Chapter 23: Problem 30
An infinitely long line of charge has linear charge density 5.00 \(\times 10^{-12}\) C\(/\)m. A proton (mass 1.67 \(\times 10^{-27}\) kg, charge \(+\)1.60 \(\times 10^{-19}\) C) is 18.0 cm from the line and moving directly toward the line at 3.50 \(\times 10^3\) m\(/\)s. (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge?
Short Answer
Step by step solution
Calculate Initial Kinetic Energy
Electric Potential Energy Difference Calculation
Set Initial and Final Energy Equal
Solve for Closest Distance r
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Charge Density
Understanding linear charge density helps in calculating the electric field created by the line of charge. The electric field strength is crucial when figuring out the interaction between the charge line and any nearby charged particles, like the proton in the problem. An infinite line of charge creates an electric field around it, and this electric field determines the potential energy and force that acts on other charges.
By recognizing the relationship between linear charge density and the resulting electric field, we can determine the potential energy changes and predict how other charges will behave in potential scenarios. This is fundamental for solving problems related to electrostatics involving lines of charge.
Kinetic Energy
- \( KE = \frac{1}{2}mv^2 \)
- Where \( m \) is the mass of the proton \( 1.67 \times 10^{-27} \text{ kg} \)
- \( v \) is the velocity of \( 3.50 \times 10^3 \text{ m/s} \)
Plugging in these values provides an initial kinetic energy of approximately \( 1.02 \times 10^{-20} \text{ J} \). Understanding kinetic energy is vital to analyzing how much of the proton's energy is converted into potential energy as it moves towards the line of charge.
In many physics problems, kinetic energy sets the scene for exploring energy transfer or transformation between different forms like potential energy. For this exercise, kinetic energy represents the initial energy the proton has as it heads towards the line of charge, before being affected by the electric field.
Conservation of Energy
- Proton's initial kinetic energy is transformed into electric potential energy as it approaches the line of charge.
- This transformation happens due to the electric field created by the line.
The equation captures this conservation:\[ KE_{initial} + PE_{initial} = PE_{final} \]The initial potential energy \( PE_{initial} \) is often assumed to be zero if infinitely far from the line. Thus, the equation simplifies such that the initial kinetic energy equals the change in potential energy as the proton gets closer.
The closest approach, or \( r_{final} \), is calculated by considering the balance in kinetic and potential energy changes. Overall, energy conservation explains why the proton slows down, comes to a halt momentarily, and eventually reverses direction when too close to the line of charge. Understanding this principle is crucial for predicting motion and energy changes in electromagnetic fields.