Chapter 23: Problem 7
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.
Short Answer
Expert verified
The maximum electric force between the protons is calculated using Coulomb's law at the distance of closest approach.
Step by step solution
01
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.
02
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.
03
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 \(K_i\).
04
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.
05
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 \(r\).
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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,
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:
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.
- \(K\) is the kinetic energy,
- \(m\) is the mass of the object,
- \(v\) is the velocity of the object,
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.