An infinitely long line of charge has linear charge den­sity 5.00x10^-12 C/m. A proton (mass 1.67x10^-27 kg, charge +1.60x10^-19 C) is 18.0 cm from the line and moving directly toward the line at 3.50x10^3 m/s. (b) How close does the proton get to the line of cha rge?

Verified step by step guidance

1. Calculate the electric field (E) generated by the infinitely long line of charge using the formula: E = \( \frac{\lambda}{2\pi \epsilon_0 r} \), where \( \lambda \) is the linear charge density, \( \epsilon_0 \) is the permittivity of free space (approximately \( 8.85 \times 10^{-12} \) C²/(N·m²)), and r is the distance from the line of charge.
2. Determine the force (F) acting on the proton due to the electric field using Coulomb's Law: F = qE, where q is the charge of the proton.
3. Set up the equation of motion for the proton under the influence of this electric force. The net force acting on the proton provides the centripetal force required to change the direction of the proton as it moves towards the line of charge. Use the formula: F = ma, where m is the mass of the proton and a is the acceleration.
4. As the proton approaches the line of charge, its kinetic energy decreases due to the work done against the electric force. Set up the conservation of energy equation: Initial Kinetic Energy = Work done by the electric force + Final Kinetic Energy. Solve this equation to find the minimum distance (r_min) from the line of charge where the velocity of the proton becomes zero.
5. Substitute all known values into the equations and solve for r_min, the closest distance the proton gets to the line of charge.

Key Concepts

Electric Field of a Line Charge

The electric field generated by an infinitely long line of charge can be calculated using the formula E = (λ / (2πε₀r)), where λ is the linear charge density, ε₀ is the permittivity of free space, and r is the distance from the line. This electric field points radially outward from the line if the charge is positive, affecting the motion of charged particles nearby.

Kinematics of Charged Particles

The motion of charged particles, such as protons, in an electric field can be analyzed using kinematic equations. The force acting on the proton due to the electric field will cause it to accelerate, and its trajectory can be predicted by applying Newton's second law (F = ma) to determine how its velocity and position change over time.

Conservation of Energy

In this scenario, the conservation of energy principle states that the total mechanical energy (kinetic plus potential) of the proton remains constant if only conservative forces are acting. As the proton moves closer to the line of charge, its kinetic energy will convert into electric potential energy, allowing us to calculate the closest approach by equating these energy forms.