34: Problem 34.22

Part A:
At what distance from a 15W point source of electromagnetic waves is the magnetic field amplitude 0.60μT ?
Express your answer to two significant figures and include the appropriate units.

SOLUTION:
givens:
P_source = 15 W
B₀ = 0.60μT
and we're trying to determine the distance from the point source, r

How can we relate the magnetic field amplitude and the distance from the point source? 
neither of our basic field formulas helps us, but we do know a series of equations relating the Electric field amplitude and electromagnetic wave intensity, and the amplitudes of the Electric field and the Magnetic field are related by the speed of light, we can arrange the following formulas to solve for r

Intensity relations~ 
I= P_source⋅1/(4πr²) = c⋅ε₀⋅E₀²⋅1/2
Magnetic & Electric field relations~
B₀⋅c = E₀
so E₀² = (B₀⋅c)² 
∴ I= c⋅ε₀⋅(B₀⋅c)²⋅1/2=c³⋅ε₀⋅(B₀)²⋅1/2 
since I also equals P_source⋅1/(4πr²), it can be simplified that
P_source⋅1/(4πr²) = c³⋅ε₀⋅(B₀)²⋅1/2
now, ε₀ = 1/(μ₀c²) = 1/[(4π⋅10^-7 H/m)c²]
so P_source⋅1/(4πr²) = c³⋅1/[(4π⋅10^-7 H/m)c²]⋅(B₀)²⋅1/2 simplifies to
P_source⋅[2⋅10^-7 H/m] / (r²) = c⋅(B₀)² which simplifies to
r²=P_source⋅(2⋅10^-7 H) ⋅1/[c⋅B₀² ⋅m]
leaving r = √(P_source⋅(2⋅10^-7 H) ⋅1/[c⋅B₀² ⋅m])
now, plugging in for all of the values for which we've arranged so nicely, we get
r = √{(15 W)⋅(2⋅10^-7 H)⋅1/[(3⋅10⁸ m/s)(3.6⋅10^-13)⋅T)² ⋅m] }
→ the first thing I'm going to do here is solve for the units & check that I'm on the right path
√{W⋅H⋅1/(m/s)⋅1/(T²)⋅1/m}
W = kg⋅m⋅m⋅1/s⋅1/s⋅1/s & H = kg⋅m⋅m⋅1/s⋅1/s⋅1/A⋅1/A & T = kg⋅1/A⋅1/s⋅1/s →1/T = A⋅s⋅s⋅1/kg
⇒√{kg⋅m⋅m⋅1/s⋅1/s⋅1/s⋅kg⋅m⋅m⋅1/s⋅1/s⋅1/A⋅1/A⋅A⋅s⋅s⋅1/kg⋅A⋅s⋅s⋅1/kg⋅1/m⋅s⋅1/m}
⇒√{kg²⋅m⁴⋅A²⋅s⁵⋅1/kg²⋅1/m²⋅1/s⁵⋅1/A²}⇒√{m²}
⇒units=m
which is right so that means our equation is set up properly, now solve
√{(15)⋅(2⋅10^-7)⋅1/[1.08⋅10^-4] = √{1/36} = 1/6 = 0.17 m

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