A deuteron particle (the nucleus of an isotope of hydrogen...

A deuteron particle (the nucleus of an isotope of hydrogen consisting of one proton and one neutron and having a mass of $3.34 \times 10^{-27} \mathrm{kg}$ ) moving horizontally enters a uniform, vertical, 0.500 T magnetic field and follows a circular arc of radius 55.6 $\mathrm{cm} .$ (a) How fast was this deuteron moving just before it entered the magnetic field and just after it came out of the field? (b) What would be the radius of the arc followed by a proton that entered the field with the same velocity as the deuteron?

Key Concepts

Lorentz Force

This is the force experienced by a charged particle moving in a magnetic field. It is given by the vector cross-product between the particle's velocity and the magnetic field. Since the force is always perpendicular to the direction of motion, it does not change the particle’s speed, only its direction, causing the particle to follow a curved path.

Uniform Magnetic Field

A magnetic field that has the same magnitude and direction at every point. In such a field, a charged particle moving perpendicular to the field lines will experience a constant magnetic force, leading to uniform circular motion.

Circular Motion of Charged Particles

When a charged particle moves in a plane perpendicular to a uniform magnetic field, the constant perpendicular force causes the particle to follow a circular path. The radius of the circular path depends on the mass, speed of the particle, and its charge, as well as the strength of the magnetic field.

Radius of Curvature Formula

The radius of the circular path followed by a charged particle in a magnetic field is given by the formula r = (mv)/(qB), where m is the mass of the particle, v is its speed, q is its charge, and B is the magnetic field strength. This formula allows one to relate the physical properties of the particle to its trajectory in the field.

Conservation of Speed in Magnetic Fields

Since the magnetic force does no work on the particle—it is always perpendicular to the velocity—the particle’s kinetic energy and speed remain constant as it moves through the magnetic field. Any change in the trajectory is due solely to the change in direction, not in speed.

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