Suppose the electric field between the plates in Fig. 27.24...

Suppose the electric field between the plates in Fig. 27.24 is $1.88 \times 10 ^ { 4 } \mathrm { V } / \mathrm { m }$ and the magnetic field in both regions is 0.682$\mathrm { T }$ . If the source contains the three isotopes of krypton, $^ { 82 } \mathrm { Kr }$ $84 \mathrm { Kr } ,$ and $86 \mathrm { Kr } ,$ and the ions are singly charged, find the distance between the lines formed by the three isotopes on the particle detector. Assume the atomic masses of the isotopes (in atomic mass units are equal to their mass numbers, $82,84 ,$ and $86 .$ (One atomic mass unit $= 1 \mathrm { u } = 1.66 \times 10 ^ { - 27 } \mathrm { kg. }$

Suppose the electric field between the plates in Fig. 27.24 is $1.88 \times 10 ^ { 4 } \mathrm { V } / \mathrm { m }$ and the magnetic field in both regions is
0.682$\mathrm { T }$ . If the source contains the three isotopes of krypton, $^ { 82 } \mathrm { Kr }$
$84 \mathrm { Kr } ,$ and $86 \mathrm { Kr } ,$ and the ions are singly charged, find the distance
between the lines formed by the three isotopes on the particle detector. Assume the atomic masses of the isotopes (in atomic mass
units are equal to their mass numbers, $82,84 ,$ and $86 .$ (One atomic
mass unit $= 1 \mathrm { u } = 1.66 \times 10 ^ { - 27 } \mathrm { kg. }$

Key Concepts

Mass Spectrometry

Mass spectrometry is an analytical technique used to separate and identify ions based on their mass-to-charge (m/z) ratios. By subjecting accelerated ions to magnetic and sometimes electric fields, their paths curve to degrees that depend on their m/z ratios, allowing for precise identification and measurement of different isotopes or compounds.

Radius of Curvature in a Magnetic Field

When a charged particle enters a uniform magnetic field perpendicularly, it undergoes circular motion due to the magnetic force acting as a centripetal force. The radius of this curvature is given by the equation r = mv/(qB), linking the particle's mass, velocity, charge, and the strength of the magnetic field, which is fundamental for understanding how particles with different masses are separated.

Charge-to-Mass Ratio

The charge-to-mass ratio (q/m) is a key parameter in determining how a charged particle will respond to electric and magnetic fields. Precision in measuring or calculating the q/m ratio is essential in techniques like mass spectrometry, as differences in this ratio directly impact the radius of curvature and the resulting spatial separation on a detector.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through electric and magnetic fields. It is the vector sum of the electric force (which causes acceleration in the direction of the field) and the magnetic force (which acts perpendicular to both the magnetic field and the particle’s velocity), and it governs the particle's trajectory in such environments.

Electric Field Acceleration

An electric field can accelerate charged particles by exerting a force on them in the direction of the field. This acceleration increases the particle’s kinetic energy and determines its velocity when it enters regions influenced by magnetic fields, making it a critical step in devices that analyze particle properties.

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