Question

A system consists of two positive point charges, q1 and q2>q1. The total charge of the system is 68.0 μC , and each charge experiences electrostatic force of magnitude 84.0 N when the separation between them is 0.270 m. Find q1 and q2.

Answer 1

Answer:

q₁ = +12.19 μC

q₂ = +55.81 μC

Explanation:

From Coulomb's law,

                                 $F = k\frac{q_{1}q_{2}}{r^{2}}$

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

                         $84 = 9 * 10^{9}\frac{q_{1}q_{2}}{0.270^{2}}$

                         $q_{1}q_{2} = \frac{84 * 0.270^{2}}{9 * 10^{9}}$  

                         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

                  $q_{1} = \frac{-b \± \sqrt{b^{2} - 4ac} }{2a}$

                 $q_{1} = \frac{-(- 68 * 10^{-6}) \± \sqrt{(- 68 * 10^{-6})^{2} - (4*1*0.6804*10^{-9}})}{2*1}$

                 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