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Chapter 27: Problem 53

Suppose the electric field between the plates in Fig. 27.22 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 \(=1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg} .\) )

Short Answer

Expert verified
The solution of the exercise involves calculation of the radii of the paths for the isotopes and the subsequent calculation the difference in these radii. This determination of the radii takes into consideration the proportionality of the radius with the mass. The final solution would be numeric after substituting the given.mass, charge and magnetic field values into the formula and performing the arithmetic operations

Step by step solution

01

Understand and apply the formula

The radius of the circular path of the particle in a magnetic field can be calculated with the formula \(r = \frac{mv}{qB}\), where \(m\) is the mass of the particle, \(v\) is the velocity, \(q\) is the charge and \(B\) is the magnetic field. The velocity picks up in the electric field and can be calculated with the formula \(v = \frac{Ed}{q}\), where \(E\) is the electric field and \(d\) is the distance between the plates. Since the charge and the electric field are the same for all isotopes, the velocity will also be the same. Hence, the radius is proportional to the mass.
02

Calculate the radius for each isotope

The isotopes have masses 82u, 84u and 86u. Since 1 atomic mass unit \(u = 1.66 脳 10^{-27}\) kg, the radius for each isotope can be calculated by multiplying \(u\) with the respective masses. Convert the masses into kg and then substitute the values into the formula for \(r\) to calculate the radius of the path for each isotope.
03

Calculate the distance between lines

The distance between the lines will be twice the difference in the radii of the paths for the isotopes, as any two lines on the screen are the corresponding points of two mutually tangential circles. Calculate the difference in the radii for each neighboring isotopes and double the value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field Particle Trajectory
The trajectory of a charged particle moving in a magnetic field deviates from a straight line due to the presence of the Lorentz force. This force is perpendicular to the direction of both the magnetic field and the velocity of the particle, causing the particle to follow a circular path. The radius of this path, or trajectory, can be determined using the formula \( r = \frac{mv}{qB} \), where \( m \) represents the mass of the particle, \( v \) its velocity, \( q \) its charge, and \( B \) the magnetic field strength.

Understanding the influence of the magnetic field on the trajectory is crucial for analyzing how isotopes separate within a mass spectrometer. Because the mass of each isotope differs, even if slightly, each type of isotope will have a unique radius of curvature when moving through the magnetic field. Consequently, this results in distinct paths for each isotope, making it possible to identify and measure them based on their respective trajectories.
Isotope Separation
Isotope separation is a process of differentiating and segregating isotopes of the same element, which have the same number of protons but vary in the number of neutrons. In mass spectrometry, this is accomplished by exploiting differences in the mass-to-charge ratio (\( m/q \) ratio) of ions. When ions of different isotopes are subjected to a magnetic field, each follows a separate trajectory based on its unique \( m/q \) ratio.

In our example involving the isotopes of krypton, identifiable trajectories appear on a particle detector because the isotopes are singly charged and have slightly different masses. The separation process allows the mass spectrometer to discriminate between \( {}^{82}\mathrm{Kr} \), \( {}^{84}\mathrm{Kr} \) and \( {}^{86}\mathrm{Kr} \) because they trace different paths and create distinct lines on the detector. The mass spectrometer captures these distinct paths as patterns of deflection, which reflects their atomic mass and the magnetic field's influence on their trajectory.
Mass-to-Charge Ratio
The mass-to-charge ratio (\( m/q \) ratio) is a critical parameter in mass spectrometry, determining how particles with different masses are deflected by a magnetic field. The trajectory radius of a particle in the presence of a magnetic field is directly proportional to its mass and inversely proportional to the charge. Therefore, particles with a higher mass-to-charge ratio will have a larger trajectory radius.

For isotopes that have the same charge, like the singly charged krypton ions in the textbook example, the differing radius of the path each isotope takes is only due to the differences in their masses. Knowing the velocity and electric field applied, we can calculate the expected radius for each isotope's path. Thus, the krypton isotopes, having nearly the same charge but different masses (82u, 84u, and 86u), will have varying mass-to-charge ratios. These differences lead to different trajectory radii in the magnetic field, enabling the isotope separation process.

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Most popular questions from this chapter

A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses (see Section 27.5 ). In one design for such an instrument, ions with mass \(m\) and charge \(q\) are accelerated through a potential difference \(V\). They then enter a uniform magnetic field that is perpendicular to their velocity, and they are deflected in a semicircular path of radius \(R .\) A detector measures where the ions complete the semicircle and from this it is easy to calculate \(R\). (a) Derive the equation for calculating the mass of the ion from measurements of \(B, V, R,\) and \(q\). (b) What potential difference \(V\) is needed so that singly ionized \({ }^{12} \mathrm{C}\) atoms will have \(R=50.0 \mathrm{~cm}\) in a 0.150 T magnetic field? (c) Suppose the beam consists of a mixture of \({ }^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C}\) ions. If \(v\) and \(B\) have the same values as in part \((\mathrm{b}),\) calculate the separation of these two isotopes at the detector. Do you think that this beam separation is sufficient for the two ions to be distinguished? (Make the assumption described in Problem 27.53 for the masses of the ions.)

An electron is moving in the \(x y\) -plane. If at time \(t\) a magnetic field \(B=0.200 \mathrm{~T}\) in the \(+z\) -direction exerts a force on the electron equal to \(F=5.50 \times 10^{-18} \mathrm{~N}\) in the \(-y\) -direction, what is the velocity (magnitude and direction) of the electron at this instant?

A straight, vertical wire carries a current of 2.60 A down- ward in a region between the poles of a large superconducting electromagnet, where the magnetic field has magnitude \(B=0.588 \mathrm{~T}\) and is horizontal. What are the magnitude and direction of the magnetic force on a \(1.00 \mathrm{~cm}\) section of the wire that is in this uniform magnetic field, if the magnetic field direction is (a) east; (b) south; (c) \(30.0^{\circ}\) south of west?

A coil with magnetic moment \(1.45 \mathrm{~A} \cdot \mathrm{m}^{2}\) is oriented initially with its magnetic moment antiparallel to a uniform 0.835 T magnetic field. What is the change in potential energy of the coil when it is rotated \(180^{\circ}\) so that its magnetic moment is parallel to the field?

The plane of a \(5.0 \mathrm{~cm} \times 8.0 \mathrm{~cm}\) rectangular loop of wire is parallel to a 0.19 T magnetic field. The loop carries a current of 6.2 A. (a) What torque acts on the loop? (b) What is the magnetic moment of the loop? (c) What is the maximum torque that can be obtained with the same total length of wire carrying the same current in this magnetic field?
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