Warm Up

Our next topic within the waves unit is light.

• What is light?

• What do you know about light?

• What would you like to know about it?

Electromagnetic Waves

By Faraday’s and Lenz’s laws, we know that a current is induced. But what actually causes the current? What force pushes the charges around the loop?

Is it the force the magnetic field exerts on the charges in the loop?

It can’t be the B field for two reasons. First, the field outside the solenoid is very weak. Second, and more importantly, the magnetic force is perpendicular to the motion of the charges, so it can never push the charges in the direction that they’re moving.

What’s happening is this—as the magnetic field changes in time, an electric field is induced. This E field is tangent to every point of the loop, and thus pushes the charges around the loop.

A changing magnetic field creates an electric field.

This electric field is called a non-Coulomb field since it is not produced by the presence of charge. So an electric field can exist without there being any electric charge present!

The induced electric field exists whether there is a conducting loop or not. It is a direct consequence of the changing magnetic field.

Maxwell’s Equations

A time-varying magnetic field creates a spatially-varying electric field.

A time-varying electric field creates a spatially-varying magnetic field.

Faraday showed that a time-varying magnetic field leads to an induced electric field. In 1865, James Maxwell proposed that a time-varying electric field would produce a magnetic field. This two-way interaction between fields is elegantly wrapped up in Maxwell’s four equations.

James Clerk Maxwell (1831 - 1879)

So when either an electric field or a magnetic field is changing with time, a field of the other kind is created in adjacent regions of space.

In this way, time-varying electric and magnetic fields can propagate through space from one region to another, even when there is no matter in the intervening space.

Such a propagating disturbance is called an electromagnetic wave.

Notice that unlike mechanical waves, which require a medium through which they move, electromagnetic waves can propagate through vacuum!

Electromagnetic waves can also travel through transparent or translucent media.

Electromagnetic waves can be a little difficult to picture. One of the simpler cases is a sinusoidal electromagnetic wave, as shown below.

Notice that this is a transverse wave—that is, both E and B are perpendicular to the direction of propagation.

Also note that E and B are perpendicular to one another.

We have yet another right-hand rule here: Right-hand fingers along E. Then swing in the direction of B.

Your right-hand thumb gives the direction the wave moves.

When we studied transverse mechanical waves, such as a wave on a string, we found that the speed with which the wave moves along the rope was given by the properties of the medium; namely, the tension in the string FT and the linear mass density μ.

TFv

Mechanical wave speed

But as we’ve mentioned, electromagnetic waves are not mechanical waves; they don’t need a medium to be “waving” through since as they move along they produce the very fields that are waving!

What we find is that for electromagnetic waves moving in vacuum (empty space), they travel with an unchanging speed of c given by

0 0

1c

EM wave speed in vacuum

Putting in the values for ε0 and μ0,we find that c = 3 × 108 m/s. This is the speed of light! Thus light must be an electromagnetic wave!

We also find that the EM wave speed is given by c = E/B, the ratio of fields.

Recall that for any type of wave, the wave speed can be written as

v = λ f.

This holds for EM waves as well: c = λ f.

c = 3 × 108 m/s

While all EM waves travel with the speed c in vacuum, they do not all have the same wavelength λ or the same frequency f.

In fact, EM waves cover a very broad spectrum of wavelengths and frequencies. We refer to this as the electromagnetic spectrum.

R O Y G B I V

The Electromagnetic Spectrum

Notice the small range of wavelengths that constitute visible light for the human eye. There is much more going on in the universe than we can see!

Visible wavelengths Ultraviolet (UV) wavelengths

Sinusoidal Electromagnetic Waves

In a sinusoidal EM wave, the E and B fields at any point in space are sinusoidal functions of time, and at any instant of time the spatial variation of the fields is also sinusoidal.

Some sinusoidal waves are spherical waves, for which the fields are uniform over any sphere perpendicular to the direction of propagation.

Far from a source of spherical waves, the wave fronts appear as flat planes. We call these plane waves.

For a plane wave, the fields are uniform over any plane that is perpendicular to the direction of propagation.

Sinusoidal Electromagnetic Waves

Just as we did with mechanical sinusoidal waves, we can write a wave function for EM waves. In this case, the wave function describes the E and B field strengths at a particular location and at a particular time.

If we have a situation like that shown below, with the wave moving to the right, the wave functions will be given by the following:

If moving to the left, the minus becomes a plus.

E = Emax sin(ωt – kx)

B = Bmax sin(ωt – kx)

ω = angular frequency = 2πf = 2π/T

k = wave number = 2π/λ

c = Emax/Bmax = ω/k

Energy in Electromagnetic Waves

Recall that the energy density in an electric field is given by uE = ½𝜖0E 2.A similar expression can be derived for the magnetic field: uB = (1/2μ0)B2.

So the total energy density u in a region of empty space where an EM wave is present is given by

u = uE + uB = ½𝜖0E 2 + (1/2μ0)B2. We can show that the E and B fields contribute equally to the energy density, and so we can write

u = 𝜖0E 2.Recall that intensity I is the energy transferred to a given area A in a given amount of time t. In other words, intensity is the power per area.

Let’s find an expression for the intensity: u = U/vol. = U/AΔx = U/AcΔt

So U/AΔt = I = cu.

In the context of EM waves, intensity is sometimes labeled S.

S = Intensity = cu = 𝜖0 c E 2.

Energy in Electromagnetic Waves

S = Intensity = cu = 𝜖0 c E 2.

This equation is for the instantaneous values of the electric and magnetic fields. Since the fields vary with time, so does S.

We can get an average intensity by averaging the values of the fields. As always happens when finding an rms value of a sinusoidal function (E and B), we get a factor of ½ times the amplitude:

Savg = Average Intensity = ½ 𝜖0 c Emax 2.

Electromagnetic waves also carry momentum, p. It can be shown that the intensity and the momentum flow are related as follows:

I = Savg = Δ(pc)/AΔt = c Δp/AΔt = (c/A) (Δp/Δt)

Recall that Newton’s 2nd law, F = ma, can also be written as F = Δp/Δt.

So we have the following: F/A = Force/Area = Pressure = I/c = Savg/c.

Radiation pressure = I/c (total absorption)

The Wave Nature of Light

Young’s Double-Slit Experiment

Young’s Double-Slit Experiment

Thin-Film Interference

The interference of light waves reflected from the two boundaries of a thin film, such as a thin layer of oil, is called thin-film interference.

The bright colors of oil slicks and soap bubbles are do to thin-film interference.

Thin-Film Interference

When light encounters an interface, most often some of the light is reflected, and some of the light is transmitted into the second material. We call these partial reflection and partial transmission.

A light wave is partially reflected from any boundary between two transparent media with different indices of refraction.

Thus light is partially reflected not only from the front surface of a sheet of glass, but from the back surface as well, as it exits from the glass into the air.

A light wave undergoes a phase change (an inversion) if it reflects from a boundary at which the index of refraction increases.

The Particle Nature of Light

Thomas Young, in 1801, and James Maxwell, in 1861, showed conclusively that light is a wave.

By its very nature, a wave is spread out in space. It is not localized like a particle.

But nature is not so simple!

The Photoelectric Effect

Max Planck and the Blackbody Spectrum