Calculating Induced EMF from AM Radio Station at 4 Miles Away

[thenewbosco]

Here is the question i am working on:

An AM radio station broadcasts with average power 4000 W in all directions. A dipole receiving antenna 65 cm long is at a location 4 miles from the transmitter. Compute the amplitude of the emp that is induced be this signal between the ends of the antenna.

I know that the intensity of power falls off as $$\frac{1}{r^2}$$ but i am not sure how to get the power at 4 miles.

i see on a website that intensity of an EM wave a distance r from the source is $$I=\frac{P_s}{4\pi r^2}$$ and also that $$I=\frac{I{E_{rms}}^2}{c\mu_0}$$. I was thinking to equate these two expressions but i do not know what the I and Erms refer to in the second equation. thanks for the help.

 

[berkeman]

Assuming an isotropic radiation pattern (as the problem implies), the 4kW of power will uniformly be going through a spherical area at the 4 mile radius. That will give you a power density, which should give you the E-field. The 65cm antenna is way the heck off resonance for the US AM broadcast band, so you will use "electrically small" dipole antenna equations to calculate the terminal voltage given that electric field intensity... Does your textbook cover electrically small dipole antennas?

 

[thenewbosco]

no i do not have the equations, in fact the only equations i have are the ones i have mentioned in the question, i would be grateful if you could show me the equations to use in this problem

 

[nrqed]

Here is the question i am working on:

An AM radio station broadcasts with average power 4000 W in all directions. A dipole receiving antenna 65 cm long is at a location 4 miles from the transmitter. Compute the amplitude of the emp that is induced be this signal between the ends of the antenna.

I know that the intensity of power falls off as $$\frac{1}{r^2}$$ but i am not sure how to get the power at 4 miles.

i see on a website that intensity of an EM wave a distance r from the source is $$I=\frac{P_s}{4\pi r^2}$$ and also that $$I=\frac{I{E_{rms}}^2}{c\mu_0}$$. I was thinking to equate these two expressions but i do not know what the I and Erms refer to in the second equation. thanks for the help.

You have a typo in the sceond equation. There is no I on the right hand side. It is $$I=\frac{{E_{rms}}^2}{c\mu_0}$$. The I on the left side is the same I appearing in P/(4 pi r^2).

As for E_RMS, it is simply $E_{max}/ {\sqrt{2}}$.


So, with the distance and the power of the source, you can find I and then calculate the rms value of the electric field.

I am not totally sure what the question is asking...the voltage induced in he antenna? Or what else?

Pat

 

[thenewbosco]

yes, the voltage induced across the antenna is what it is asking i believe

 

[nrqed]

yes, the voltage induced across the antenna is what it is asking i believe

Ok. Then the rms voltage across the antenna is I think simply the value of the rms electric field at that point times the length of the antenna. The only thing I am not 100% sure about is whether there is a factor of two in this relation, but the voltage is for sure the Efield times the length of the antenna (apart possibly a factor of 2).

Patrick

 

[thenewbosco]

thanks for the help, i have gotten the correct answer