Max Distance to Detect EM Wave: Find r w/ 0.0050 V Induced EMF

[stanli121]

Homework Statement

A radio transmission tower radiates electromagnetic waves uniformly in all directions
with an average total power Pav = 70, 000 W at a frequency f = 98 MHz. A radio
receiver uses the induced emf in a single circular wire loop of radius r = 5.0 cm to
detect a radio signal.
If the maximum induced emf must exceed 0.0050 V to be detectable, find the maximum
distance d the receiver can be from the transmission tower and detect the wave.
(The distance d is much greater than the wavelength  of the wave.)

Homework Equations

S = EB/uo
I = P/4(pi)r^2
I = E(rms)^2/c*uo
E or B are functions given by E(x,t)=Esin(kx-wt) or B(x,t)=Bsin(kx-wt)
c=f(wavelength) and k = 2(pi)/(wavelength)

The Attempt at a Solution

I started with the end and went for V = -dflux/dt. I found the flux of the B field through the loop and then differentiated B(x,t) wrt time. I have an expression for this but I'm not sure what the B coefficient in the B(x,t) equation (max value of B) should be or what to do about the cos(kx - wt) term. Any help on where to go for this problem?

 

[anathema]

Hello there,

Thank you for sharing your approach to this problem. It seems like you are on the right track by using Faraday's Law to calculate the induced emf in the loop. However, there are a few things to consider in your solution.

Firstly, the B coefficient in the equation for B(x,t) will depend on the distance d from the transmission tower. As the distance increases, the B field will decrease, and therefore the B coefficient will also decrease. You will need to take this into account in your calculation.

Secondly, the cos(kx-wt) term represents the direction of the B field. This term will not affect the magnitude of the induced emf, but it will affect its direction. Therefore, you can ignore this term when calculating the maximum induced emf.

Finally, once you have an expression for the induced emf, you can use the fact that it must exceed 0.0050 V to find the maximum distance d. This will involve solving for d in terms of the other variables in the equation.

I hope this helps guide you towards a solution. Good luck!