Final answer:
To find the linear charge density (λ) of the wire, we equate the magnetic force providing centripetal force to the proton in circular motion around the wire to the required centripetal force and solve for λ. The calculation involves using the proton's velocity, which is acquired from its frequency of revolution and radius of orbit.
Step-by-step explanation:
To calculate the wire's linear charge density, we need to analyze the motion of the proton around the wire using concepts from electromagnetism and circular motion in physics.
Since the proton is orbiting around the wire, the magnetic force acting on it provides the necessary centripetal force to keep it in circular motion. This force is given by F = qvB, where q is the charge of the proton (q = 1.6 x 10-19 C), v is the velocity of the proton, and B is the magnetic field produced by the current in the wire.
The magnetic field around a long straight wire with a linear charge density λ can by found using Amp\u00e8re's Law, leading to B = (μ0λ)/2πr, where μ0 is the permeability of free space (μ0 = 4π x 10-7 H/m) and r is the radius of the orbit.
By setting the magnetic force equal to the centripetal force required for circular motion (mv2/r), we can solve for λ, taking into account that the proton makes 1.80 x 106 revolutions per second and the radius of the orbit is 1.20 cm.
Therefore, using v = 2πr × frequency and substituting the values into the equation, we can find λ, the linear charge density of the wire. Remember to convert the radius to meters for calculations.