Chapter 32 - Electromagnetic Waves

I.We will start with Maxwell's equations and examine what they predict about electric and magnetic fields as they travel through space.

Gauss's Law for Electricity

Gauss's Law for Magnetism

Faraday's Law

Ampere's Law

If we apply these equations to fields in space -- a vacuum:

II.Production of Traveling Electric and Magnetic Fields. Look at an AC source (like a function generator) connected to a simple antenna. Notice that time varying electric and magnetic fields will be produced by the movement of the charges in the antenna.


A.Look at the electric and magnetic fields after they have traveled a long distance away from the antenna. The fields will have less curvature, and at a great enough distance they can be considered to be plane. If the fields are traveling in the +x-direction and the electric field points in the y-direction, then the magnetic field points in the z-direction.

We will look at a thin, square slab of space through which the fields are traveling, see the consequences of Maxwell’s equations, and examine the transport of energy by the fields.

  1. Apply Ampere's Law over the top face:

as :equation 1

2.Apply Faraday's Law over the front face:

as :equation 2

3.By taking the derivative of equation 1 with respect to t, and the derivative of equation 2 with respect to x, we get:

and .

Setting them equal to each other, we get

equation 3

4.A similar result can be obtained for the magnetic field by taking the derivative of equation 1 with respect to x, and the derivative of equation 2 with respect to t.

, equation 4

  1. Both of these equations are called "wave equations," the same form as the wave equation for waves on strings. The solution is of the form f(x - vt) or f(x + vt). For equation 3 a solution can be written as:

E = Em sin [k(x – vt)] = Em sin (kx – kvt) = Em sin (kx - t) , equation 5

where k is the propagation constant,  is the angular frequency = 2f, f is the frequency, and v is the velocity of the wave. Note that the velocity is also v = /k. A similar equation is found for the magnetic field:

B = Bm sin (kx - t) .equation 6

  1. Look at some of the characteristics:
  • direction of motion
  • maximum value
  • frequency and angular frequency
  • propagation constant and wavelength
  • speed of the wave

7.Substitute equation 5 into the wave equation 3. This will give the speed of the wave.

B.Look at the relationship between the electric field and the magnetic field. Given E, find B using equation 1. We will get time varying electric and magnetic fields propagating through space - these are called electromagnetic waves or EM waves.

C.Look at the energy that is transported by the electromagnetic wave and the rate at which it is transferred. Refer back to the diagram showing a thin, square slab of volume where the electric and magnetic fields are present. The energy equals the electric and magnetic field energy densities multiplied by the volume of the slab.

Electric field energy density:

Magnetic field energy density:

The energy in the slab volume is:

U =

The instantaneous power (rate of transfer of energy) is:

=

D.When looking at the strength of the electromagnetic wave, the quantity that is usually measured is its intensity. The intensity is defined as the average energy delivered per unit area per unit time = average power delivered per unit area.

In this case, we will first find the instantaneous power delivered per unit area (called the Poynting Vector, ), and then find its average value. From above:

S = ,

which can be written in vector form as:

, where the Poynting vector points in the direction of energy travel.

The intensity of the electromagnetic radiation is the average value of the Poynting vector:

Intensity = Savg

1.Example: The intensity of the light reaching the earth's surface from the sun is approximately 1.4 kW/m2. Find the amplitude of the electric and magnetic fields due to this light.

2.Example: If all the electrical energy supplied to a 100 watt light bulb is changed to light, the find the intensity of the electromagnetic wave and the amplitudes of the electric and magnetic fields one meter from the light bulb.

III.Pressure exerted by EM radiation. When EM radiation hits a surface, pressure is exerted by that radiation. Find that pressure.

A.In order to do so, we need to find the momentum and the energy associated with the radiation:

Momentum = mv = (effective mass of the radiation) x (velocity of the radiation)

= mv = (meff)c

Energy = U = (effective mass of the radiation) x (velocity of the radiation)2

= U = (meff)c2

So the effective mass of the radiation, , and the momentum of the radiation becomes .

B.Now find the pressure exerted by the radiation:

Average pressure = Pavg = .

The average force can be found using the impulse-momentum equation:

,

.

So the average pressure becomes:

.

Since energy per unit time per unit area is the intensity or the Poynting vector, we get

.

Question: Is more pressure exerted when the radiation is absorbed or when it is reflected?

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