Chapter 32

Electromagnetic waves

Lecture by Dr. Hebin Li

PHY 2049, Dr. Hebin Li

Goals for Chapter 32

• To learn why a light wave contains both electric and

magnetic fields

• To relate the speed of light to the fundamental

constants of electromagnetism

• To describe electromagnetic waves

• To determine the power carried by electromagnetic

waves

PHY 2049, Dr. Hebin Li

Maxwell’s equations and electromagnetic waves

Maxwell’s equations predicted that an oscillating

charge emits electromagnetic radiation in the form of

electromagnetic waves.

PHY 2049, Dr. Hebin Li

Generating EM radiation by a moving charge

https://phet.colorado.edu/sims/radiating-charge/radiating-charge_en.html

PHY 2049, Dr. Hebin Li

The electromagnetic spectrum

• The electromagnetic spectrum includes electromagnetic waves of all frequencies and wavelengths.

PHY 2049, Dr. Hebin Li

Plane electromagnetic waves

A plane wave has a planar wave front.

A wave such as this, in which at any instant the fields are uniform over any plane perpendicular to the direction of propagation, is called a plane wave.

PHY 2049, Dr. Hebin Li

Plane electromagnetic waves

To satisfy Maxwell’s first and second equations, the electric and magnetic fields must be perpendicular to the direction of propagation; that is, the wave must be transverse.

PHY 2049, Dr. Hebin Li

Plane electromagnetic waves

To satisfy Faraday’s law

For the rectangle chosen in the figure, we have

During a time interval 𝑑𝑡 the wave front moves a distance 𝑐𝑑𝑡, the magnetic flux thought the box increases by

So, we have

PHY 2049, Dr. Hebin Li

Plane electromagnetic waves

To satisfy Ampere’s law

For the rectangle chosen in the figure, we have

During a time interval 𝑑𝑡 the wave front moves a distance 𝑐𝑑𝑡, the electric flux thought the box increases by

So, we have

PHY 2049, Dr. Hebin Li

Key properties of electromagnetic waves

• The magnitudes of the fields in vacuum are related by

E = cB.

• The speed of the waves is c = 3.00 × 108 m/s in vacuum.

• The waves are transverse. Both fields are perpendicular to the direction of propagation and to each other.

PHY 2049, Dr. Hebin Li

Electromagnetic wave equation

The wave equation can be derived from Maxwell’s equations

PHY 2049, Dr. Hebin Li

Fields of a sinusoidal wave

In a sinusoidal electromagnetic wave, 𝑬and 𝑩 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.

PHY 2049, Dr. Hebin Li

Electromagnetic waves in matter

The speed v of an electromagnetic wave in a material depends on the dielectric constant of the material.

The index of refraction of a material is n = c/v.

PHY 2049, Dr. Hebin Li

Energy in electromagnetic waves

The magnitude of the Poyntingvector is the power per unit area in the wave, and it points in the direction of propagation.

The intensity of a sinusoidal electromagnetic wave is the time average of the Poynting vector.

PHY 2049, Dr. Hebin Li

Example:

A carbo dioxide laser emits a sinusoidal electromagnetic wave that

travels in vacuum in the negative x-direction. The wavelength is 10.6

𝜇m and the 𝑬 field is parallel to the z-axis, with 𝐸𝑚𝑎𝑥 = 1.5 MV/m.

(a)Write vector equations for 𝑬 and 𝑩 as functions of time and

position. (Example 32.1 on page 1062)

(b)Find the intensity of this wave in vacuum.