Question

A popular demonstration of eddy currents involves dropping a magnet down a long metal tube and a long glass or plastic tube. As the magnet falls through a tube, there is changing flux as the magnet falls toward or away from each part of the tube. a) Which tube has the larger voltage induced in it? b) Which tube has the larger eddy currents induced in it?

Short answer

Answer: The metal tube has the larger induced voltage and larger eddy currents compared to the glass or plastic tube.

Step-by-step solution

Faraday's law of electromagnetic induction

According to Faraday's law: emf (induced voltage) = -dΦ/dt where emf is the induced voltage, Φ is the magnetic flux in the tube, and t is the time. The negative sign indicates that the induced voltage opposes the change in the magnetic flux.

Analyzing the induced voltage in metal and glass/plastic tubes

Since the magnet is falling through both tubes, the change in the magnetic flux with time (dΦ/dt) is the same in both cases. The main difference is that the metal tube is a conductor, and the glass or plastic tube is an insulator. When the magnet falls through the metal tube, due to its conductive properties, the change in magnetic flux induces a voltage and generates eddy currents inside the tube. However, for the glass or plastic tube, due to its insulating properties, the change in magnetic flux doesn't cause significant eddy currents and induced voltage. Thus, the metal tube has larger induced voltage compared to the glass or plastic tube. Answer for (a): The metal tube has the larger voltage induced in it.

Analyzing the induced eddy currents in metal and glass/plastic tubes

Eddy currents are generated when there is a change in magnetic flux and the material has good conductivity. The metal has this property, and as a result, eddy currents form and flow in a circular pattern inside the metal tube. On the other hand, glass or plastic tubes have low conductivity. So, the change in magnetic flux during the magnet's fall through the glass or plastic tube doesn't cause significant eddy currents. Answer for (b): The metal tube has the larger eddy currents induced in it.

Key concepts

Faraday's Law of Electromagnetic Induction

Understanding Faraday's Law of Electromagnetic Induction is essential to grasping many concepts in physics and engineering, particularly when studying how electric generators and transformers work. This law, discovered by Michael Faraday in 1831, states that an electromotive force (commonly referred to as an EMF or induced voltage) is generated in a conductor when there is a change in magnetic flux through it. In simple terms, when a magnetic field changes around a loop or coil of wire, electricity is produced.

The formula given by Faraday's Law is expressed as emf = -dΦ/dt, where emf represents the induced voltage, and dΦ/dt symbolizes the rate of change of the magnetic flux Φ over time, t. The negative sign in this equation, known as Lenz's Law, tells us that the induced voltage will always work to oppose the change that created it; this is foundational to understanding how eddy currents operate.

In our exercise involving a magnet falling through different types of tubes, it is Faraday's Law that explains why an induced voltage appears in the first place. It also helps us reason why the metal tube, being conductive, would experience a greater induced voltage than a non-conductive tube made of glass or plastic.

Magnetic Flux

The magnetic flux (Φ) is a concept that quantifies the number of magnetic field lines passing through a given area; think of it like measuring the flow of water through a net. The magnetic flux is highest when the field lines are perpendicular to the area's surface and diminishes as the angle increases until it is parallel and the flux is zero. The formula for magnetic flux is Φ = B * A * cos(θ), where B is the magnetic field (measured in teslas), A is the area through which the field lines pass, and θ is the angle between the magnetic field lines and the perpendicular to the area.

In the exercise where a magnet drops through tubes of different materials, the magnetic flux changes as the magnet moves, creating an area of varying magnetic field strength within the tubes. The changing flux is the key event that leads us to analyze the induction of voltages and hence, the formation of eddy currents.

Induced Voltage

The induced voltage, as the name implies, is not a source of voltage that's inherently present in a system, but rather one that's produced as a result of changing magnetic fields. The magnitude of this voltage depends on several factors, including the rate of change of magnetic flux and the properties of the material within the magnetic field.

In practical terms, if the material is a good electrical conductor—like the metal tube in our exercise—then it allows for the generation of significant induced voltage and hence strong eddy currents. The induced voltage in the metal is what essentially brakes or slows down the magnet as it falls. In contrast, a glass or plastic tube which is a poor conductor will have an almost negligible induced voltage due to its inability to support free movement of electrical charges.

Exercising Knowledge:

When considering improvements on textbook exercises around induced voltage, it's beneficial for students to manipulate variables such as the speed of the magnet's descent, the tube's material, or the strength of the magnet. Each change could dynamically illustrate the principles of electromagnetism and offer a more vivid understanding of how induced voltage operates in different scenarios.