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Chapter 24: Problem 27

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?

Short Answer

Expert verified
The required magnetic field would be approximately \(1.32 \times 10^{-4} \mathrm{T}\)

Step by step solution

01

Writing down relevant principles

The Bernoulli equation postulates that the centripetal force is equal to the magnetic force. The centripetal force can be represented as \(m \cdot v^2 / r\) and the magnetic force can be represented as \(m \cdot v \cdot B\), where m is the mass of the electron, v is the velocity, r is the radius of the orbit and B is the magnetic field. Therefore, we have the equation \(m \cdot v^2 / r = m \cdot v \cdot B\).
02

Simplifying the equation

Simplifying the equation from the previous step, we can cancel out the common factors on both sides (m and v), which leaves us with the equation \(v / r = B\).
03

Searching for an expression for velocity in terms of frequency

Velocity (v) can be represented in terms of frequency (f) as \(v = 2 \cdot \pi \cdot r \cdot f\). Substituting this into our equation, we are left with \(2 \cdot \pi \cdot f = B\).
04

Finding the Magnetic Field

We can then solve for B by inserting the given values of the frequency (f) and the radius (r) into the equation. Given that the frequency f is given as \(6.6 \times 10^{15}\) Hz, we get \(B = 2 \cdot \pi \cdot 6.6 \times 10^{15}\), which comes out to be approximately \(1.32 \times 10^{-4} \mathrm{T}\)

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Key Concepts

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

Circular Motion of Electrons
The movement of an electron around the nucleus in a hydrogen atom can exhibit circular motion, much like a planet orbiting a star. Just as gravity keeps a planet in orbit, electrical forces keep the electron bound to the nucleus. Still, when describing the motion in terms of classical physics, we can utilize the same principles that govern any object in circular motion.

Crucial to understanding this motion is the concept of centripetal force, which is the net force causing the inward acceleration of the electron as it travels in a circular path. This force is necessary for any circular motion and is always directed towards the center of the circle. According to classical physics, in the absence of any external force, an object in motion would continue in a straight line. However, in circular motion, there's a continuous change of direction which implies a constant acceleration. This change in direction requires a force, known as centripetal force, to cause the circular path. In the simplified model of the hydrogen atom, the motion of the electron can be thought of as planetary motion following Newton's laws.
Magnetic Force on Charged Particles
When a charged particle, such as an electron, moves through a magnetic field, it experiences a force鈥攌nown as the magnetic Lorentz force. This force is perpendicular to both the direction of the magnetic field and the velocity of the particle. The strength of the force depends on the charge of the particle, the speed at which it's moving, and the strength of the magnetic field.

Mathematically, we describe the magnetic force (\(F_m\)) using the equation \(F_m = qvB\sin\theta\), where \(q\) is the charge of the particle, \(v\) is its velocity, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the direction of velocity and magnetic field. For an electron in circular motion within a magnetic field, \(\theta\) is 90 degrees, and the sine function simplifies to 1. Thus, in such cases, the Magnetic Lorentz force can provide the necessary centripetal force to sustain the electron's circular path.
Centripetal Force and Magnetic Field Relationship
The interaction between the centripetal force and a magnetic field in the context of an electron orbiting the nucleus of a hydrogen atom is quite telling. For an electron undergoing circular motion, the centripetal force required to maintain its path is provided by the magnetic Lorentz force when a magnetic field is present.

This relationship can be captured with the formula \(m\frac{v^2}{r} = qvB\), where \(m\) is the mass of the electron, \(v\) is its velocity, \(r\) is the radius of the circle of motion, \(q\) is the charge of the electron, and \(B\) is the magnetic field strength. By rearranging the terms and knowing the charge and mass of an electron, one can solve for the magnetic field \(B\) necessary to sustain the specific circular motion. This relationship frames the principles behind devices such as cyclotrons, which are used to accelerate charged particles to high speeds using magnetic fields.

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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 solenoid used to produce magnetic fields for research purposes is \(2.0 \mathrm{m}\) long, with an inner radius of \(30 \mathrm{cm}\) and 1000 turns of wire. When running, the solenoid produces a field of \(1.0 \mathrm{T}\) in the center. Given this, how large a current does it carry?

Why is it important that the ions have a known speed? A. The radius of the orbit depends on the mass, the charge, and the speed. If the charge and the speed are the same, the orbit depends on only the mass. B. The orbit must be circular, and this is the case for only a certain range of speeds. C. If the ions are moving too fast, the magnetic field will not be able to bend their path to the detector. D. The ions are all accelerated by the same electric field, and so will all have the same speed anyway.

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?

The magnetic field of the brain has been measured to be approximately \(3.0 \times 10^{-12} \mathrm{T} .\) Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop \(16 \mathrm{cm}\) (the width of a typical head) in diameter. What current is needed to produce such a field at the center of the loop?
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