Chapter 17, Problem 36P

* Evaluate the solution Metal sphere 1 has charge q and is attached to an insulating stand that is fixed on a table. Metal sphere 2 has charge 3q and is attached to an insulating stand that can be moved along the table. The distance between the centers of the spheres is initially a = 35 cm. If we slowly move sphere 1 toward sphere 2 until the distance between their centers is equal to b = 25 cm, we do ${10}^{-3}$

J work on the system consisting of the two spheres. (a) Determine the magnitudes and the signs of the charges on the spheres. (b) Your friend provides the following solution for (a): The signs of the charges are either both positive or both negative because work is done on the system as they are moved closer together. The magnitudes of the charges can be determined using the generalized work-energy principle:

${\kappa }_{\text{C}}\frac{q\cdot 3q}{a^2}+W={\kappa }_C\frac{q\cdot 3q}{b^2}$

$\sqrt{\Rightarrow q=\frac{W}{3{\kappa }_{\text{C}}}{\left(\frac{1}{b^2}-\frac{1}{a^2}\right)}^{-1}}=\sqrt{\frac{{10}^{-3}\text{J }\cdot {\text{C}}^2}{3(9\times {10}^9\text{N}\cdot {\text{m}}^2)}{(16.0{\text{ m}}^{\text{-2}}-8.2{\text{ m}}^{-2})}^{-1}}$

$=6.9\times {10}^{-8}\text{C}$

Evaluate the solution and correct any errors you find.