and the formula for electric force.
The principle of superposition states that the total force on a charge is the vector sum of the individual forces from all other charges.
The formula for the magnitude of the electric force between two charges is given by Coulomb's law:
F = (k * |q1| * |q2|) / r^2
Where:
F is the magnitude of the force
k is the electrostatic constant (k ≈ 8.99 x 10^9 N m^2/C^2)
q1 and q2 are the magnitudes of the charges
r is the distance between the charges
Let's calculate the forces between q3 and q1, and between q3 and q2 separately.
First, let's calculate the force between q3 and q1.
Given:
q1 = -4.43 nC
x1 = 199 mm = 0.199 m (distance from q3 to q1)
Using Coulomb's law, the force between q3 and q1 is:
F1 = (k * |q3| * |q1|) / x1^2
Next, let's calculate the force between q3 and q2.
Given:
q2 = 2.33 nC
x2 = -302 mm = -0.302 m (distance from q3 to q2)
Using Coulomb's law, the force between q3 and q2 is:
F2 = (k * |q3| * |q2|) / x2^2
The net force on q3 will be the vector sum of F1 and F2:
|F_net| = sqrt(F1^2 + F2^2)
We are given that |F_net| = 4.50 µL = 4.50 x 10^-6 N
Solving the above equation for |F_net|, we can substitute the values of F1 and F2 and solve for |q3|.
To find the value of q3, we can use the principle of superposition. The net force on a point charge in the presence of multiple charges is the vector sum of the individual forces due to each charge.
Let's break down the problem step by step:
1. Calculate the force between q1 and q3:
The force between two point charges can be found using Coulomb's Law: F = (k * |q1 * q3|) / r^2, where F is the force, k is the electrostatic constant (9 * 10^9 Nm^2/C^2), q1 and q3 are the charges, and r is the distance between them.
Given:
q1 = -4.43 nC
q3 = unknown
x1 = 199 mm (convert to meters: 0.199 m)
Let's calculate the force between q1 and q3:
F1 = (9 * 10^9 * |-4.43 * q3|) / (0.199)^2
2. Calculate the force between q2 and q3:
Given:
q2 = 2.33 nC
q3 = unknown
x2 = -302 mm (convert to meters: -0.302 m)
Let's calculate the force between q2 and q3:
F2 = (9 * 10^9 * |2.33 * q3|) / (-0.302)^2
3. Calculate the net force on q3:
Since the net force has a magnitude of 4.50 µN, we can write it as:
|F1 + F2| = 4.50 * 10^-6 N
4. Substitute the calculated forces into the equation and solve for q3:
|F1 + F2| = (9 * 10^9 * |-4.43 * q3|) / (0.199)^2 + (9 * 10^9 * |2.33 * q3|) / (-0.302)^2
Solve for q3:
4.50 * 10^-6 = (9 * 10^9 * (|-4.43 * q3| / (0.199)^2 + |2.33 * q3| / (-0.302)^2
Now, this equation is an absolute value equation, so we need to consider both positive and negative values of q3. We solve this equation separately for q3 > 0 and q3 < 0.
Once we have the possible values of q3, we can check which one satisfies the given conditions.