Final answer:
In order to keep the proton traveling in a straight line between the parallel metal plates, a potential difference is necessary. The potential difference creates an electric field that can counteract the magnetic force on the proton, ensuring its straight path. The magnitude of the potential difference can be calculated using the formulas for the magnetic force and the centripetal force.
Step-by-step explanation:
In order to keep the proton traveling in a straight line between the parallel metal plates, a potential difference needs to be applied across the plates. This potential difference creates an electric field that can counteract the magnetic force on the proton, ensuring its straight path.
The force experienced by a charged particle moving through a magnetic field is given by the equation:
F = qvBsinθ
Where F is the magnetic force, q is the charge of the particle, v is its velocity, B is the magnetic field strength, and θ is the angle between the velocity and the magnetic field.
In this case, the proton is traveling due east, so its velocity is in the positive x-direction, while the magnetic field is directed north. Since the proton is moving perpendicular to the magnetic field, θ is 90 degrees, and sinθ is equal to 1.
Using the formula above, we can find the magnitude of the magnetic force on the proton, which is:
F = (1.6 x 10^-19 C)(4.58 x 10^5 m/s)(0.428 T)
Given that the force is perpendicular to the velocity, it must be equal to the centripetal force required to keep the proton moving in a straight line:
F = ma
The acceleration can be calculated using the formula:
a = v^2 / r
Where a is the acceleration and r is the radius of the circular path.
Setting the two forces equal to each other:
F = ma
(1.6 x 10^-19 C)(4.58 x 10^5 m/s)(0.428 T) = (1.6 x 10^-19 C) * (4.58 x 10^5 m/s)^2 / r
Simplifying the equation:
(1.6 x 10^-19 C)(0.428 T) = (1.6 x 10^-19 C) * (4.58 x 10^5 m/s)^2 / r
For the proton to move in a straight line, the radius of the circular path should be equal to the separation between the plates. Therefore, we can solve for the potential difference between the plates:
V = (1.6 x 10^-19 C) * (4.58 x 10^5 m/s)^2 / (5.828 x 10^-3 m)
Calculating the potential difference:
V = 7.14 x 10^3 V
Therefore, a potential difference of 7.14 x 10^3 V is necessary to keep the proton traveling in a straight line between the metal plates.