Suppose some ^235 U^+ ions are present in the beam. They...

Suppose some ${ }^{235} \mathrm{U}^{+}$ ions are present in the beam. They have the same charge as the ${ }^{238} \mathrm{U}^{+}$ ions but a smaller mass (approximately 0.98737m). (a) With what speed do the ${ }^{23} \mathrm{U}^{+}$ ions emerge from the accelerating plates, assuming ${ }^{238} \mathrm{U}^{+}$ ions emerge with speed $v ?$ (b) Sketch the trajectory of ${ }^{235} \mathrm{U}^{+}$ ions inside the velocity selector. (c) Now the velocity selector is removed. ${ }^{238} \mathrm{U}^{+}$ ions move in a circular path of diameter $D$ in the uniform magnetic field. What is the diameter of the path of the ${ }^{235} \mathrm{U}^{+}$ ions?

Suppose some ${ }^{235} \mathrm{U}^{+}$ ions are present in the beam. They have the same charge as the ${ }^{238} \mathrm{U}^{+}$ ions but a smaller mass (approximately 0.98737m). (a) With what speed do the ${ }^{23} \mathrm{U}^{+}$ ions emerge from the accelerating plates, assuming ${ }^{238} \mathrm{U}^{+}$ ions emerge with speed $v ?$
(b) Sketch the trajectory of ${ }^{235} \mathrm{U}^{+}$ ions inside the velocity selector.
(c) Now the velocity selector is removed. ${ }^{238} \mathrm{U}^{+}$ ions move in a circular path of diameter $D$ in the uniform magnetic field. What is the diameter of the path of the ${ }^{235} \mathrm{U}^{+}$ ions?

Key Concepts

Conservation of Energy for Charged Particles

This concept states that when a charged particle is accelerated through a potential difference, the work done by the electric field is converted into the particle’s kinetic energy. The relationship is commonly expressed as qV = ½mv², where q is the charge, V the potential difference, m the mass, and v the particle’s speed. This key concept underlies the determination of particle speeds after acceleration.

Mass Dependence of Kinetic Energy and Velocity

For ions gaining the same kinetic energy due to acceleration (because they have the same charge and are subject to the same potential difference), the resulting speed is inversely proportional to the square root of the mass. In other words, a lighter ion will emerge at a higher speed than a heavier one when both receive the same energy boost. This scaling factor is crucial when comparing ions of different masses in a beam.

Velocity Selector Operation

A velocity selector is an apparatus that uses perpendicular electric and magnetic fields to filter charged particles by speed. Particles moving at a specific speed experience equal electric and magnetic forces, allowing them to pass through undeflected. Particles with speeds differing from this selected value are deflected by the net force, thereby being excluded. Understanding this mechanism helps explain the trajectories of ions in devices designed to isolate particles by their velocities.

Charged Particle Motion in a Uniform Magnetic Field

When a charged particle moves through a uniform magnetic field, it experiences a Lorentz force that is perpendicular to both its velocity and the magnetic field. This force causes the particle to move in a circular path. The radius of this circle is determined by the equation r = mv/(qB), where m is the particle’s mass, v its velocity, q its charge, and B the magnetic field strength. Thus, for a given magnetic field and same charge, a lighter particle or one moving faster will have a different circular orbit compared to a heavier or slower particle.

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