an arrangement of four charged particles, with angle θ = 34.0 ˚ and distance d = 1.50 cm. Particle 2 has charge q2 = 8.00 × 10-19 C; particles 3 and 4 have charges q3 = q4 = -1.60 × 10-19 C. What is the distance D between the origin and particle 2 if the net electrostatic force on particle 1 due to the other particles is zero?
The set up of the particles is q1(negative d) and q2(positive D) are on the x axis with q3(positive) and q4(negative)with the angle theta with the x axis.
To find the distance D between the origin and particle 2, we need to use Coulomb's Law and calculate the net electrostatic force on particle 1 due to the other particles. The net force will be zero since the question states that the net electrostatic force on particle 1 is zero.
Coulomb's Law states that the magnitude of the electrostatic force between two charged particles is given by the equation:
F = (k * |q1 * q2|) / r^2
Where:
- F is the magnitude of the electrostatic force
- k is Coulomb's constant, approximately equal to 9.0 x 10^9 N·m^2/C^2
- q1 and q2 are the charges of the particles
- r is the distance between the particles
In this case, particle 1 is negatively charged and is located at a distance of d. Particle 2 is positively charged and positioned at a distance D. Particles 3 and 4 also have charges and positions, but we are only concerned with their effect on particle 1.
Since the net electrostatic force on particle 1 is zero, the magnitudes of the forces between particle 1 and particles 3, 4, and 2 must be equal. Let's call these magnitudes F₁₃, F₁₄, and F₁₂ respectively.
Using Coulomb's Law, we can set up the following equations:
F₁₃ = (k * |q1 * q3|) / (distance₁₃)^2
F₁₄ = (k * |q1 * q4|) / (distance₁₄)^2
F₁₂ = (k * |q1 * q2|) / (distance₁₂)^2
Since we are only interested in the distance D between the origin and particle 2, we need to find an equation involving D only. From the given setup, we can deduce that:
distance₁₂ = D
distance₁₃ = distance₁₄ = d
We can set up the following equation:
F₁₂ = F₁₃ + F₁₄
Substituting the respective equations:
(k * |q1 * q2|) / D^2 = (k * |q1 * q3|) / d^2 + (k * |q1 * q4|) / d^2
Simplifying, we get:
|q1 * q2| / D^2 = |q1 * q3| / d^2 + |q1 * q4| / d^2
Now we can plug in the given values:
q1 = -1.60 × 10^-19 C
q2 = 8.00 × 10^-19 C
q3 = q4 = -1.60 × 10^-19 C
d = 1.50 cm (convert to meters by dividing by 100)
Solving for D, we can rearrange the equation:
D^2 = (|q1 * q2| / (|q1 * q3| + |q1 * q4|)) * (d^2)
Taking the square root of both sides:
D = √[(|q1 * q2| / (|q1 * q3| + |q1 * q4|)) * (d^2)]
Finally, plug in the values and calculate.