A Stubbornly Persistent Illusion

by Stephen Hawking

  The exact expression for the force of a current acting on a magnetic pole is quite complicated, much more so, indeed, than the expression for gravitational forces. We can, however, attempt to visualize the actions just as we did in the case of a gravitational force. Our question is: with what force does the current act upon a magnetic pole placed somewhere in its vicinity? It would be rather difficult to describe this force in words. Even a mathematical formula would be complicated and awkward. It is best to represent all we know about the acting forces by a drawing, or rather by a spatial model, with lines of force. Some difficulty is caused by the fact that a magnetic pole exists only in connection with another magnetic pole, forming a dipole. We can, however, always imagine the magnetic needle of such length that only the force acting upon the pole nearer the current has to be taken into account. The other pole is far enough away for the force acting upon it to be negligible. To avoid ambiguity we shall say that the magnetic pole brought nearer to the wire is the positive one.

  The character of the force acting upon the positive magnetic pole can be read from our drawing.

  FIG. 2.

  First we notice an arrow near the wire indicating the direction of the current, from higher to lower potential. All other lines are just fines of force belonging to this current and lying on a certain plane. If drawn properly, they tell us the direction of the force vector representing the action of the current on a given positive magnetic pole as well as something about the length of this vector. Force, as we know, is a vector and to determine it we must know its direction as well as its length. We are chiefly concerned with the problem of the direction of the force acting upon a pole. Our question is: how can we find, from the drawing, the direction of the force, at any point in space?

  The rule for reading the direction of a force from such a model is not as simple as in our previous example, where the lines of force were straight. In our next diagram only one line of force is drawn in order to clarify the procedure. The force vector lies on the tangent to the line of force, as indicated. The arrow of the force vector and the arrows on the line of force point in the same direction. Thus this is the direction in which the force acts on a magnetic pole at this point. A good drawing, or rather a good model, also tells us something about the length of the force vector at any point. This vector has to be longer where the lines are denser, i.e., near the wire, shorter where the lines are less dense, i.e., far from the wire.

  FIG. 3.

  In this way, the lines of force, or in other words, the field, enable us to determine the forces acting on a magnetic pole at any point in space. This, for the time being, is the only justification for our elaborate construction of the field. Knowing what the field expresses, we shall examine with a far deeper interest the lines of force corresponding to the current. These lines are circles surrounding the wire and lying on the plane perpendicular to that in which the wire is situated. Reading the character of the force from the drawing we come once more to the conclusion that the force acts in a direction perpendicular to any line connecting the wire and the pole, for the tangent to a circle is always perpendicular to its radius. Our entire knowledge of the acting forces can be summarized in the construction of the field. We sandwich the concept of the field between that of the current and that of the magnetic pole in order to represent the acting forces in a simple way.

  Every current is associated with a magnetic field, i.e., a force always acts on a magnetic pole brought near the wire through which a current flows. We may remark in passing that this property enables us to construct sensitive apparatus for detecting the existence of a current. Once having learned how to read the character of the magnetic forces from the field model of a current, we shall always draw the field surrounding the wire through which the current flows, in order to represent the action of the magnetic forces at any point in space. Our first example is the so-called solenoid. This is, in fact, a coil of wire as shown in the drawing. Our aim is to learn, by experiment, all we can about the magnetic field associated with the current flowing through a solenoid and to incorporate this knowledge in the construction of a field. A drawing represents our result. The curved lines of force are closed, and surround the solenoid in a way characteristic of the magnetic field of a current.

  FIG. 4.

  The field of a bar magnet can be represented in the same way as that of a current. Another drawing shows this. The lines of force are directed from the positive to the negative pole. The force vector always lies on the tangent to the line of force and is longest near the poles because the density of the lines is greatest at these points. The force vector represents the action of the magnet on a positive magnetic pole. In this case the magnet and not the current is the “source” of the field.

  FIG. 5.

  Our last two drawings should be carefully compared. In the first, we have the magnetic field of a current flowing through a solenoid; in the second, the field of a bar magnet. Let us ignore both the solenoid and the bar and observe only the two outside fields. We immediately notice that they are of exactly the same character; in each case the lines of force lead from one end of the solenoid or bar to the other.

  The field representation yields its first fruit! It would be rather difficult to see any strong similarity between the current flowing through a solenoid and a bar magnet if this were not revealed by our construction of the field.

  The concept of field can now be put to a much more severe test. We shall soon see whether it is anything more than a new representation of the acting forces. We could reason: assume, for a moment, that the field characterizes all actions determined by its sources in a unique way. This is only a guess. It would mean that if a solenoid and a bar magnet have the same field, then all their influences must also be the same. It would mean that two solenoids, carrying electric currents, behave like two bar magnets, that they attract or repel each other depending, exactly as in the case of bars, on their relative positions. It would also mean that a solenoid and a bar attract or repel each other in the same way as two bars. Briefly speaking, it would mean that all actions of a solenoid through which a current flows, and of a corresponding bar magnet are the same, since the field alone is responsible for them, and the field in both cases is of the same character. Experiment fully confirms our guess!

  How difficult it would be to find those facts without the concept of field! The expression for a force acting between a wire through which a current flows and a magnetic pole is very complicated. In the case of two solenoids we should have to investigate the forces with which two currents act upon each other. But if we do this, with the help of the field, we immediately notice the character of all those actions at the moment when the similarity between the field of a solenoid and that of a bar magnet is seen.

  We have the right to regard the field as something much more than we did at first. The properties of the field alone appear to be essential for the description of phenomena; the differences in source do not matter. The concept of field reveals its importance by leading to new experimental facts.

  The field proved a very helpful concept. It began as something placed between the source and the magnetic needle in order to describe the acting force. It was thought of as an “agent” of the current, through which all action of the current was performed. But now the agent also acts as an interpreter, one who translates the laws into a simple, clear language, easily understood.

  The first success of the field description suggests that it may be convenient to consider all actions of currents, magnets and charges indirectly, i.e., with the help of the field as an interpreter. A field may be regarded as something always associated with a current. It is there even in the absence of a magnetic pole to test its existence. Let us try to follow this new clew consistently.

  The field of a charged conductor can be introduced in much the same way as the gravitational field, or the field of a current or magnet. Again only the simplest example! To design the field of a positively charged sphere, we must ask what ki
nd of forces are acting on a small positively charged test body brought near the source of the field, the charged sphere. The fact that we use a positively and not a negatively charged test body is merely a convention, indicating in which direction the arrows on the line of force should be drawn. The model is analogous to that of a gravitational field (figure 1) because of the similarity between Coulomb’s law and Newton’s. The only difference between the two models is that the arrows point in opposite directions. Indeed, we have repulsion of two positive charges and attraction of two masses. However, the field of a sphere with a negative charge will be identical with a gravitational field since the small positive testing charge will be attracted by the source of the field.

  FIG. 6.

  FIG. 7.

  If both electric and magnetic poles are at rest, there is no action between them, neither attraction nor repulsion. Expressing the same fact in the field language we can say: an electrostatic field does not influence a magnetostatic one and vice versa. The words “static field” mean a field that does not change with time. The magnets and charges would rest near one another for an eternity if no external forces disturbed them. Electrostatic, magnetostatic and gravitational fields are all of different character. They do not mix; each preserves its individuality regardless of the others.

  Let us return to the electric sphere which was, until now, at rest, and assume that it begins to move due to the action of some external force. The charged sphere moves. In the field language this sentence reads: the field of the electric charge changes with time. But the motion of this charged sphere is, as we already know from Rowland’s experiment, equivalent to a current. Further, every current is accompanied by a magnetic field. Thus the chain of our argument is:

  We, therefore, conclude: The change of an electric field produced by the motion of a charge is always accompanied by a magnetic field.

  Our conclusion is based on oersted’s experiment but it covers much more. It contains the recognition that the association of an electric field, changing in time, with a magnetic field is essential for our further argument.

  As long as a charge is at rest there is only an electrostatic field. But a magnetic field appears as soon as the charge begins to move. We can say more. The magnetic field created by the motion of the charge will be stronger if the charge is greater and if it moves faster. This also is a consequence of Rowland’s experiment. Once again using the field language, we can say: the faster the electric field changes, the stronger the accompanying magnetic field.

  We have tried here to translate familiar facts from the language of fluids, constructed according to the old mechanical view, into the new language of fields. We shall see later how clear, instructive, and far-reaching our new language is.

  THE TWO PILLARS OF THE FIELD THEORY

  “The change of an electric field is accompanied by a magnetic field.” If we interchange the words “magnetic” and “electric,” our sentence reads: “The change of a magnetic field is accompanied by an electric field.” Only an experiment can decide whether or not this statement is true. But the idea of formulating this problem is suggested by the use of the field language.

  Just over a hundred years ago, Faraday performed an experiment which led to the great discovery of induced currents.

  The demonstration is very simple. We need only a solenoid or some other circuit, a bar magnet, and one of the many types of apparatus for detecting the existence of an electric current. To begin with, a bar magnet is kept at rest near a solenoid which forms a closed circuit. No current flows through the wire, for no source is present. There is only the magnetostatic field of the bar magnet which does not change with time. Now, we quickly change the position of the magnet either by removing it or by bringing it nearer the solenoid, whichever we prefer. At this moment, a current will appear for a very short time and then vanish.

  FIG. 8.

  Whenever the position of the magnet is changed, the current reappears, and can be detected by a sufficiently sensitive apparatus. But a current—from the point of view of the field theory—means the existence of an electric field forcing the flow of the electric fluids through the wire. The current, and therefore the electric field, too, vanishes when the magnet is again at rest.

  Imagine for a moment that the field language is unknown and the results of this experiment have to be described, qualitatively and quantitatively, in the language of old mechanical concepts. Our experiment then shows: by the motion of a magnetic dipole a new force was created, moving the electric fluid in the wire. The next question would be: upon what does this force depend? This would be very difficult to answer. We should have to investigate the dependence of the force upon the velocity of the magnet, upon its shape, and upon the shape of the circuit. Furthermore, this experiment, if interpreted in the old language, gives us no hint at all as to whether an induced current can be excited by the motion of another circuit carrying a current, instead of by motion of a bar magnet.

  It is quite a different matter if we use the field language and again trust our principle that the action is determined by the field. We see at once that a solenoid through which a current flows would serve as well as a bar magnet. The drawing shows two solenoids: one, small, through which a current flows, and the other, in which the induced current is detected, larger. We could move the small solenoid, as we previously moved the bar magnet, creating an induced current in the larger solenoid. Furthermore, instead of moving the small solenoid, we could create and destroy a magnetic field by creating and destroying the current, that is, by opening and closing the circuit. Once again, new facts suggested by the field theory are confirmed by experiment!

  FIG. 9.

  Let us take a simpler example. We have a closed wire without any source of current. Somewhere in the vicinity is a magnetic field. It means nothing to us whether the source of this magnetic field is another circuit through which an electric current flows, or a bar magnet. Figure 10 shows the closed circuit and the magnetic lines of force.

  FIG. 10.

  The qualitative and quantitative description of the induction phenomena is very simple in terms of the field language. As marked on the drawing, some lines of force go through the surface bounded by the wire. We have to consider the lines of force cutting that part of the plane which has the wire for a rim. No electric current is present so long as the field does not change, no matter how great its strength. But a current begins to flow through the rim-wire as soon as the number of lines passing through the surface surrounded by wire changes. The current is determined by the change, however it may be caused, of the number of lines passing the surface. This change in the number of lines of force is the only essential concept for both the qualitative and the quantitative descriptions of the induced current. “The number of lines changes” means that the density of the lines changes and this, we remember, means that the field strength changes.

  These then are the essential points in our chain of reasoning: change of magnetic field → induced current → motion of charge → existence of an electric field.

  Therefore: a changing magnetic field is accompanied by an electric field.

  Thus we have found the two most important pillars of support for the theory of the electric and magnetic field. The first is the connection between the changing electric field and the magnetic field. It arose from Oersted’s experiment on the deflection of a magnetic needle and led to the conclusion: a changing electric field is accompanied by a magnetic field.

  The second connects the changing magnetic field with the induced current and arose from Faraday’s experiment. Both formed a basis for quantitative description.

  Again the electric field accompanying the changing magnetic field appears as something real. We had to imagine, previously, the magnetic field of a current existing without the testing pole. Similarly, we must claim here that the electric field exists without the wire testing the presence of an induced current.

  In fact, our two-pillar structure could be reduced to only
one, namely, to that based on Oersted’s experiment. The result of Faraday’s experiment could be deduced from this with the law of conservation of energy. We used the two-pillar structure only for the sake of clearness and economy.

  One more consequence of the field description should be mentioned. There is a circuit through which a current flows, with for instance, a voltaic battery as the source of the current. The connection between the wire and the source of the current is suddenly broken. There is, of course, no current now! But during this short interruption an intricate process takes place, a process which could again have been foreseen by the field theory. Before the interruption of the current there was a magnetic field surrounding the wire. This ceased to exist the moment the current was interrupted. Therefore, through the interruption of a current, a magnetic field disappeared. The number of lines of force passing through the surface surrounded by the wire changed very rapidly. But such a rapid change, however it is produced, must create an induced current. What really matters is the change of the magnetic field making the induced current stronger if the change is greater. This consequence is another test for the theory. The disconnection of a current must be accompanied by the appearance of a strong, momentary induced current. Experiment again confirms the prediction. Anyone who has ever disconnected a current must have noticed that a spark appears. This spark reveals the strong potential differences caused by the rapid change of the magnetic field.

  The same process can be looked at from a different point of view, that of energy. A magnetic field disappeared and a spark was created. A spark represents energy, therefore, so also must the magnetic field. To use the field concept and its language consistently, we must regard the magnetic field as a store of energy. Only in this way shall we be able to describe the electric and magnetic phenomena in accordance with the law of conservation of energy.