34: Enhanced EOC: Problem 34.14

INTRO:
The magnetic field of an electromagnetic wave in a vacuum is B_z=(2.0μT)sin((9.00×10⁶)x−ωt), where x is in m and t is in s.

Part A:
What is the wave's wavelength?
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
a value for k is given in the formula
k=9.00×10⁶
since λ=2π⋅1/k, 
λ = 2π⋅1/9.00×10⁶ = 6.98×10^-7 m= 698 nm

Part B:
What is the wave's frequency?
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
using the formula c=fλ, we can easily solve for frequency 
f = c/λ = 3.0×10⁸m / 6.98×10^-7 m⋅s
f = 4.3×10¹⁴ Hz

Part C:
What is the wave's electric field amplitude?
Express your answer to two significant figures and include the appropriate units.

SOLUTION:
E₀=cB₀
→c(2.0μT) 
E₀ = 599.6 N/C ≈ 600 V/m

34: Electromagnetic Waves Ranking Task

Part A:
Rank these electromagnetic waves on the basis of their speed (in vacuum).
Rank from fastest to slowest. To rank items as equivalent, overlap them.

AM radio wave, yellow light, X-ray, infrared light, green light, FM radio wave

SOLUTION:
all the light waves have the same speed, c; The speed of light in a vacuum
v_{AM radio} = v_{FM radio} = v_infrared = v_yellow = v_green = v_X-ray 

Part B:
Rank these electromagnetic waves on the basis of their wavelength.
Rank from longest to shortest. To rank items as equivalent, overlap them.

AM radio wave, yellow light, X-ray, infrared light, green light, FM radio wave

SOLUTION:
looking at a spectrum displaying the varying wavelengths, one can rank the waves accordingly

λ_{AM radio} ≥ λ_{FM radio} ≥ λ_infrared ≥ λ_yellow ≥ λ_green ≥ λ_X-ray

Part C:
Rank these electromagnetic waves on the basis of their frequency.
Rank from largest to smallest. To rank items as equivalent, overlap them.

AM radio wave, yellow light, X-ray, infrared light, green light, FM radio wave

SOLUTION:
looking at a spectrum displaying the varying frequencies, one can rank the waves accordingly
f_{AM radio} ≤ f_{FM radio} ≤ f_infrared ≤ f_yellow ≤ f_green ≤ f_X-ray

34: Traveling Electromagnetic Wave

INTRO:
Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves. Electromagnetic waves comprise combinations of electric and magnetic fields that are mutually compatible in the sense that the changes in one generate the other.
The simplest form of a traveling electromagnetic wave is a plane wave. For a wave traveling in the x direction whose electric field is in the y direction, the electric and magnetic fields are given by

E =E₀sin(kx−ωt)j^
B =B₀sin(kx−ωt)k^

This wave is linearly polarized in the y direction.

Part A:
In these formulas, it is useful to understand which variables are parameters that specify the nature of the wave. The variables E₀ and B₀ are the __________ of the electric and magnetic fields.
Choose the best answer to fill in the blank.

A) maxima
B) amplitudes
C) wavelengths
D) velocities

Part B:
The variable ω is called the __________ of the wave.
Choose the best answer to fill in the blank.

A) velocity
B) angular frequency
C) wavelength

Part C:
The variable k is called the __________ of the wave.
Choose the best answer to fill in the blank.

A) wavenumber 
B) wavelength
C) velocity
D) frequency

Part D:
What is the mathematical expression for the electric field at the point x=0, y=0, z at time t?

A) E = E₀sin(−ωt)j^
B) E = E₀sin(−ωt)k^
C) E = 0
D) E = E₀sin(kz−ωt)i^
E) E = E₀sin(kz−ωt)j^

Part E:
For a given wave, what are the physical variables to which the wave responds?

A) x only
B) t only
C) k only
D) ω only
E) x and t
F) x and k
G) ω and t
H) k and ω

Part F:
What is the wavelength λ of the wave described in the problem introduction?
Express the wavelength in terms of the other given variables and constants like π.

SOLUTION:
λ = 2π⋅1/k

Part G:
What is the period T of the wave described in the problem introduction?
Express the period of this wave in terms of ω and any constants

SOLUTION:
frequency = 1/period ↔ period = 1/frequency
f = 1/T ↔ T = 1/f
f = ω/[2π] → T = 2π⋅1/ω

Part H:
What is the velocity v of the wave described in the problem introduction?
Express the velocity in terms of quantities given in the introduction (such as ω and k) and any useful constants.

SOLUTION:
v = fλ & λ = 2π⋅1/k & f = ω/[2π] 
so... v = 2π⋅1/k⋅ω/[2π] = ω/k