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.