Textbook knowledge revised

For around 200 years, numerous textbooks have stated that the magnetic field outside of a long coil is zero. But when this claim was questioned by the student Jonathan Bollig during a lecture held by Prof. Dr Gerhard G. Paulus in the 2020 summer semester, a subsequent experiment proved him right: The statement is not accurate. In this interview, the physicists explain what this is all about.

Interview by Marco Körner

Mr Bollig, it was your question that got the ball rolling – what exactly did you want to know?

Bollig: One of our online physics lectures in the 2020 summer semester was about the long coil through which an electric current flows. Using an online polling tool, Prof. Paulus asked us about the magnetic field outside the coil. There were different possible answers. Instinctively, I said that the magnetic field would decrease linearly with distance. But then I was told that there was no magnetic field outside of the coil, at least not along its axis. I gave it some thought and came to the conclusion that there should still be a magnetic field, namely along the winding. As if the coil were simply a wire. I then asked Prof. Paulus about it.

Paulus: With regard to Ampère’s Law, there are only two possibilities that are easy to calculate: One is an infinitely long wire and the other is an infinitely long coil. And for an infinitely long coil, we get the following answer: Outside of the long coil, the magnetic field is zero. That’s what it says in all the textbooks and that’s what I’ve always taught – for over ten years. I was taught the same as a student. And then Mr Bollig turns up at my tutorial one evening and says: »But a small component of the current must flow in the direction of the coil’s axis!«.

I was taken aback at first and then it really bugged me that I hadn’t noticed it myself over the past 30 years. I recognized immediately what the magnetic field must look like outside of the coil, but I still sat down on my patio the weekend afterwards, wound a coil and measured the magnetic field. Mr Bollig was right!

So why did we previously assume that the magnetic field outside of the coil was zero?

Paulus: That wasn’t just an assumption – it was a consequence of the fundamental fact that magnetic field lines are closed curves. And that fact, in turn, comes from the well-known fact that magnetic north and south poles come in pairs.

Apart from the subtle detail discussed in our article, the magnetic field lines in a coil run parallel to the coil axis. The denser the magnetic field lines in the coil, the stronger the magnetic field. When the field lines emerge at the ends of the coil, they have to return to the other end by going around the coil – otherwise the field lines wouldn’t be closed curves. Now you can imagine what happens when the coil gets longer and longer: The density of the magnetic field lines – and therefore the strength of the magnetic field outside of the coil – gets smaller and smaller.

Now, the coil you built on your patio wasn’t infinitely long!

Paulus: Well, it’s half a metre long. And its diameter is two centimetres. So, in comparison to the diameter, it’s – almost – infinitely long.

Bollig: The set-up is very simple: The wire has to be insulated to prevent short circuits. It’s wrapped around a tube and then an electrical current is passed through it.

Paulus: In order to measure the magnetic field quantitatively, we built a frame for the coil and a magnetized needle that allowed us to determine the magnetic field at different distances from the coil. The frame is oriented in such a way that the axis of the coil points north. When the current is not flowing, the magnetized needle aligns itself parallel to the axis of the coil – so simply according to the Earth’s magnetic field. If I now turn on the power and regulate it until the magnetized needle is at a 45° angle (to the north-west or north-east), the magnetic field outside of the coil will be just as large as the magnetic field of the Earth. If I increase the distance between the magnetized needle and the coil, I have to increase the current flow accordingly to achieve the same deflection of the magnetized needle.

In your publication you write that you can also determine the strength of the earth's magnetic field with this setup.

Paulus: Yes, to do this, the whole thing is turned upside down, so to speak. If you assume that you have understood the physics of this, you can use it to determine the earth's magnetic field and ultimately also your own position on Earth. But given the good result I achieved, I'm not sure if I wasn't just lucky. I would have to repeat the measurement a few more times. But that was not the point. It was only a test measurement.

In your introduction you mention Mr Ørstedt's romantic view of nature, who did not consider distinguishing the measured magnetic field from the Earth's magnetic field. What exactly was his conception?

Paulus: This is an interesting story that also has to do with Jena. Johann Wilhelm Ritter, the discoverer of UV rays and many other things, also conducted experiments on electricity theory. At that time, people had strange ideas about electricity that came from Galvanism. The tale is that Luigi Galvani dried frogs' legs to study the nerve pathways. He had hung these on an iron railing in his garden. Then a thunderstorm came and the frogs' legs twitched. He thought he had discovered the secret of life through electricity.

This was probably also the inspiration for the novel "Frankenstein; or, The Modern Prometheus" by Mary Shelley.

Paulus: Frankenstein was Galvani's nephew, so to speak. At the time, people thought: "If electricity can do this to a frog, what must it be like to a human being?" It was assumed that electricity must be much greater in humans. So the Italian physicist Giovanni Aldini carried out experiments on bodies of executed people. There are scary stories about this and it was probably also received controversially at the time. Such experiments were also banned in our country. Other scientists, especially Alessandro Volta, were quick to show that this was all nonsense and that electricity did not originate in life. Volta in fact built a battery that consisted entirely of inorganic materials. Nevertheless, Galvanism had its followers for a long time. So here we already see a romantic idea of physics, which I also wrote about.

Ritter also carried out experiments on himself. He even pierced electrodes into his eyes and then saw strange flashes. Ritter, for his part, also worked with Johann Wolfgang von Goethe and was a close friend of Hans Christian Ørsted. Ørsted also visited Jena and Weimar and demonstrated his experiments to Goethe, as far as I know.

Have you been showing your experiment in your lectures?

Paulus: I have not done that so far due to time constraints. It’s a subtle detail and could cause confusion if I don’t take enough time to explain it. But it could be a good practical exercise.

So, everyone who answers according to the textbooks would be wrong?

Paulus: We can learn from our mistakes! And the exercises won’t be graded.

In the publication, you say that you’re probably not the first to notice this. But you’re amazed that such a mistake has remained in textbooks for so long. Apart from this publication, do you have any ideas how the textbooks could be corrected?

Paulus (laughing): The first step would be for me to bring up the issue in my lectures… 

What exactly is meant by »textbooks«? Is Ampère’s Law is taught in schools?

Paulus: Partially. In my 40-year-old collection of formulas from school, there is a formula to work out the size of the magnetic field inside a long coil. But I don’t think the derivation is done in schools.

In your publication, you write that the Danish physicist Hans Christian Oersted observed as early as 1820 – two years before Ampère – that an electrical current in a wire can deflect a magnetized needle; and this had even been published 18 years earlier by the Italian Gian Domenico Romagnosi.

Paulus: Romagnosi was a jurist and, as far as I’m aware, he published his observation in a daily newspaper, which obviously failed to attract a lot of attention. He probably didn’t realize the significance of his discovery. Oersted, on the other hand, was »electrified« and had his observations immediately printed at his own expense. He sent the manuscript to all academies in Europe. That’s how he became famous. For a while, the unit of magnetic field strength was even named after him – the ultimate honour for a physicist.