Two positively charged spheres, each with a charge of \(2.0 \times\) $10^{-5} \mathrm{C}\(, a mass of \)1.0 \mathrm{~kg}\(, and separated by a distance of \)1.0 \mathrm{~cm}$, are held in place on a frictionless track. (a) What is the electrostatic potential energy of this system? (b) If the spheres are released, will they move toward or away from each other? (c) What speed will each sphere attain as the distance between the spheres approaches infinity? [Section 5.1]

The electrostatic potential energy of a system with two point charges is given by the formula: \(U = \dfrac{kq_1q_2}{r}\) where \(U\) is the electrostatic potential energy, \(k\) is Coulomb's constant (\(k = 8.99 \times 10^9 \mathrm {Nm^2/C^2}\)), \(q_1\) and \(q_2\) are the charges on the spheres respectively, and \(r\) is the distance between the centers of the spheres. We are given that \(q_1 = q_2 = 2.0 \times 10^{-5} \mathrm{C}\) and \(r = 1.0 \mathrm{cm}\). Now, let's find the electrostatic potential energy U: \(U = \dfrac{(8.99 \times 10^9 \mathrm {Nm^2/C^2}) (2.0 \times 10^{-5} \mathrm{C}) (2.0 \times 10^{-5} \mathrm{C})}{1.0 \times 10^{-2} \mathrm{m}}\) By calculating the above expression, we get the electrostatic potential energy of the system.

Both spheres have the same positive charge, therefore they will exert repulsive forces on each other according to Coulomb's Law. Since the spheres are on a frictionless track, these repulsive forces will make the spheres move away from each other.

We can use the conservation of mechanical energy principle to find the speed of each sphere as the distance approaches infinity. Initially, the spheres have both potential energy \(U\) and no kinetic energy. As they move away from each other, their potential energy decreases and gets converted into kinetic energy. When the distance between the spheres approaches infinity, their electrostatic potential energy becomes negligible, and almost all of it is converted into kinetic energy. Since each sphere has the same mass and charge, they will have equal kinetic energy. The conservation of mechanical energy equation can be written as: \(U_{initial} + K_{initial} = U_{final} + K_{final}\) Since the initial kinetic energy is 0 and the final potential energy is negligible: \(U_{initial} = K_{final}\) The kinetic energy can be written as: \(K = \dfrac{1}{2}mv^2\) Where \(m\) is the mass of the sphere and \(v\) is its speed. We can now solve for \(v\): \(\dfrac{1}{2}(1.0 \mathrm{kg})v^2 = U_{initial}\) \(v^2 = \dfrac{2U_{initial}}{1.0 \mathrm{kg}}\) Finally, we can find the speed of each sphere by taking the square root of the above equation: \(v = \sqrt{\dfrac{2U_{initial}}{1.0 \mathrm{kg}}}\) By substituting the value of \(U_{initial}\) found in part (a) and calculating the above expression, we will get the speed of each sphere as the distance between them approaches infinity.