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