Three point charges lie in a straight line along the y-axis. A charge of q1 = -9.50 µC is at y = 6.30 m, and a charge of q2 = -7.20 µC is at y = -4.00 m. The net electric force on the third point charge is zero. Where is this charge located?
y = m
To find the location of the third point charge where the net electric force is zero, we need to consider the forces due to q1 and q2 on the third charge.
The magnitude of the electric force between two point charges is given by Coulomb's Law: F = k * |q1| * |q2| / r^2, where F is the force, k is the electrostatic constant (k ≈ 8.99 × 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 the net electric force on the third charge is zero, it means that the forces due to q1 and q2 are equal in magnitude but opposite in direction.
Let's denote the distance between the third charge and the q1 charge as r1, and the distance between the third charge and the q2 charge as r2.
The force due to q1 on the third charge is F1 = k * |q1| * |q3| / r1^2.
The force due to q2 on the third charge is F2 = k * |q2| * |q3| / r2^2.
Since the forces are equal in magnitude but opposite, we have:
|F1| = |F2|
Using the equations for F1 and F2:
k * |q1| * |q3| / r1^2 = k * |q2| * |q3| / r2^2
Simplifying, we have:
|q1| / r1^2 = |q2| / r2^2
Rearranging, we get:
|r1 / r2| = √(|q2| / |q1|)
|r1 / r2| = √(7.20 µC / 9.50 µC)
|r1 / r2| ≈ 0.880
Since r1 and r2 are distances, their ratio must be positive. Therefore, we can remove the absolute value symbols:
r1 / r2 ≈ 0.880
Since the charges lie on the y-axis, we know that the distance from the third charge to q1 is equal to the distance from the third charge to q2:
r1 = r2
r1 / r1 ≈ 0.880
1 ≈ 0.880
This contradicts our assumption that the forces are equal in magnitude but opposite. Therefore, it is not possible for the net electric force on the third charge to be zero in this configuration.
As a result, there is no specific location for the third charge where the net electric force is zero.
To find the position of the third point charge where the net electric force is zero, we need to consider the forces that each of the charges exerts on the third charge.
Let's assume that the third charge has a charge of q3 and is located at position y = y3.
The electric force between any two charges can be calculated using Coulomb's law:
F = k * |q1 * q3| / r1^2 (1)
F' = k * |q2 * q3| / r2^2 (2)
where F is the force between q1 and q3, F' is the force between q2 and q3, k is the Coulomb's constant (8.99 x 10^9 N m^2/C^2), and r1 and r2 are the distances between the charges.
Since the net electric force on the third point charge is zero, the magnitudes of the forces (F and F') must be equal and opposite:
|q1 * q3| / r1^2 = |q2 * q3| / r2^2
First, let's solve for r1^2 and r2^2:
r1^2 = (|q1 * q3| * r2^2) / |q2 * q3| = (|q1 * q3| * r2^2) / |q2|
Next, we substitute the values given for q1, q2, r2, and solve for r1^2:
r1^2 = (|(-9.50 µC) * q3| * (-4.00 m)^2) / |-7.20 µC| = (9.50 * q3 * 16) / 7.20
r1^2 = (152 * q3) / 7.20
r1^2 = 21.11 * q3
Now, let's solve for r2^2:
r2^2 = (|q2 * q3| * r1^2) / |q1 * q3| = (|(-7.20 µC) * q3| * r1^2) / |-9.50 µC|
r2^2 = (7.20 * q3 * r1^2) / 9.50
r2^2 = (76.11 * q3) / 9.50
r2^2 = 8.02 * q3
Since the forces are equal and opposite, the magnitudes of q3 * r1^2 and q3 * r2^2 must also be equal:
|r1^2| = |r2^2|
Substituting the expressions for r1^2 and r2^2:
|r1^2| = |21.11 * q3| = |8.02 * q3|
Simplifying:
21.11 * q3 = 8.02 * q3
13.09 * q3 = 0
q3 = 0
Since q3 is zero, it means that the third charge has no charge. In other words, it is an uncharged object.
Therefore, the position of the third charge (y3) can be anywhere on the y-axis since an uncharged object doesn't experience a net electric force from the other charges.
set up the two force equations:
kq1q/(6.40-x)^2= kq2q/(4+x)^2
solve for x