Answer:
q₁ = +12.19 μC
q₂ = +55.81 μC
Explanation:
From Coulomb's law,
![F = k\frac{q_{1}q_{2}}{r^{2}}](https://tex.z-dn.net/?f=F%20%3D%20k%5Cfrac%7Bq_%7B1%7Dq_%7B2%7D%7D%7Br%5E%7B2%7D%7D)
Where,
F is the electrostatic force
k is the constant of proportionality 9 × 10⁹ Nm²/C²
q₁ and q₂ are the point charges
r is the distance between the charges
![84 = 9 * 10^{9}\frac{q_{1}q_{2}}{0.270^{2}}](https://tex.z-dn.net/?f=84%20%3D%209%20%2A%2010%5E%7B9%7D%5Cfrac%7Bq_%7B1%7Dq_%7B2%7D%7D%7B0.270%5E%7B2%7D%7D)
q₁q₂ = 6.804 × 10 ⁻¹⁰ C
q₁q₂ = 0.6804 × 10 ⁻⁹ C -------(1)
Also, the total charge of the system is 68.0 μC
⇒ q₁ + q₂ = 68.0 μC
For simplicity, q₁ + q₂ = 68 × 10⁻⁶ C --------(2)
Solving equation 1 and 2 simultaneously using substitution method. From equation 2,
q₂ = 68 × 10⁻⁶ - q₁ ---------(3)
Substituting equation (3) into (1)
q₁(68 × 10⁻⁶ - q₁) = 0.6804 × 10 ⁻⁹
68 × 10⁻⁶q₁ - q₁² = 0.6804 × 10 ⁻⁹
q₁² - 68 × 10⁻⁶q₁ + 0.6804 × 10 ⁻⁹ = 0
solving the quadratic equation using quadratic formula,
a = 1, b = - 68 × 10⁻⁶, c = 0.6804 × 10 ⁻⁹
![q_{1} = \frac{-b \± \sqrt{b^{2} - 4ac} }{2a}](https://tex.z-dn.net/?f=q_%7B1%7D%20%3D%20%5Cfrac%7B-b%20%5C%C2%B1%20%5Csqrt%7Bb%5E%7B2%7D%20-%204ac%7D%20%7D%7B2a%7D)
![q_{1} = \frac{-(- 68 * 10^{-6}) \± \sqrt{(- 68 * 10^{-6})^{2} - (4*1*0.6804*10^{-9}})}{2*1}](https://tex.z-dn.net/?f=q_%7B1%7D%20%3D%20%5Cfrac%7B-%28-%2068%20%2A%20%2010%5E%7B-6%7D%29%20%5C%C2%B1%20%5Csqrt%7B%28-%2068%20%2A%2010%5E%7B-6%7D%29%5E%7B2%7D%20-%20%284%2A1%2A0.6804%2A10%5E%7B-9%7D%7D%29%7D%7B2%2A1%7D)
q₁ = +12.19 μC or +55.81 μC
Testing for the two q₁ to get q₂
When q₁ = +12.19 μC
12.19 μC + q₂ = 68.0 μC
q₂ = 68.0 μC - 12.19 μC
q₂ = +55.81 μC
When q₁ = +55.81 μC
55.81 μC + q₂ = 68.0 μC
q₂ = 68.0 μC - 55.81 μC
q₂ = +12.19 μC
We have two possible solutions for the question.
When q₁ = +12.19 μC, q₂ = +55.81 μC and
When q₁ = +55.81 μC, q₂ = +12.19 μC
Considering the conditions given in the system,
(1) Two positive point charges q₁ and q₂
(2) Charge q₂ is greater than charge q₁, q₂>q₁
The solution that satisfies the condition is q₁ = +12.19 μC, q₂ = +55.81 μC