Problem 7 - University Physics with Modern Physics

Question

Two protons, starting several meters apart, are aimed directly at each other with speeds of \(2.00 \times 10^5\) m\(/\)s, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Step-by-Step Solution

Answer

The maximum electric force between the protons is calculated using Coulomb's law at the distance of closest approach.

Step 1: Define the Variables

The speed of each proton is given as \(v = 2.00 \times 10^5\ \text{m/s}\) and this is the initial speed relative to earth. Two protons have equal mass \(m_p = 1.67 \times 10^{-27}\ kg\) and charge \(e = 1.60 \times 10^{-19}\ C\). We want to find the maximum electric force that they exert on each other.

Step 2: Apply Conservation of Energy

The initial kinetic energy \(K_i\) of the system comes from both protons. Therefore, \(K_i = 2 \times \frac{1}{2}mv^2 = mv^2\). This energy will be converted into electrical potential energy (\(U\)) at the point of closest approach.

Step 3: Calculate Initial Kinetic Energy

The initial kinetic energy for one proton is \(K = \frac{1}{2} mv^2\). Adding up for both protons gives us \(K_i = mv^2 = (1.67 \times 10^{-27}\ kg)(2.00 \times 10^5\ \text{m/s})^2\). Calculate this to find \(K_i\).

Step 4: Set Kinetic Energy Equal to Potential Energy

The point at which the two protons stop moving toward each other and start to move away is when their kinetic energy is fully converted into electric potential energy, which is given by \(U = \frac{k e^2}{r}\), where \(k = 8.99 \times 10^9\ \text{N m}^2/\text{C}^2\) is Coulomb's constant, \(e\) is the charge of a proton, and \(r\) is the separation distance.

Step 5: Solve for Distance of Closest Approach

Set the calculated initial kinetic energy equal to the potential energy expression: \[ mv^2 = \frac{k e^2}{r} \]Solve for \(r\).

Step 6: Calculate Maximum Electric Force

The force between the protons at the distance of closest approach can be calculated using Coulomb's law:\[ F = \frac{k e^2}{r^2} \]Substitute the value of \(r\) obtained from the previous step into the equation to find the maximum force.

Key Concepts

Conservation of Energy

The conservation of energy is a foundational concept in physics that states that energy can neither be created nor destroyed, only transformed from one form to another. In the exercise involving two protons, this principle is crucial in understanding how their energy transitions between kinetic and electric potential energy.

• Initially, each proton has a certain amount of kinetic energy due to its motion.
• As they approach each other, this kinetic energy is gradually converted into electric potential energy.
• The point at which the protons cease their approach marks the total conversion of kinetic energy into potential energy.

Conservation of energy ensures that the sum of kinetic and electric potential energy remains constant throughout the protons' motion. This is fundamental to solving the exercise, allowing us to set these energies equal at the closest point to find the distance of closest approach.

Coulomb's Law

Coulomb's Law is a key equation used to describe the electric force between two charged objects. It is given by:\[ F = \frac{k e^2}{r^2} \]where:

• \(F\) is the force between the charges,
• \(k\) is Coulomb's constant \(8.99 \times 10^9 \text{N m}^2/\text{C}^2\),
• \(e\) is the charge of a proton \(1.60 \times 10^{-19}\),
• \(r\) is the distance between the charges,

Coulomb's Law tells us that the electric force is inversely proportional to the square of the distance between the charges and exponentially greater as they are closer. This is significant in the exercise, as the maximum electric force occurs at the closest distance between the protons. By determining this minimum distance from the conservation of energy, we can calculate the maximum force exerted.

Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion, and is calculated with the formula:\[ K = \frac{1}{2} mv^2 \]where:

• \(K\) is the kinetic energy,
• \(m\) is the mass of the object,
• \(v\) is the velocity of the object,

In the given problem, each proton starts with a significant kinetic energy due to their high speed of \(2.00 \times 10^5\ \text{m/s}\). The total initial kinetic energy contributes to the electric potential energy at the point of closest approach.
This conversion illustrates how energy cannot just disappear. Instead, it transforms into a different form that influences the behavior of the system, like repelling forces that stop the two protons as they approach each other.

Electric Potential Energy

Electric potential energy is the energy stored between two charged objects due to their positions in an electric field. It is expressed as:\[ U = \frac{k e^2}{r} \]where:

• \(U\) is the electric potential energy,
• \(k\) is Coulomb's constant,
• \(e\) is the charge of a proton,
• \(r\) is the separation distance between the charges.

In the context of this exercise, as the protons move closer, their initial kinetic energy converts into electric potential energy until motion stops entirely. This is when the conversion is complete and the electric potential energy is at its maximum, describing the point of closest approach.Understanding electric potential energy helps visualize at which point two charged particles will repel each other enough to stop moving closer, an integral aspect of solving the problem at hand.