Lectures on Chapter 24: Electromagnetic Waves

Absence of Magnetic Monopoles

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Recall that for electrostatics, a positive point charge has electric field lines that all point out. It acts as a ``source" of field lines. A negative point charge has electric field lines that all point toward it. It acts as a ``sink" of field lines.

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We can regard point charges as electric ``monopoles." If we put ``+" and ``-" charge near each other, we get an electric dipole where the electric field lines begin on the ``+" and end on the ``-."

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Are there magnetic monopoles or magnetic charges? The answer is no, though scientists have searched for them. The simplest magnetic structure is a dipole. The field lines of a bar magnet are those of a dipole. They start at N and end at S.

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We could try to isolate magnetic charges by breaking the magnet up into tiny pieces, but we would just get tiny dipoles.

To express this mathematically, recall that Gauss' law

(1)

states that the flux of electric field through a closed surface is proportional to the charge enclosed. Since there are no magnetic charges, there can be no net magnetic flux through any closed surface:

(2)

This is one of Maxwell's equations. Any closed surface will have equal amounts of outgoing and incoming magnetic flux.

Changing produces

Faraday's law tells us that a changing magnetic field can induce an electric field.

(3)

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So if is increasing and pointing into the page, is counterclockwise. It turns out that the converse is also true. Namely, a changing field can produce a field. Think of a parallel plate capacitor that is charging up.

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As the charge on the plates increases, the field between the plates increases. This changing (increasing) field produces a field. Just as field lines circle around a current carrying wire, so the induced field circles around the changing field.

Mathematically, we describe what's going on by adding a term to Ampere's law.

Here the surface S is an imaginary surface pierced by the field lines. The closed curve C is the boundary of S. For the parallel plate capacitor, the surface S lies between the plates and is parallel to the plates.

Notice that the added term has a ``+" sign in front: . (This is in contrast to Faraday's law where we had a minus sign and Lenz's law.) The ``+" sign means that the field goes in the same direction around the capacitor as it does around the current carrying wire.

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So if is increasing into the page, is clockwise. (Notice that is opposite to in the first figure which illustrated Faraday's law.) We can get the direction of from the right hand rule: if your thumb is in the direction of increasing , your fingers curl in the direction of .

Displacement Current:

We can write Ampere's law

in the form

where the displacement current I_d is given by

(4)

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Let's calculate I_d for a parallel plate capacitor where the plates each have area A. Then the flux through a surface S between the plates is

(5)

So

(6)

It turns out that this I_d is equal to the conduction current I that is flowing in the wires connected to the capacitor. To see this, note that

where q is the charge flowing through the wire and onto the capacitor plate.

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We can relate q and the field between the plates with Gauss' law:

where and a is the area of the face of the Gaussian box.

(or have the Gaussian surface cover the inner surface of the capacitor plate to get , ignoring fringing fields). So the current in the wires is

There is another way to see the importance of the displacement current. Suppose we use

(7)

where = current density, to write Ampere's law in the form

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Ampere's law tells us that the line integral of the magnetic field around any closed loop C is equal to times the current passing through a surface S bounded by the loop plus times the rate of change of the electric field integrated over the surface S. Now we have a choice of surfaces. If we choose S₁, which lies in the plane of the loop, then at S₁ but . (If the current I is steady in the wire, then in the wire is time independent.) So

(8)

But if we choose the bag-shaped surface S₂ that encloses one plate of the capacitor, then because no electrons go through S₂. However, , so

(9)

Since we've already shown that I=I_d, we get the same result for , which is good. Ampere's law holds no matter which surface we choose.

Maxwell's Equations

It was James Clerk Maxwell who added the term to Ampere's law. He realized that a changing electric field generates a magnetic field. We can now write down the four basic equations of electricity and magnetism. These are called Maxwell's equations:

Gauss' law and Faraday's law tell us that electric charges and changing magnetic fields generate electric fields. Ampere's law tells us that electric currents and changing electric fields generate magnetic fields. The coefficient in Ampere's law is related to the speed of light:

In fact, from Maxwell's equations one can derive a wave equation that describes electromagnetic waves and light. Maxwell discovered that light consists of electromagnetic waves. Radio waves, microwaves, and x-rays are all forms of electromagnetic waves described by Maxwell's equations. An electromagnetic wave consists of oscillating and fields. where points in the direction that the wave is travelling. The changing field generates a changing field and vice-versa. One example of a (linearly polarized) electromagnetic wave traveling in the direction is

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Notice that and oscillate in space and time and that they are perpendicular to each other and to the direction that the wave is travelling in. Maxell's equations are incredibly powerful. They are the basis for the operation of all electromagnetic and optical devices such as electric motors, telescopes, cyclotrons, eyeglasses, television transmitters and receivers, telephones, electromagnets, radar, and microwave ovens. These equations help to explain natural phenomena such as rainbows and lightening. That so much can be summed up in just four ``simple" equations is the epitome of elegance.