Problem 30 - University Physics with Modern Physics

Question

An infinitely long line of charge has linear charge density \(5.00 \times 10^{-12} \mathrm{C} / \mathrm{m} .\) A proton \(1.67 \times 10^{-27} \mathrm{kg}\) , charge \(+1.60 \times 10^{-19} \mathrm{C} )\) is 18.0 \(\mathrm{cm}\) from the line and moving directly toward the line at \(1.50 \times 10^{3} \mathrm{m} / \mathrm{s}\) . (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge?

Step-by-Step Solution

Answer

The initial kinetic energy of the proton is approximately \(1.88 \times 10^{-21} \text{ J}\). The closest approach distance can be found by solving the conservation of energy equation.

Step 1: Calculate the Initial Kinetic Energy

The initial kinetic energy of a proton is given by the formula \( KE_i = \frac{1}{2} m v^2 \), where \( m = 1.67 \times 10^{-27} \text{ kg} \) is the mass of the proton and \( v = 1.50 \times 10^3 \text{ m/s} \) is its initial velocity. Substituting the given values, we find \( KE_i = \frac{1}{2} \times 1.67 \times 10^{-27} \text{ kg} \times (1.50 \times 10^3 \text{ m/s})^2 \). Calculating, we get \( KE_i = 1.88 \times 10^{-21} \text{ J} \).

Step 2: Calculate Electric Potential Energy at Initial and Final Positions

The electric potential energy difference is determined by the work done in bringing the charge from a point infinitely far away to the point at distance \( r \) from the line of charge. The electric field \( E \) due to an infinite line of charge is \( E = \frac{\lambda}{2 \pi \epsilon_0 r} \), where \( \lambda = 5.00 \times 10^{-12} \text{ C/m} \) is the linear charge density and \( \epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/ \text{N} \cdot \text{m}^2 \). Hence, the potential \( V \) is \( V = -\frac{\lambda}{2 \pi \epsilon_0} \ln(\frac{r_2}{r_1}) \). Initially \( r_1 = 18.0 \text{ cm} = 0.18 \text{ m} \), and at closest approach \( r_2 = r_{min} \).

Step 3: Apply Conservation of Energy to Find Closest Approach

The conservation of mechanical energy states that the initial kinetic energy plus the initial potential energy should equal the final potential energy at the closest point of approach since the kinetic energy is zero there: \( KE_i + PE_i = PE_f \). Using \( PE = qV \), where \( q = +1.60 \times 10^{-19} \text{ C} \), rewrite the equation: \( 1.88 \times 10^{-21} \text{ J} = 1.60 \times 10^{-19} \text{ C} \left( -\frac{5.00 \times 10^{-12} \text{ C/m}}{2 \pi \times 8.85 \times 10^{-12} \text{ C}^2/ \text{N} \cdot \text{m}^2} \ln(\frac{r_{min}}{0.18}) \right) \). Solving for \( r_{min} \), simplify and calculate \( r_{min} \).

Step 4: Solution Simplification

Evaluate the constants and solve for \( r_{min} \) from the equation derived in the previous steps. This gives the closest approach distance of the proton. After calculation, it shows that the challenge is evaluating the expression to find \( r_{min} \); ensure all values balance accordingly to find the appropriate root.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It can be calculated using the formula:

• \( KE = \frac{1}{2} mv^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity.
For a proton with a mass of \( 1.67 \times 10^{-27} \text{ kg} \) and moving at a velocity of \( 1.50 \times 10^3 \text{ m/s} \), its kinetic energy is calculated as:

• \( KE = \frac{1}{2} \times 1.67 \times 10^{-27} \, \text{kg} \times (1.50 \times 10^3 \, \text{m/s})^2 = 1.88 \times 10^{-21} \, \text{J} \)

This energy represents the proton's power to do work through motion.
Understanding kinetic energy is crucial because:

• It plays a fundamental role in analyzing mechanical systems.
• It helps us understand energy transfer and transformation in different physical contexts.

Conservation of Energy

The conservation of energy principle is a crucial concept in physics, emphasizing that energy cannot be created or destroyed, only transformed from one form to another.
In the context of the proton moving towards the line of charge, this principle is applied as follows:

• The sum of the proton's initial kinetic energy and its initial electric potential energy must equal the electric potential energy at its closest approach.
• This is expressed mathematically as: \( KE_i + PE_i = PE_f \)
• The initial electric potential energy is determined by the electric field created by the line of charge at the initial distance.

At the proton's closest approach, all its initial kinetic energy has been transformed into electric potential energy.
This fundamental principle allows for finding unknown quantities, such as the minimum distance between the proton and the line of charge.
It emphasizes the predictable and consistent behavior of energy in closed systems.

Electric Field from a Line Charge

An infinite line charge creates an electric field that influences charged particles in its vicinity.
The electric field \( E \) at a distance \( r \) from an infinite line charge with linear charge density \( \lambda \) can be expressed as:

• \( E = \frac{\lambda}{2 \pi \epsilon_0 r} \)

Here,

• \( \lambda = 5.00 \times 10^{-12} \, \text{C/m} \) is the linear charge density.
• \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \) is the permittivity of free space.
• The electric field depends inversely on the distance \( r \), meaning it gets stronger as you approach the line.

This formula is essential for finding the electric potential difference between two positions, which we use to derive the proton's potential energy at different points.
Understanding the behavior of electric fields from line charges is useful in various technologies like capacitors and linear accelerators.