We are tasked with finding the electrostatic force on particle 3 due to particles 1 and 2. The forces between charged particles are governed by Coulomb’s Law, which is given by:
F = k_e * (|Q1 * Q2|) / r² * r̂
Where:
- F is the magnitude of the electrostatic force between two charges.
- k_e is Coulomb’s constant, approximately 8.99 × 10⁹ N·m²/C².
- Q1, Q2 are the charges of the two particles.
- r is the distance between the two charges.
- r̂ is the unit vector pointing from the charge producing the force to the charge experiencing the force.
Now, let’s break down the problem step by step:
Step 1: Identify the Positions of the Particles
The coordinates of the particles are as follows:
- Particle 1: Q1 = +80.0 nC at (0, +3.00 mm) = (0, +3.00 × 10⁻³ m)
- Particle 2: Q2 = +80.0 nC at (0, -3.00 mm) = (0, -3.00 × 10⁻³ m)
- Particle 3: q = +18.0 nC at (+4.00 mm, 0) = (+4.00 × 10⁻³ m, 0)
Step 2: Calculate the Force on Particle 3 due to Particle 1
First, we calculate the distance between particle 3 and particle 1:
r₁₃ = √[(x₃ - x₁)² + (y₃ - y₁)²]
Substitute the coordinates:
r₁₃ = √[(4.00 × 10⁻³ m - 0)² + (0 - 3.00 × 10⁻³ m)²]
r₁₃ = √[(4.00 × 10⁻³)² + (-3.00 × 10⁻³)²] = √[(16 × 10⁻⁶) + (9 × 10⁻⁶)] = √(25 × 10⁻⁶)
r₁₃ = 5.00 × 10⁻³ m
Now we use Coulomb's Law to find the force between particle 1 and particle 3:
F₁₃ = k_e * (|Q1 * q|) / r₁₃²
Substitute the values:
F₁₃ = (8.99 × 10⁹ N·m²/C²) * (|80.0 × 10⁻⁹ C * 18.0 × 10⁻⁹ C|) / (5.00 × 10⁻³ m)²
Simplify:
F₁₃ = (8.99 × 10⁹) * (1.44 × 10⁻¹⁵) / (2.50 × 10⁻⁵)
F₁₃ = 5.17 × 10⁻¹ N
Next, we need to find the direction of the force. The position vector of particle 1 relative to particle 3 is:
r₁₃ = (0 - 4.00 × 10⁻³) î + (3.00 × 10⁻³ - 0) ĵ = (-4.00 × 10⁻³) î + (3.00 × 10⁻³) ĵ
The unit vector pointing from particle 1 to particle 3 is:
r̂₁₃ = r₁₃ / r₁₃ = (-4.00 × 10⁻³) î + (3.00 × 10⁻³) ĵ / 5.00 × 10⁻³
Simplify:
r̂₁₃ = (-0.80) î + (0.60) ĵ
So, the force vector on particle 3 due to particle 1 is:
F₁₃ = 5.17 × 10⁻¹ N * (-0.80 î + 0.60 ĵ)
This gives:
F₁₃ = (-4.14 × 10⁻¹ î + 3.10 × 10⁻¹ ĵ) N
Step 3: Calculate the Force on Particle 3 due to Particle 2
The distance between particle 3 and particle 2 is calculated in the same way:
r₂₃ = √[(4.00 × 10⁻³)² + (0 - (-3.00 × 10⁻³))²] = 5.00 × 10⁻³ m
Using Coulomb's Law, we find the force:
F₂₃ = k_e * (|Q2 * q|) / r₂₃²
Substitute the values:
F₂₃ = (8.99 × 10⁹ N·m²/C²) * (|80.0 × 10⁻⁹ C * 18.0 × 10⁻⁹ C|) / (5.00 × 10⁻³ m)²
Simplify:
F₂₃ = 5.17 × 10⁻¹ N
The position vector of particle 2 relative to particle 3 is:
r₂₃ = (0 - 4.00 × 10⁻³) î + (-3.00 × 10⁻³ - 0) ĵ = (-4.00 × 10⁻³) î - (3.00 × 10⁻³) ĵ
The unit vector pointing from particle 2 to particle 3 is:
r̂₂₃ = (-0.80) î - (0.60) ĵ
So, the force vector on particle 3 due to particle 2 is:
F₂₃ = 5.17 × 10⁻¹ N * (-0.80 î - 0.60 ĵ)
This gives:
F₂₃ = (-4.14 × 10⁻¹ î - 3.10 × 10⁻¹ ĵ) N
Step 4: Total Force on Particle 3
Now, we add the forces from particle 1 and particle 2:
F₃ = F₁₃ + F₂₃
Substitute the values:
F₃ = (-4.14 × 10⁻¹ î + 3.10 × 10⁻¹ ĵ) + (-4.14 × 10⁻¹ î - 3.10 × 10⁻¹ ĵ)
Simplifying:
F₃ = (-8.28 × 10⁻¹ î + 0) N
Thus, the total force on particle 3 in unit-vector notation is:
F₃ = (-8.28 × 10⁻¹ î) N
The electrostatic force on particle 3 is directed along the negative x-axis with a magnitude of 0.828 N.
