A singly ionized (one electron removed) ^40 K atom passes through a velocity selector consisting

step 1 This is Chapter 27 problem number 30. We have ionized potassium atom, and it is under the influe

A singly ionized (one electron removed) ^40 K atom passes...

A singly ionized (one electron removed) $^{40} \mathrm{K}$ atom passes through a velocity selector consisting of uniform perpendicular electric and magnetic fields. The selector is adjusted to allow ions having a speed of 4.50 $\mathrm{km} / \mathrm{s}$ to pass through undeflected when the magnetic field is 0.0250 T. The ions next enter a second uniform magnetic field $\left(B^{\prime}\right)$ oriented at right angles to their velocity. 40 contains 19 protons and 21 neutrons and has a mass of $6.64 \times 10^{-26} \mathrm{kg} .$ (a) What is the magnitude of the electric field in the velocity selector? (b) What must be the magnitude of $B^{\prime}$ so that the ions will be bent into a semicircle of radius 12.5 $\mathrm{cm} ?$

Key Concepts

Centripetal Force and Cyclotron Motion

When a charged particle enters a magnetic field with its velocity perpendicular to the field, it undergoes circular motion due to the magnetic force acting as a centripetal force. The balance between this magnetic force and the required centripetal force (qvB = mv²/r) determines the radius of the particle's circular trajectory, which is central to applications like mass spectrometry and particle bending.

Lorentz Force

The Lorentz force law, expressed as F = q(E + v x B), describes the total force on a charged particle in the presence of both electric and magnetic fields. In the velocity selector, it explains how the forces can cancel for particles with the correct speed, and in the subsequent magnetic field, it describes how the particle is forced into circular (or semicircular) motion.

Velocity Selector

A velocity selector is an arrangement of perpendicular electric and magnetic fields that allows only particles of a specific speed to pass through undeflected. The electric and magnetic forces cancel for a particle whose speed meets the condition v = E/B, ensuring that only particles moving at that speed will continue along a straight path.

Electric Field

In charged particle motion, the electric field exerts a force on a particle proportional to its charge (F = qE) and acts in the direction of the field. In a velocity selector, this force is balanced by the magnetic force for particles moving with the selected speed.

Magnetic Field

A magnetic field exerts a force on a moving charged particle perpendicular to both its velocity and the field direction (F = qvB). This force causes the particle to bend, and its magnitude is critical for both selecting particles of certain speeds and determining the radius of curvature in circular motion.

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