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Chapter 24: Problem 24

An electromagnetic flow meter applies a magnetic field of \(0.20 \mathrm{T}\) to blood flowing through a coronary artery at a speed of \(15 \mathrm{cm} / \mathrm{s} .\) What force is felt by a chlorine ion with a single negative charge?

Short Answer

Expert verified
The chlorine ion feels a force of \( -4.8 脳 10^{-21} \,N \)

Step by step solution

01

Understand the Relevant Formula

The Lorentz force law dictates the force experienced by a charged particle moving in a magnetic field. The formula for the force is \( F = qvB \), where \( F \) is the force, \( q \) is the charge of the particle, \( v \) is the velocity of the particle, and \( B \) is the strength of the magnetic field.
02

Convert Velocity to Appropriate Units

The velocity of the blood is given in centimeters per second, but the standard unit of velocity in physics is meters per second. So, first convert 15 cm/s to m/s by dividing by 100, which gives 0.15 m/s.
03

Calculate the Force

Now, substitute the values into the formula. The charge of a single electron is \( -1.6 脳 10^{-19} \,C \), the velocity is 0.15 m/s, and the magnetic field strength is 0.2 T. Therefore the force is \( F = (-1.6 脳 10^{-19}\,C)(0.15\,m/s)(0.2\,T) =-4.8 脳 10^{-21} \,N \). The negative sign indicates the force is in the opposite direction of the velocity.

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Key Concepts

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

Understanding Magnetic Field
A magnetic field is an invisible force field surrounding magnets or moving charges. It's characterized by the symbol \( B \) and is measured in Tesla (T). The strength of the magnetic field can influence how charged particles move. For any charged particle moving within a magnetic field like in our exercise, the magnetic field will exert a force on the particle and this is particularly essential in applications such as electromagnetic flow meters. These devices use magnetic fields to measure the flow rate of fluids, including blood as seen in this case, by detecting the Lorentz force on charged particles in the fluid.
Velocity Conversion for Accurate Calculations
When working with velocity in physics exercises, it's crucial to use the proper units for consistent and correct calculations. Velocity is a vector quantity, meaning it has both magnitude and direction, and is typically measured in meters per second (m/s). However, sometimes, it's provided in centimeters per second (cm/s), which requires conversion for equations like the one seen in the Lorentz force formula. To convert from \(15\,\mathrm{cm/s} \) to meters per second, divide by 100, converting it to \(0.15\,\mathrm{m/s} \). This ensures that when you insert values into formulas, the units are compatible and results are accurate.
Explaining Electric Charge
Electric charge (\(q\)) is a fundamental property of particles that causes them to experience a force in the presence of electric or magnetic fields. Charges are measured in Coulombs (C). In our exercise, we consider a chlorine ion with a single negative charge, which is equivalent to the charge of an electron. This charge is \(-1.6 脳 10^{-19}\, C\). Understanding the nature of electric charge is critical because it's one of the key variables that determine the magnitude of the force experienced by a particle in a magnetic field. The sign of the charge also affects the direction of the force, an essential detail highlighted in applications like blood flow meters.
Steps in Force Calculation Using Lorentz Force Formula
Calculating the force exerted by a magnetic field on a charged particle requires the Lorentz force formula: \( F = qvB \). This formula combines the charge (\(q\)) of the particle, its velocity (\(v\)), and the magnetic field strength (\(B\)). First, ensure all measurements are in consistent units, particularly with velocity being in meters per second. Then, input these variables into the formula. For instance, substituting a charge of \(-1.6 脳 10^{-19}\, C\), a velocity of \(0.15 \mathrm{m/s}\), and a magnetic field of \(0.2 \mathrm{T}\), the resulting force calculation is: \[ F = (-1.6 脳 10^{-19} \, C)(0.15 \, m/s)(0.2 \, T) = -4.8 脳 10^{-21} \, N \]The negative sign clearly conveys the force's direction is opposite to the particle's velocity.
This insightful understanding into each step is critical for successfully managing similar physics problems.

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Most popular questions from this chapter

The ocean is salty because it contains many dissolved ions. As these charged particles move with the water in strong ocean currents, they feel a force from the earth's magnetic field. Positive and negative charges are separated until an electric field develops that balances this magnetic force. This field produces measurable potential differences that can be monitored by ocean researchers. The Gulf Stream moves northward off the east coast of the United States at a speed of up to \(3.5 \mathrm{m} / \mathrm{s}\). Assume that the current flows at this maximum speed and that the earth's field is \(50 \mu \mathrm{T}\) tipped \(60^{\circ}\) below horizontal. What is the direction of the magnetic force on a singly ionized negative chlorine ion moving in this ocean current? A. East B. West C. Up D. Down

A researcher would like to perform an experiment in zero magnetic field, which means that the field of the earth must be canceled. Suppose the experiment is done inside a solenoid of diameter \(1.0 \mathrm{m},\) length \(4.0 \mathrm{m},\) with a total of 5000 turns of wire. The solenoid is oriented to produce a field that opposes and exactly cancels the \(52 \mu\) T local value of the earth's field. What current is needed in the solenoid's wire?

II A square current loop \(5.0 \mathrm{cm}\) on each side carries a \(500 \mathrm{mA}\) current. The loop is in a \(1.2 \mathrm{T}\) uniform magnetic field. The axis of the loop, perpendicular to the plane of the loop, is \(30^{\circ}\) away from the field direction. What is the magnitude of the torque on the current loop?

Bats are capable of navigating using the earth's field-a plus for an animal that may fly great distances from its roost at night. If, while sleeping during the day, bats are exposed to a field of a similar magnitude but different direction than the earth's field, they are more likely to lose their way during their next lengthy night flight. Suppose you are a researcher doing such an experiment in a location where the earth's field is \(50 \mu \mathrm{T}\) at a \(60^{\circ}\) angle below horizontal. You make a \(50-\mathrm{cm}-\) diameter, 100 -turn coil around a roosting box; the sleeping bats are at the center of the coil. You wish to pass a current through the coil to produce a field that, when combined with the earth's field, creates a net field with the same strength and dip angle \(\left(60^{\circ}\right.\) below horizontal) as the earth's field but with a horizontal component that points south rather than north. What are the proper orientation of the coil and the necessary current?

In a simplified model of the hydrogen atom, its electron moves in a circular orbit with a radius of \(5.3 \times 10^{-10} \mathrm{m}\) at a frequency of \(6.6 \times 10^{15}\) Hz. What magnetic field would be required to cause an electron to undergo this same motion?
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