A long, straight wire lies on a horizontal table and carries a current of 3.05 µA. In a vacuum, a proton moves parallel to the wire (opposite the current) with a constant speed of 3.00 104 m/s at a distance d above the wire. Ignoring the magnetic field due to the Earth, determine the value of d.
To determine the value of d, we need to use the concept of the magnetic force on a moving charged particle due to a current-carrying wire.
The magnetic force on a moving charged particle can be calculated using the formula:
F = |q| * v * B * sin(θ)
Where:
F is the magnetic force,
|q| is the magnitude of the charge of the particle,
v is the velocity of the particle,
B is the magnetic field strength, and
θ is the angle between the velocity vector and the magnetic field vector.
In this case, the proton is moving parallel to the wire, so the angle θ between the velocity vector and the magnetic field vector is 90 degrees, leading to sin(θ) = 1.
Since we are trying to find the value of d, which is the distance above the wire, we can set up the equation for the magnetic force as follows:
F = |q| * v * B * sin(θ) = (3.20 x 10^-19 C) * (3.00 x 10^4 m/s) * B
The force on the proton in this case is due to the magnetic field generated by the current in the wire. The magnetic field produced by a straight wire can be calculated using Ampere's law:
B = (μ₀ * I) / (2π * r)
Where:
B is the magnetic field,
μ₀ is the permeability of free space (4π x 10^-7 T*m/A),
I is the current in the wire, and
r is the distance from the wire.
Substituting the values into the equation, we have:
B = (4π x 10^-7 T*m/A) * (3.05 x 10^-6 A) / (2π * r)
B = 6.10 x 10^-13 T / r
Now, substitute this expression for B into the equation for the magnetic force:
F = (3.20 x 10^-19 C) * (3.00 x 10^4 m/s) * (6.10 x 10^-13 T / r)
Since the proton is moving at a constant velocity and not accelerating vertically, the upward electric force on the proton is balanced by the downward gravitational force acting on it. Therefore, the magnetic force must be equal and opposite to the gravitational force. The gravitational force on a proton is given by:
F_gravity = m * g
Where:
m is the mass of the proton (1.67 x 10^-27 kg), and
g is the acceleration due to gravity on Earth (9.8 m/s^2).
So, we can set the magnetic force equal to the gravitational force:
(3.20 x 10^-19 C) * (3.00 x 10^4 m/s) * (6.10 x 10^-13 T / r) = (1.67 x 10^-27 kg) * (9.8 m/s^2)
Now, solve for r:
r = ((1.67 x 10^-27 kg) * (9.8 m/s^2)) / ((3.20 x 10^-19 C) * (3.00 x 10^4 m/s) * (6.10 x 10^-13 T))
Calculating this expression will give you the value of r, which is the distance d above the wire.