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Chapter 20: Problem 39

A 2 -m-long copper pipe is held vertically. When a marble is dropped down the pipe, it falls through in about 0.7 s. A magnet of similar size and shape takes much longer to fall through the pipe. (a) As the magnet is falling through the pipe with its north pole below its south pole, what direction do currents flow around the pipe above the magnet? Below the magnet (CW or CCW as viewed from the top)? (b) Sketch a graph of the speed of the magnet as a function of time. [Hint: What would the graph look like for a marble falling through honey?]

Short Answer

Expert verified
Question: Determine the direction of the induced currents above and below the falling magnet when the magnet's north pole is below it, and describe the shape of the graph representing the speed of the magnet as a function of time. Answer: The induced current above the magnet will flow counterclockwise (CCW), and the induced current below the magnet will flow clockwise (CW) when viewed from the top. The graph representing the speed of the magnet as a function of time will be a gradually increasing concave curve that levels off at a constant value for the terminal velocity.

Step by step solution

01

Application of Lenz's Law above the magnet

To determine the direction of the current above the magnet, consider the change in the magnetic field above the magnet as it falls. The north pole is below the magnet, so the magnetic field is moving downward. According to Lenz's Law, the induced current will flow in a direction to oppose this change. This means that the current will flow in a direction such that it will generate a magnetic field moving upward. So, using the right-hand rule, we're able to deduce that the current will flow counterclockwise (CCW) when viewed from the top.
02

Application of Lenz's Law below the magnet

Similar to step 1, consider the change in the magnetic field below the magnet as it falls. The south pole is above the magnet, which means the magnetic field is moving upward. The induced current will flow in a direction to oppose this change. According to Lenz's Law, this means that the current will flow in a direction such that it will generate a magnetic field moving downward. Using the right-hand rule, the current will flow clockwise (CW) when viewed from the top.
03

Understanding the motion of the magnet

In order to sketch the graph representing the speed of the magnet as a function of time during its fall through the pipe, consider the magnet's motion. It starts from rest and falls under the influence of gravity. However, due to the induced currents, there will be a magnetic force acting against the motion of the magnet, analogous to a resistive force like air resistance or friction. These magnetic forces will cause the magnet's speed to increase more slowly than free-falling due to gravity. Eventually, the magnetic forces will balance out the gravitational force, causing the magnet to reach a constant terminal velocity.
04

Sketching the graph of the speed of the magnet as a function of time

In the graph, the x-axis represents time, and the y-axis represents the speed of the magnet. At the beginning of the fall, the magnet's speed is zero. As the magnet falls, the speed increases, but at a slower rate due to magnetic forces acting against it. This results in a gradually increasing concave curve. The graph will eventually level off at a constant value for the terminal velocity, indicating that the magnet has reached a constant speed during its descent, as the magnetic and gravitational forces are in balance. Thus, the graph will resemble a marble falling through honey, as the shape of the graph and terminal velocity concept will be similar.

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