Three charges q1, q2 and q3 lie along the x-axis. Charge q1=2.5 micro coulomb is at x= 1.5 m, while charge q2= 6.3 micro coulomb is at the origin. Where must the third charge q3 be placed on the axis so that the resultant force on it is zero?

origion

190 cm

Negative

give solution

To find the position where the third charge q3 must be placed on the x-axis so that the resultant force on it is zero, we need to consider the forces exerted by charges q1 and q2 on q3 and find the point where these forces balance out.

The force between two charges can be calculated using Coulomb's Law:

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

Where F is the force between the charges, k is the Coulomb's constant (9 x 10^9 Nm²/C²), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.

Let's assume q3 is located at position x on the x-axis. The distance between q3 and q1 is (x - 1.5) m, while the distance between q3 and q2 is x m.

The force exerted by q1 on q3 is F1 = k * |q1 * q3| / (x - 1.5)^2

The force exerted by q2 on q3 is F2 = k * |q2 * q3| / x^2

For the resultant force on q3 to be zero, we need F1 and F2 to cancel each other out. In other words, F1 = -F2.

By substituting the values, we can set up the equation:

k * |q1 * q3| / (x - 1.5)^2 = -k * |q2 * q3| / x^2

Since k, q1, and q2 are constants, we can cancel them out:

|q1 * q3| / (x - 1.5)^2 = -|q2 * q3| / x^2

Now, we can simplify the equation:

|q1 * q3| * x^2 = -|q2 * q3| * (x - 1.5)^2

Taking the square root of both sides:

q1 * q3 * x = -q2 * q3 * (x - 1.5)

Simplifying:

q1 * x = -q2 * (x - 1.5)

Expanding:

q1 * x = -q2 * x + 1.5 * q2

Rearranging the terms:

(q1 + q2) * x = 1.5 * q2

Finally, solving for x:

x = (1.5 * q2) / (q1 + q2)

Plugging in the values of q1 = 2.5 μC and q2 = 6.3 μC:

x = (1.5 * 6.3) / (2.5 + 6.3)

x = 9.45 / 8.8

x ≈ 1.073 m

Therefore, the third charge q3 must be placed approximately 1.073 meters from the origin on the positive x-axis so that the resultant force on it is zero.