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Electromagnetism and relative motion
The theory of electric and magnetic fields constructed up to this point contains a paradox. One of the most basic principles of physics, dating back to Newton and Galileo and still going strong today, states that motion is relative, not absolute. Thus the laws of physics should not function any differently in a moving frame of reference, or else we would be able to tell which frame of reference was the one in an absolute state of rest. As an example from mechanics, imagine that a child is tossing a ball up and down in the back seat of a moving car. In the child's frame of reference, the car is at rest and the landscape is moving by; in this frame, the ball goes straight up and down, and obeys Newton's laws of motion and Newton's law of gravity. In the frame of reference of an observer watching from the sidewalk, the car is moving and the sidewalk isn't. In this frame, the ball follows a parabolic arc, but it still obeys Newton's laws.
When it comes to electricity and magnetism, however, we have a problem, which was first clearly articulated by Einstein: if we state that magnetism is an interaction between moving charges, we have apparently created a law of physics that violates the principle that motion is relative, since different observers in different frames would disagree about how fast the charges were moving, or even whether they were moving at all. The incorrect solution that Einstein was taught (and disbelieved) as a student around the year 1900 was that the relative nature of motion applied only to mechanics, not to electricity and magnetism. The full story of how Einstein restored the principle of relative motion to its rightful place in physics involves his theory of special relativity, which we will not take up until book 6 of this series. However, a few simple and qualitative thought experiments will suffice to show how, based on the principle that motion is relative, there must be some new and previously unsuspected relationships between electricity and magnetism. These relationships form the basis for many practical, everyday devices, such as generators and transformers, and they also lead to an explanation of light itself as an electromagnetic phenomenon.
Let's imagine an electrical example of relative motion in the same spirit as the story of the child in the back of the car. Suppose we have a line of positive charges, (a). Observer A is in a frame of reference which is at rest with respect to these charges, and observes that they create an electric field pattern that points outward, away from the charges, in all directions, like a bottle brush. Suppose, however, that observer B is moving to the right with respect to the charges. As far as B is concerned, she's the one at rest, while the charges (and observer A) move to the left. In agreement with A, she observes an electric field, but since to her the charges are in motion, she must also observe a magnetic field in the same region of space, exactly like the magnetic field made by a current in a long, straight wire.
Who's right? They're both right. In A's frame of reference, there is only an E, while in B's frame there is both an E and a B. The principle of relative motion forces us to conclude that depending on our frame of reference we will observe a different combination of fields. Although we will not prove it (the proof requires special relativity, which we get to in book 6), it is true that either frame of reference provides a perfectly self-consistent description of things. For instance, if an electron passes through this region of space, both A and B will see it swerve, speed up, and slow down. A will successfully explain this as the result of an electric field, while B will ascribe the electron's behavior to a combination of electric and magnetic forces.
Thus, if we believe in the principle of relative motion, then we must accept that electric and magnetic fields are closely related phenomena, two sides of the same coin.
Now consider figure (b). Observer A is at rest with respect to the bar magnets, and sees the particle swerving off in the z direction, as it should according to the rule given in section 6.2 (sighting along the force vector, i.e. from behind the page, the B vector is clockwise from the v vector). Suppose observer B, on the other hand, is moving to the right along the x axis, initially at the same speed as the particle. B sees the bar magnets moving to the left and the particle initially at rest but then accelerating along the z axis in a straight line. It is not possible for a magnetic field to start a particle moving if it is initially at rest, since magnetism is an interaction of moving charges with moving charges. B is thus led to the inescapable conclusion that there is an electric field in this region of space, which points along the z axis. In other words, what A perceives as a pure B field, B sees as a mixture of E and B.
In general, observers who are not at rest with respect to one another will perceive different mixtures of electric and magnetic fields.
The principle of induction
So far everything we've been doing might not seem terribly useful, since it seems that nothing surprising will happen as long as we stick to a single frame of reference, and don't worry about what people in other frames think. That isn't the whole story, however, as was discovered experimentally by Faraday in 1831and explored mathematically by Maxwell later in the same century. Let's state Faraday's idea first, and then see how something like it must follow inevitably from the principle that motion is relative:
the principle of induction
Any electric field that changes over time will produce a magnetic field
in the space around it.
The induced field tends to have a whirlpool pattern, as shown in figure (c), but the whirlpool image is not to be taken too literally; the principle of induction really just requires a field pattern such that, if one inserted a paddlewheel in it, the paddlewheel would spin. All of the field patterns shown in the following figure are ones that could be created by induction; all have a counterclockwise "curl" to them:
Figures (d) and (e) show an example of the fundamental reason why a changing B field must create an E field. The electric field would be inexplicable to observer B if she believed only in Coulomb's law, and thought that all electric fields are made by electric charges. If she knows about the principle of induction, however, the existence of this field is to be expected.
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