Final answer:
The proton gets as close as 2.80 * 10^-5 meters to the line of charge.
Step-by-step explanation:
To determine how close the proton gets to the line of charge, we need to calculate the electric field generated by the line of charge at the proton's initial position.
The electric field due to an infinitely long line of charge can be calculated using the following formula:
E = k * λ / r
Where E is the electric field, k is the Coulomb's constant (8.99 * 10^9 N*m^2/C^2), λ is the linear charge density of the line of charge, and r is the distance from the line of charge.
Substituting the given values, we find:
E = (8.99 * 10^9 N*m^2/C^2) * (4.00 * 10^-12 C/m) / (0.19 m)
Simplifying the expression, we get:
E = 1.88 * 10^14 N/C
The electric field exerts a force on the proton, causing it to change its trajectory. To find how close the proton gets to the line of charge, we can use the equation for the force on a charged particle in an electric field:
F = q * E
Where F is the force, q is the charge of the particle, and E is the electric field.
The force experienced by the proton is equal to the centripetal force required to keep it moving in a circle. Using the formula for centripetal force:
F = m * (v^2 / r)
Where F is the force, m is the mass of the proton, v is the velocity of the proton, and r is the distance from the line of charge.
Equating the two expressions for force, we can solve for r:
m * (v^2 / r) = q * E
Substituting the given values, we find:
(1.67 * 10^-27 kg) * (2400 m/s)^2 / r = (1.60 * 10^-19 C) * (1.88 * 10^14 N/C)
Simplifying the expression and solving for r, we get:
r = 2.80 * 10^-5 m
Therefore, the proton gets as close as 2.80 * 10^-5 meters to the line of charge.