- A
**charged particle**produces an electric field. This electric field exerts a force on other charged particles. Positive charges accelerate in the direction of the field and negative charges accelerate in a direction opposite to the direction of the field. - A
**moving charged particle**produces a magnetic field. This magnetic field exerts a force on other moving charges. The force on these charges is always perpendicular to the direction of their velocity and therefore only changes the direction of the velocity, not the speed. - An
**accelerating charged particle**produces an electromagnetic (EM) wave. Electromagnetic waves are electric and magnetic fields traveling through empty space with the speed of light c. A charged particle oscillating about an equilibrium position is an accelerating charged particle. If its frequency of oscillation is f, then it produces an electromagnetic wave with frequency f. The wavelength λ of this wave is given by λ = c/f. Electromagnetic waves transport energy through space. This energy can be delivered to charged particles a large distance away from the source.

Accelerating charges produce changing electric and magnetic fields.
Changing electric fields produce magnetic fields and changing magnetic fields
produce electric fields.** ** This interplay between induced electric and magnetic fields leads to
propagating electromagnetic waves. Electromagnetic
waves can propagate through free space.

Assume a charge q located near the origin is accelerating. It therefore produces electromagnetic
radiation. At some position **r** in space and at some time t, the
electric field of the electromagnetic wave produced by the accelerating charge
is given by

E_{rad}(**r**,t) = -[1/(4πε_{0})]*[q/(c^{2}r')]***a**_{⟂}(t
- r'/c).

Let us analyze this expression. The electric field is proportional to the
charge q. The bigger the accelerating charge, the bigger is the field. It
decreases as the inverse of the distance r', which is the distance between the
accelerating charge and the position where the field is observed. But it is not
the distance at the time the field is observed, but the distance at some earlier
time, called the **retarded time**,
when the radiation field was produced. All electromagnetic waves travel with
the speed of light c = 3*10^{8} m/s. It takes them a time interval ∆t
= ∆r/c to travel a distance ∆r. The electric field is also
proportional to the acceleration of the charge. The larger the
acceleration, the larger is the field. In the above expression
**E**_{rad}(**r**,t) is proportional to** ****a**_{⟂}, the component of
the acceleration perpendicular to the line of sight between **r**
and the retarded position of the charge. The direction of **E**_{rad}(**r**,t)
is perpendicular to to this line of sight and its
magnitude is proportional to the component of the acceleration perpendicular to
this line of sight.

The figure on the right illustrates that point. The electric field is zero along a line of sight in the direction of the acceleration, largest along a line of sight perpendicular to the direction of the acceleration, and always perpendicular to the line of sight.

The magnitude of **a**_{⟂} is a*sinθ,
and the magnitude of the radiation field therefore is

E_{rad}(**r**,t) = -[1/(4πε_{0})]*[q/(c^{2}r)]*sinθ*a(t - r/c).

Here θ is the angle between the line of sight and the direction of the acceleration.

The magnetic field of the electromagnetic wave is perpendicular
to the electric field and has magnitude B_{rad} = E_{rad}/c.
For electromagnetic waves **E** and **B** are
always perpendicular to each other and perpendicular to the direction of
propagation. The direction of propagation is the direction of
**E** × **B**.

The radiation field E_{rad} decreases as 1/r, while the
static Coulomb field decreases as 1/r^{2}. The static field
decreases with distance much faster than the radiation field, and therefore the
radiation field will dominate at large distance for accelerating charge
distributions. In addition, radiation fields are often produced by
accelerating electrons, while the static fields are produced by all charges
(positive nuclei and negative electrons) and cancel each other out.

Far from the source of an electromagnetic wave, we often treat the EM wave as a plane wave. A sinusoidal plane EM wave traveling in the x-direction is of the form

**E**(x,t) = **E**_{max}sin(kx - ωt + φ),

**B**(x,t) = **B**_{max}sin(kx - ωt + φ).

**E** is the electric field vector, and **B** is the magnetic
field vector of the EM wave. For electromagnetic waves **E** and **B** are always perpendicular to
each other and perpendicular to the direction of propagation. The
direction of propagation is the direction of **E** × **B**.

If, for a wave traveling in the x-direction **E** = E**j**,
then **B **= B**k** and **j **× **k** = **i**. Electromagnetic
waves are transverse waves.

The **wave vector** **k**
points into the direction of propagation, and its magnitude k = 2π/λ, where λ
is the wavelength of the wave. The frequency f of the wave is f = ω/2π, ω
is the angular frequency. The speed of any sinusoidal wave is the product of its
wavelength and frequency.

v = λf.

Maxwell's equations require that the speed v of any electromagnetic wave is
v = 1/(μ_{0}ε_{0})^{½} = 3*10^{8 }m/s in
free space. The speed of any electromagnetic waves in free space is **the speed of light** c = 3*10^{8
}m/s.

Electromagnetic waves in free space can have any wavelength λ or frequency f as long as λf = c.

Visible light is any electromagnetic wave with wavelength λ between approximately 400 nm and 750 nm.

An electromagnetic wave in vacuum has an electric field amplitude of
E_{max} = 220 V/m. Calculate the amplitude B_{max} of the corresponding magnetic field.

Solution:

B_{max} = E_{max}/c = (220 N/C)/(3*10^{8
}m/s) = (7.33*10^{-7 }N/Am) = 7.33*10^{-7 }T.

The eye is most sensitive to light having a wavelength of 5.5*10^{-7 }m, which is in the green-yellow region of the electromagnetic
spectrum. What is the frequency of this light?

Solution:

f = c/λ = (3*10^{8}m/s)/(5.5*10^{-7 }m) = 5.455*10^{14 }Hz.

A plane electromagnetic wave is propagating through
space. In some plane at some time t the fields are uniform and
oriented as shown. What is the direction of propagation of the
plane wave?

Solution:

The direction of propagation is the direction of
**E **× **B**.
Using the right hand rule we find that the direction of propagation
is into the page.

Electromagnetic waves can travel through transparent media, such as water and
glass. In a medium, they interact with the atoms or molecules, and due to
this interaction have an apparent speed different from c. When electromagnetic waves travel through a medium, the speed of the waves in
the medium is v = c/n, where n is the **index of refraction of the medium**. When an
EM wave travels from one medium with index of refraction n_{1} into
another medium with a different index of refraction n_{2}, then its
frequency remains the same, but its speed changes, and therefore it wavelength
changes. For air n is nearly equal to 1, for water n is 1.33, and for most
glasses n is around 1.5.

What is the apparent speed of light in water?

Solution:

v = c/n = (3*10^{8} m/s)/1.33 = 2.26*10^{8} m/s.

Most radio waves are emitted by charges oscillating in antennas. The
direction of the acceleration of the charges is along the antenna. A radio
wave propagates from the antenna to the receiver along a straight-line
path called the line of sight. The direction
of the electric field **E** of the electromagnetic radiation emitted by the antenna
lies in a plane that contains the antenna and the line of sight to the receiver,
and is perpendicular to the line of sight. The wave is
**polarized**, which means that **E** has a well defined direction.

The electric field is strongest and the intensity highest in the directions
perpendicular to the antenna and goes to zero in the direction along the
antenna.** **You get very poor reception if you stand under the antenna.

To carry information, the electromagnetic wave must be modulated. The information carried by a radio wave is sound. The amplitude of an AM (amplitude modulated) radio wave represents the pressure variations, which make up the sound. The frequency of FM (frequency modulated) radio waves can be shifted slightly from their nominal carrier frequency. The amount of shift is proportional to the variations in the pressure, which make up the sound.