Three point charges lie in a straight line along the y-axis. A charge of q1 = -9.20 µC is at y = 6.60 m, and a charge of q2 = -8.00 µC is at y = -3.90 m. The net electric force on the third point charge is zero. Where is this charge located?

The third point charge is located at y = 0 m.

To find the location of the third point charge where the net electric force is zero, we can apply the principle of superposition. The net force on the third charge is the vector sum of the forces due to the other two charges.

Given:
Charge q1 = -9.20 µC located at y1 = 6.60 m
Charge q2 = -8.00 µC located at y2 = -3.90 m

Let's assume the third charge is q3 located at y3.

The electric force between two charges is given by Coulomb's Law:

F = k * |q1| * |q2| / r^2

where F is the force between the charges, |q1| and |q2| are the magnitudes of the charges, r is the distance between the charges, and k is the electrostatic constant.

Considering the forces acting on the third charge:

Force due to q1 on q3 = F1 = k * |q1| * |q3| / (y3 - y1)^2
Force due to q2 on q3 = F2 = k * |q2| * |q3| / (y3 - y2)^2

Since the net electric force on the third charge is zero, the sum of these forces must be zero:

F1 + F2 = 0

Substituting the values:

k * |q1| * |q3| / (y3 - y1)^2 + k * |q2| * |q3| / (y3 - y2)^2 = 0

Now we can rearrange this equation to solve for y3, the position of the third charge:

|q1| / (y3 - y1)^2 + |q2| / (y3 - y2)^2 = 0

Plugging in the known values:

|-9.20 µC| / (y3 - 6.60 m)^2 + |-8.00 µC| / (y3 - (-3.90 m))^2 = 0

Simplifying this equation, we can solve for y3.

To find the location of the third charge where the net electric force is zero, we need to consider the electric forces exerted by the first two charges.

Since the third charge is in equilibrium (net electric force is zero), the magnitudes of the electric forces from the first two charges must be equal.

Let's assume the third charge, q3, is located at position y = y3.

The distance between the first charge and the third charge is given by d1 = |y3 - y1| = |y3 - 6.60 m|.

The distance between the second charge and the third charge is given by d2 = |y3 - y2| = |y3 - (-3.90 m)| = |y3 + 3.90 m|.

The electric force between charges is given by Coulomb's Law:

F = k * |q1 * q3 / d1^2| = k * |q2 * q3 / d2^2|

Here, k is the electrostatic constant, which is approximately k = 8.99 * 10^9 N m^2/C^2.

Setting the two electric forces equal to each other, we have:

k * |q1 * q3 / d1^2| = k * |q2 * q3 / d2^2|

Simplifying, we get:

|q1 * q3 / d1^2| = |q2 * q3 / d2^2|

Substituting the known values:

|-9.20 µC * q3 / (y3 - 6.60 m)^2| = |-8.00 µC * q3 / (y3 + 3.90 m)^2|

Now, we can solve this equation for y3 to find the location of the third charge.

|-9.20 * 10^(-6) * q3 / (y3 - 6.60)^2| = |-8.00 * 10^(-6) * q3 / (y3 + 3.90)^2|

Dividing both sides of the equation by -q3 (assuming q3 is non-zero):

|-9.20 * 10^(-6) / (y3 - 6.60)^2| = |-8.00 * 10^(-6) / (y3 + 3.90)^2|

Taking the absolute value away, we have:

9.20 * 10^(-6) / (y3 - 6.60)^2 = 8.00 * 10^(-6) / (y3 + 3.90)^2

Now, we can solve this equation to find the value of y3.