particle 1 of charge q1 = 1.04 ìC and particle 2 of charge q2 = -2.99 ìC, are held at separation L = 10.8 cm on an x axis. If particle 3 of unknown charge q3 is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the (a)x and (b)y coordinates of particle 3?

The point where electric field is zero may be either to the left from q1 or to the right from q2; y-coordinate of the ‘zero point’ is zero.

Let ‘x’ be the distance to the “zero point” located to the left from q1. Then
E1=k•q1/x², (the field deiected to the left)
E2=k•q2/(10.8+x)² ,(to the right)
E1=E2
k•q1/x²= k•q2/(10.8+x)²
1.04/ x² = 2.99/(10.8+x)².
Solve for “x”
I obtained x1=18.35 cm and x2=-15.64 cm . Check my calculations

To find the x-coordinate of particle 3, we need to consider the force exerted by particle 1 and particle 2 on particle 3.

Step 1: Calculate the force exerted by particle 1 on particle 3.
The electrostatic force between two charges is given by Coulomb's law:
F₁₃ = k * |q₁| * |q₃| / r₁₃²,
where F₁₃ is the force exerted by particle 1 on particle 3, k is Coulomb's constant (k = 8.99 * 10⁹ Nm²/C²), |q₁| and |q₃| are the magnitudes of the charges, and r₁₃ is the distance between the charges.

Given:
q₁ = 1.04 μC = 1.04 * 10⁻⁶ C
r₁₃ = L = 10.8 cm = 10.8 * 10⁻² m

Substituting the values into Coulomb's law, we get:
F₁₃ = (8.99 * 10⁹ Nm²/C²) * (1.04 * 10⁻⁶ C) * |q₃| / (10.8 * 10⁻² m)²

Step 2: Calculate the force exerted by particle 2 on particle 3.
We use the same formula as above, but substitute the values for particle 2:
F₂₃ = (8.99 * 10⁹ Nm²/C²) * (2.99 * 10⁻⁶ C) * |q₃| / (10.8 * 10⁻² m)²

Step 3: Set the net force on particle 3 to zero.
Since we want the net force on particle 3 to be zero, the sum of the forces exerted by particle 1 and particle 2 should equal zero:
F₁₃ + F₂₃ = 0

Substituting the expressions for F₁₃ and F₂₃, we have:
(8.99 * 10⁹ Nm²/C²) * (1.04 * 10⁻⁶ C) * |q₃| / (10.8 * 10⁻² m)² + (8.99 * 10⁹ Nm²/C²) * (2.99 * 10⁻⁶ C) * |q₃| / (10.8 * 10⁻² m)² = 0

Now, solve this equation for |q₃|.

To find the y-coordinate of particle 3, we need to consider the electric field due to particle 1 and particle 2.

Step 1: Calculate the electric field due to particle 1 at particle 3.
The electric field due to a point charge is given by:
E₁ = k * |q₁| / r₁²,
where E₁ is the electric field, k is Coulomb's constant, |q₁| is the magnitude of the charge, and r₁ is the distance from the charge.

Given:
r₁ = L = 10.8 cm = 10.8 * 10⁻² m

Substituting the values into the equation, we get:
E₁ = (8.99 * 10⁹ Nm²/C²) * (1.04 * 10⁻⁶ C) / (10.8 * 10⁻² m)²

Step 2: Calculate the electric field due to particle 2 at particle 3.
Use the same formula as above, but substitute the values for particle 2:
E₂ = (8.99 * 10⁹ Nm²/C²) * (2.99 * 10⁻⁶ C) / (10.8 * 10⁻² m)²

Step 3: Set the net electric field at particle 3 to zero.
Since we want the net electric field at particle 3 to be zero, the sum of the electric fields due to particle 1 and particle 2 should equal zero:
E₁ + E₂ = 0

Substituting the expressions for E₁ and E₂, we have:
(8.99 * 10⁹ Nm²/C²) * (1.04 * 10⁻⁶ C) / (10.8 * 10⁻² m)² + (8.99 * 10⁹ Nm²/C²) * (2.99 * 10⁻⁶ C) / (10.8 * 10⁻² m)² = 0

Solve this equation for |q₃|.

Once you know the value of |q₃|, you can find the (a)x-coordinate and (b)y-coordinate of particle 3 using the distance L = 10.8 cm on the x-axis.