Magnetic Flux Linkage

Introduction

Magnetic induction between coils is the mechanism by which transformers work. It was discovered by Michael Faraday in 1831 and is usually known as Faraday’s law of induction. James Maxwell correctly quantified the law and made it part of the group of equations known by his name. In Faraday’s experiment, he used an iron ring with windings around both sides, as seen here:

This shows a power source connected to turns of wire around the shaft of an iron ring. At a different point on the ring coils of wire are connected to a galvanometer.

Each time Faraday opened or closed the switch, the galvanometer would register a reading, which did not occur when the switch was kept open or closed. It is the change in the magnetic field that is important and causes the induction. This arrangement is reflected in the modern transformer that we use to change alternating current (AC) voltages by using different numbers of windings on the two coils to either step up or step down the voltage.

This experiment establishes the relationship between the induced voltage in a coil when at various angles to another coil. It uses an audio signal as the varying input voltage.

The objective

To establish the relationship between the induced voltage in a coil when at various angles to an outer coil.

The apparatus

You will need:

• An outer coil of 80 turns

• An inner coil of 40 turns set in the middle of the outer coil and that can be rotated through a measurable angle. This coil is sometimes called the search coil

• An audio signal generator

• An oscilloscope

This shows the concentric coils on the left. The inner coil is connected to a knob with a pointer to a protractor so that it can be rotated by a given number of degrees. The outer coil is connected to a signal generator, and the inner coil is connected to an oscilloscope.

The variables

The independent variable is the angle of the inner coil. The dependent variable is the voltage induced in the inner coil.

The outer coil needs to be connected to the output of the signal generator, while the output of the inner coil is connected to the input of the oscilloscope.

The Physics

The Biot–Savart law gives us the magnetic field due to a small line segment of wire carrying a current i as:

dB = (μ0/4π ) i (dl x r) /r²

A schematic showing a small section of wire of length delta l carrying a current i. The magnetic field at a vector r from the line segment is labelled as delta b which is also a vector.

In other words, the line segment dl carrying a current i generates a magnetic field db at the point indicated, which is r from the line segment, where μ0 is the magnetic constant.

The Biot–Savart law can be used to find the equation of the magnetic field along the axis through the middle of the coil (as you are measuring in this experiment). This involves integration, and the workings can be found online. The result is:

B(d) = (μ0 i / 2) N R² / (R² + d²)^(3/2)

You can use this formula to calculate the theoretical values for the magnetic field for the values you have used in your experiment by substituting for N (the number of turns), for i (the current in your circuit), and for R (the radius of the coil).

The method

Start with both coils aligned as shown in Figure 34.2. Switch on the audio signal generator and turn up the amplitude. Adjust the vertical sensitivity on the oscilloscope so that the wave fills most of the screen. Adjust the sweep time to fit one or two entire wavelengths onto the screen. You should be looking at something like that shown here:

This shows the characteristic sine wave pattern on the screen of the oscilloscope.

You may find that you need to go back and adjust the amplitude to get the best trace. You could measure the voltage by counting the squares, but this oscilloscope allows you to do it in a better way. Different oscilloscopes work in different ways; however, they all basically have the same functionality. The Tektronix oscilloscope shown i above is common in education and provides a very nice and easy-to-use interface. Obviously, if you are using a different oscilloscope, you are going to need to find out how it is controlled so that this experiment can be followed.

There are two cursors that can be used for measuring voltage. To display these requires you to use some controls that are not particularly intuitive the first time you use them. To start with, you need to press the Cursor button, as indicated here:

This displays the top-level Cursor Menu, as shown here:

This shows the options ‘Type Off’ and ‘Source Ch1’ on the oscilloscope screen.

The two options are aligned with the top two buttons to the right of the oscilloscope screen.

Note the line of buttons alongside the right-hand edge of the screen. These are for selecting menu options that are displayed to the left of the buttons on the actual screen of the oscilloscope. Select the top-most button to display the Cursor submenu as shown here indicated by the arrow.

This shows the options of the sub-menu which are: ‘Off’, ‘Amplitude’, and ‘Time’.

Note that the top-most submenu entry is currently selected, indicating that cursors are switched off.

Rotate the large multipurpose knob (just to the right of the menu button you have just used to change the selection to ‘Amplitude’), it's the big round button at the top right of the above screen shot.

Now click on the multipurpose button without rotating it. This will cause both cursors to be displayed, as shown here:

This shows the two horizontal cursors on the screen of the oscilloscope, one above the sine wave and one just below.

You can adjust the positions of the two horizontal cursors by using the multipurpose knob when the cursor you want to move has been selected from the Cursor menu that is still displayed.

The image below shows the cursors when they have been adjusted to align with the maximum and minimum voltages of the trace. Note that the menu entries are indicating the various voltages, the most useful of which is the dV entry. This displays the voltage difference between the top cursor and the bottom cursor. This is the voltage you need to measure in the fashion just described.

The Video

Watch a video for the Magnetic Flux Linkage e-practical here.

This shows how to use it and how to collect the data.

Clicking on either of the controls to the right of the menu entries ‘Cursor 1’ or ‘Cursor 2’ will select that cursor, which you can then adjust the position of with the large multipurpose knob.

The E-Practical

Perform the experiment yourself, collect your own data, make mistakes and be able to correct them. The e-practical requires that your browser can run WebGL 2 (usually found on Windows browsers, safari on iOS, and various Mobile browsers, test with https://get.webgl.org/webgl2/). This link is for students and evaluation only, schools should purchase a site licence.

All the e-practicals will run on devices as small as a mobile 'phone. However, the best experience is on a PC using a mouse which gives very precise control, but if you are limited to a small device, consider using a stylus or a blue tooth mouse.

The signal generator is switched on by clicking on the On/Off button. The frequency can be changed by using the mouse wheel with the cursor over the large red knob (or dragging when using the onscreen controls). The amplitude of the signal can be changed by using the mouse wheel (or dragging when using the onscreen controls).

The angle of the inner coil is varied by using the knob at the left central edge of the protractor.

Note that not all of the functionality of the Tektronix oscilloscope has been implemented, but it is sufficient for this experiment using the controls described previously.

Try it yourself

The Results

Take readings at 10-degree intervals and complete this table:

When you have completed this table plot a graph of the cosine of the angle against the voltage. You should get a straight line that indicates that the induced voltage is proportional to the cosine of the angle between the coils.

Further Discussion

Find out what a Hall probe is and how it works. Then investigate how the same experiment can be done using a Hall probe.

What are the advantages?

Are there any disadvantages?