Two fixed particles of changes 1 micro coulomb and 3micro coulomb are 10cm apart. How far apart from each other should a third charge be placed so that no net electrostatic force acts on it.

To find the distance at which no net electrostatic force acts on the third charge, we need to use the concept of Coulomb's Law. According to Coulomb's Law, the electrostatic force between two charges is given by the equation:

F = (k * q1 * q2) / r^2

where F is the electrostatic force, k is Coulomb's constant (9 × 10^9 Nm^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between them.

In this case, we have two fixed particles with charges of 1 microcoulomb (1 × 10^-6 C) and 3 microcoulombs (3 × 10^-6 C), separated by a distance of 10 cm (0.1 m).

Let's assume the third charge is q3 and the distance from each fixed charge is x. To find the distance x at which no net electrostatic force acts on the third charge, we need to equate the forces exerted by the two fixed charges on it.

Using Coulomb's Law, the force exerted by the first fixed particle on the third charge is:

F1 = (k * q1 * q3) / x^2

The force exerted by the second fixed particle on the third charge is:

F2 = (k * q2 * q3) / (0.1 - x)^2

Since the forces are equal, we can equate these two equations:

(k * q1 * q3) / x^2 = (k * q2 * q3) / (0.1 - x)^2

We can cancel out the common terms and rearrange to solve for x:

q1 / x^2 = q2 / (0.1 - x)^2

Substituting the given values, we get:

(1 × 10^-6) / x^2 = (3 × 10^-6) / (0.1 - x)^2

Cross-multiplying, we get:

(1 × 10^-6) * (0.1 - x)^2 = (3 × 10^-6) * x^2

Expanding and simplifying, we have:

(0.1 - x)^2 = 3 * x^2

Taking the square root of both sides, we get:

0.1 - x = sqrt(3) * x

Rearranging and solving for x, we find:

x = 0.1 / (1 + sqrt(3))

Calculating this value, we get:

x ≈ 0.0346 m

Therefore, the third charge should be placed approximately 0.0346 meters (or 3.46 cm) away from each fixed charge to experience no net electrostatic force.