Question

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]

Short answer

(a) The electrostatic potential energy of the system is: \(U = \dfrac{(8.99 \times 10^9 \mathrm {Nm^2/C^2}) (2.0 \times 10^{-5} \mathrm{C}) (2.0 \times 10^{-5} \mathrm{C})}{1.0 \times 10^{-2} \mathrm{m}}\) (b) Since both spheres have the same positive charge, they will exert repulsive forces on each other, making them move away from each other. (c) The speed of each sphere as the distance between them approaches infinity can be found using the conservation of mechanical energy: \(v = \sqrt{\dfrac{2U_{initial}}{1.0 \mathrm{kg}}}\) Substitute the value of \(U_{initial}\) found in part (a) and calculate the expression to find the speed of each sphere.

Step-by-step solution

a) Finding the electrostatic potential energy

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.

b) Movement of the spheres

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.

c) Finding the speed of each sphere when the distance approaches infinity

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.

Key concepts

Coulomb's Law

Coulomb's Law describes the force between two point charges. According to the law, the electrostatic force exerted by two charged objects on each other is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as: \[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} \] where:

• \( F \) is the electrostatic force between the charges
• \( k \) is the Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges
• \( r \) is the distance between the center of the two charges

In the context of the given exercise, since the spheres are positively charged, and like charges repel, the force between them is a repulsive one, causing the spheres to move away when released. Coulomb's Law helps determine the magnitude of this repulsion.

Conservation of Mechanical Energy

The conservation of mechanical energy principle indicates that the total mechanical energy of a system remains constant if it is subject only to conservative forces such as electrostatic forces. It is expressed through the equation: \[ U_{\text{initial}} + K_{\text{initial}} = U_{\text{final}} + K_{\text{final}} \] where:

• \( U \) is the potential energy
• \( K \) is the kinetic energy
• "initial" and "final" denote states at different points in time

In our exercise, initially, the spheres have maximum potential energy and no kinetic energy since they are at rest. As they repel each other due to like charges, the potential energy converts into kinetic energy, causing the spheres to move faster. As the spheres move far apart, the potential energy becomes negligible, and the energy is completely transformed into their motion energy, i.e., kinetic energy. Understanding this principle allows us to calculate the final speeds of the spheres as they approach infinity.

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion. For an object with mass \( m \) moving with speed \( v \), kinetic energy is given by the formula: \[ K = \frac{1}{2} m v^2 \] In the scenario with our charged spheres, as they move further apart due to the repulsive force, all the initial potential energy gets converted into kinetic energy. Initially, this energy was stored as electrostatic potential energy. When released, this stored energy leads to an increase in motion, providing the spheres with a certain speed. The kinetic energy helps us find the speed of the spheres when their potential energy approaches zero, at a hypothetical infinite separation distance.