Question

Chapter 14 discussed damped harmonic oscillators, in which the damping force is velocity dependent and always opposes the motion of the oscillator. One way of producing this type of force is to use a piece of metal, such as aluminum, that moves through a nonuniform magnetic field. Explain why this technique is capable of producing a damping force.

Short answer

Explain why a piece of metal moving through a nonuniform magnetic field can produce a damping force on a damped harmonic oscillator. When a piece of metal moves through a nonuniform magnetic field, an electromotive force (EMF) is induced in the metal according to Faraday's Law of Electromagnetic Induction, creating an induced electric current. Lenz's Law states that this induced electric current will create a magnetic field that opposes the change in magnetic flux that produced it. The interaction between the induced magnetic field and the nonuniform magnetic field generates an induced force that opposes the motion of the oscillator. This induced force acts as a damping force, causing the amplitude of oscillations to decay over time.

Step-by-step solution

A damped harmonic oscillator is a system that experiences an oscillatory motion, such as a mass attached to a spring or a pendulum, but also subject to an external damping force that opposes its motion. This damping force causes the amplitude of oscillation to gradually decay over time. The damping force is typically proportional to the velocity of the oscillator. #Step 2: Faraday's Law of Electromagnetic Induction#

When a piece of metal, such as aluminum, moves through a nonuniform magnetic field, a change in magnetic flux occurs, causing an electromotive force (EMF) to be induced in the metal, according to Faraday's Law of Electromagnetic Induction. The EMF generated leads to the creation of an induced electric current in the metal. Mathematically, Faraday's Law can be expressed as: \(\text{EMF} = -\dfrac{d\Phi_B}{dt}\), where \(\Phi_B\) is the magnetic flux and \(t\) is time. #Step 3: Lenz's Law and Induced Force#

Lenz's Law states that the induced electric current in the metal will create a magnetic field that opposes the change in magnetic flux that produced it, which means that the induced magnetic field will always oppose the motion of the oscillator. The interaction between the induced magnetic field and the nonuniform magnetic field will produce a force, called the "induced force," which opposes the motion of the oscillator as well. #Step 4: Connecting Induced Force to Damping Force#

Since the induced force opposes the motion of the oscillator, it can be considered as a damping force acting on the oscillator. This damping force, which is the result of the velocity of the metal moving through the nonuniform magnetic field, will cause the amplitude of oscillations to decay over time. Thus, a nonuniform magnetic field can produce a damping force on a damped harmonic oscillator when a piece of metal moves through it.