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: Electric and Magnetic Field Vectors Conceptual Question

INTRO:

Figure 1

Figure 2

Figure 4

Figure 3

all of the questions will refer to the following answer choices:

a) +x
b) –x
c) +y
d) –y
e) +z
f) –z
g) at a +45∘ angle in the xy plane

Part C:
The electric and magnetic field vectors at a specific point in space and time are illustrated. (Figure 1) Based on this information, in what direction does the electromagnetic wave propagate?

SOLUTION:
the Poynting Vector S points in the direction of E×B
which in this instance would point in the positive z direction, or option e)
~ thumb is E, pointer is B → middle finger points +z

Part B:
The electric and magnetic field vectors at a specific point in space and time are illustrated. (Figure 2) (E and B are in the xy plane. Both vectors make45∘ angles with the +y axis.) Based on this information, in what direction does the electromagnetic wave propagate?

SOLUTION:
Since E & B are both in the xy plane, their cross product will be along the z-axis
In this case, E×B points in a negative z direction, or option f)
~thumb is E, pointer is B, middle points -z

Part C:
The magnetic field vector and the direction of propagation of an electromagnetic wave are illustrated. (Figure 3) Based on this information, in what direction does the electric field vector point?

SOLUTION:
Basically just work backwards
~middle is v, pointer is B, thumb points +y → option c)

Part D: 
The electric field vector and the direction of propagation of an electromagnetic wave are illustrated. (Figure 4) (E is in xz plane and makes a 45° angle with the +x axis.) Based on this information, in what direction does the magnetic field vector point?

SOLUTION:
work backwards again but be mindful of angles
~thumb is E, middle is v, pointer points at 45° between +x & -z, or option g)

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