A proton with a speed of 3.5 × 10
6
m/s is shot into a region between two plates that are
separated by a distance of 0.23 m. As indicated in the figure, a magnetic field exists between the
plates, and it is perpendicular to the velocity of the proton. What must be the magnitude of the
magnetic field so that the proton just misses colliding with the opposite plat
wouldn't
m v^r/r=Bqv
where r= .23/2 ?
To determine the magnitude of the magnetic field required for the proton to just miss colliding with the opposite plate, we can use the principles of magnetic force and centripetal force.
The magnetic force on a charged particle moving through a magnetic field is given by the equation:
F = qvBsinθ
where F is the 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, we know that the proton is moving perpendicular to the magnetic field, so sinθ is equal to 1. Therefore, the equation simplifies to:
F = qvB
The centripetal force required for the proton to move in a circular path without colliding with the opposite plate is given by:
Fc = mv^2 / r
where Fc is the centripetal force, m is the mass of the proton, v is its velocity, and r is the radius of the circular path.
In this problem, the radius of the circular path can be approximated by the distance between the plates, since the proton just misses colliding with the opposite plate. Therefore, r = 0.23 m.
Since the proton is not accelerating in the direction perpendicular to the magnetic field, the magnetic force and the centripetal force must be equal:
qvB = mv^2 / r
Simplifying the equation by canceling out v:
qB = mv / r
Now we can rearrange the equation to solve for the magnitude of the magnetic field B:
B = (mv) / (qr)
Given that the speed of the proton, v, is 3.5 × 10^6 m/s, the charge of a proton, q, is 1.6 × 10^-19 C, and the mass of a proton, m, is 1.67 × 10^-27 kg, we can substitute these values into the equation:
B = ((1.67 × 10^-27 kg)(3.5 × 10^6 m/s)) / ((1.6 × 10^-19 C)(0.23 m))
Calculating this expression will give you the magnitude of the magnetic field B required for the proton to just miss colliding with the opposite plate.