Question

In Fig. 16-61, three objects, O1 and O3 are charged. And O2 is neutral. Suppose the flux through the surface A1 that encloses all three objects is -4.60 105 N·m2/C, and the flux through the surface A2 that encloses the third object only is 2.40 105 N·m2/C. Find the charges O1, O2, O3 carry respectively.

Answer 1

Given data

Let the charge of the three objects be q₁,q_{2 ,}q₃

The flux through A₁ is, φ₁ = - 4.60 * 10^{5 _N.m2} / C

The flux through A₂ is, φ₂ = 2.40 * 10^{5 _N.m2} / C

The charge of the second particle is, q₂ = 0

Solution:

According to Gauss's law, the net flux can be given as,

                           φ = q / ε₀

First we consider the flux through A₂, that is,

                          φ₂= q₃/ ε₀

2.40 * 10^{5 _N.m2} / C = q₃/ 8.85 * 10^-12 C² / N.m²

                         q₃   = 2.124 * 10^-6 C or 2.124 μC

Now, the flux through A₁ can be written as,

                             φ₁= ( q₁ + q₂ + q₃ ) / ε₀

- 4.60 * 10^{5 _N.m2} / C = (q₁ + q_{2+ q3 ) /  8.85 * 10^-12 C² / N.m²}

             q₁ + q_{2+ q3 = - 4.071 * 10^-6                                              ...... (1)}

Substituting the value of q₃ in equation (1), we get

         q₁ + q_{2+ 2.124 * 10^-6 C =  - 4.071 * 10^-6}

                                  ^{_{q1 + q2   = - 6.195 * 10-6}}

                                             q₁ = - 6.195 * 10^-6 - q₂    

                                                 = - 6.195 * 10^-6 - 0

                                                 = - 6.195 * 10^-6 C or - 6.195 μC