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Chapter 21: Problem 13

In the operating room, anesthesiologists use mass spectrometers to monitor the respiratory gases of patients undergoing surgery. One gas that is often monitored is the anesthetic isoflurane (molecular mass \(=\) \(3.06 \times 10^{-25} \mathrm{kg}\) ). In a spectrometer, a singly ionized molecule of isoflurane (charge \(=+e\) ) moves at a speed of \(7.2 \times 10^{3} \mathrm{m} / \mathrm{s}\) on a circular path that has a radius of \(0.10 \mathrm{m} .\) What is the magnitude of the magnetic field that the spectrometer uses?

Short Answer

Expert verified
The magnetic field magnitude is approximately \( 1.38 \times 10^{-3} \text{ T} \).

Step by step solution

01

Understanding the Problem

We need to find the magnitude of the magnetic field that is required for a singly ionized molecule of isoflurane to move in a circular path with a given speed and radius. The mass, charge, speed, and radius are provided.
02

Identify Known Values

Let's list the given values: \( m = 3.06 \times 10^{-25} \text{ kg} \) is the molecular mass, \( v = 7.2 \times 10^{3} \text{ m/s} \) is the speed, \( r = 0.10 \text{ m} \) is the radius of the path, and \( q = e = 1.6 \times 10^{-19} \text{ C} \) for a singly ionized molecule.
03

Formulate the Key Equation

In a magnetic field, a charged particle moves in a circular path due to the Lorenz force, which can be equated to the centripetal force: \[ F_m = F_c \] \[ qvB = \frac{mv^2}{r} \]where \( B \) is the magnetic field strength. We need to solve for \( B \).
04

Rearrange the Equation to Solve for B

Rearrange the equation to solve for the magnetic field:\[ B = \frac{mv}{qr} \]
05

Substitute Known Values and Calculate B

Now, substitute the known values into the equation:\[ B = \frac{3.06 \times 10^{-25} \times 7.2 \times 10^3}{1.6 \times 10^{-19} \times 0.10} \]Calculate to find the value of \( B \).
06

Perform the Calculation

Carrying out the calculation:\[ B = \frac{(3.06 \times 10^{-25}) \times (7.2 \times 10^3)}{(1.6 \times 10^{-19}) \times (0.10)} = 1.377 \times 10^{-3} \text{ T} \]
07

Conclude with the Answer

We have determined that the magnitude of the magnetic field required in the spectrometer is approximately \( 1.38 \times 10^{-3} \text{ T} \).

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Key Concepts

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

Mass Spectrometry
Mass spectrometry is a powerful analytical technique used to measure the mass-to-charge ratio of ions. In the medical field, it plays a crucial role in monitoring the gases a patient breathes during surgery, such as the anesthetic isoflurane. By doing this, it ensures the accurate dosage and safety of the patient.

The main components of a mass spectrometer include an ion source, a mass analyzer, and a detector. Here's how it works:
  • Ion Source: This is where molecules are ionized, meaning they are given a positive or negative charge. For example, in the exercise, isoflurane becomes singly ionized.
  • Mass Analyzer: Ions are separated based on their mass-to-charge ratio. Charged particles are deflected by a magnetic field, causing them to move along curved paths. The radius of these paths helps determine their mass.
  • Detector: The ions are detected and measured, which allows for the identification and quantification of substances.
Mass spectrometry provides highly accurate measurements and can analyze complex mixtures. Moreover, it's a fundamental technique in biochemistry and pharmacology, aiding in the development of new drugs and understanding metabolism.
Lorentz Force
The Lorentz force is the force exerted on a charged particle when it moves through a magnetic field. Named after the physicist Hendrik Lorentz, it is key to understanding how particles like ions in mass spectrometers move.

When a charged particle, such as an ionized isoflurane molecule, enters a magnetic field, it experiences a force due to both its electric charge and its velocity. This can be expressed by the formula:
  • Force (\( F \) ) = Charge (\( q \) ) 脳 Velocity (\( v \) ) 脳 Magnetic Field (\( B \) ), or \( F = qvB \).
The direction of the Lorentz force is perpendicular to both the velocity of the particle and the magnetic field. This creates a centripetal force, causing the particle to follow a circular path.
  • The Lorentz force provides the centripetal force needed for the ion to undergo circular motion within the mass spectrometer.
  • This mechanism allows the separation of ions based on their mass-to-charge ratio, essential in identifying various particles in mass spectrometry.
Circular Motion
Circular motion involves any object moving along a circular path, and it requires a centripetal force to maintain such motion. In the context of the mass spectrometry exercise, this concept helps us understand how ions move within the device.

For an ion like isoflurane moving in a spectrometer, several factors determine its motion:
  • Centripetal Force: Necessary to keep the ion moving in a circle, provided by the Lorentz force (\( F_c = \frac{mv^2}{r} \) ).
  • Mass (\( m \) ): The heavier the ion, the more force required to hold it in a circular path.
  • Speed (\( v \) ): The speed affects the force needed since higher speeds demand more force to maintain circular motion.
  • Radius (\( r \) ): The curvature of the path influences the centripetal force; smaller radii require higher force.
These factors collectively determine the behavior of ions in mass spectrometry, aiding in their identification by comparing their paths under the influence of a magnetic field. Understanding this can simplify the optimization and calibration needed for precise measurement in scientific applications.

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Most popular questions from this chapter

At a certain location, the horizontal component of the earth's magnetic field is \(2.5 \times 10^{-5} \mathrm{T},\) due north. A proton moves eastward with just the right speed for the magnetic force on it to balance its weight. Find the speed of the proton.

A horizontal wire of length \(0.53 \mathrm{m},\) carrying a current of \(7.5 \mathrm{A},\) is placed in a uniform external magnetic field. When the wire is horizontal, it experiences no magnetic force. When the wire is tilted upward at an angle of \(19^{\circ},\) it experiences a magnetic force of \(4.4 \times 10^{-3} \mathrm{N} .\) Determine the magnitude of the external magnetic field.

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio \(\tau_{\text {square }} / \tau_{\text {rectangle }}\) of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

The magnetic field produced by the solenoid in a magnetic resonance imaging (MRI) system designed for measurements on whole human bodies has a field strength of \(7.0 \mathrm{T}\), and the current in the solenoid is \(2.0 \times 10^{2} \mathrm{A} .\) What is the number of turns per meter of length of the solenoid? Note that the solenoid used to produce the magnetic field in this type of system has a length that is not very long compared to its diameter. Because of this and other design considerations, your answer will be only an approximation.

A magnetic field has a magnitude of \(1.2 \times 10^{-3} \mathrm{T}\), and an electric field has a magnitude of \(4.6 \times 10^{3} \mathrm{N} / \mathrm{C} .\) Both fields point in the same direction. A positive \(1.8 \mu \mathrm{C}\) charge moves at a speed of \(3.1 \times 10^{6} \mathrm{m} / \mathrm{s}\) in a direction that is perpendicular to both fields. Determine the magnitude of the net force that acts on the charge.
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