What is the distance at which an antenna can receive signal?

[Shostakovich]

Homework Statement

Assume that the power radiated by the television transmitter uniformly fills the upper hemisphere. A UHF television with a single-turn circular loop antenna of radius 8 cm requires a maximum induced voltage above 24 mV for operation.

The speed of light is 2.99792 × 108 m/s.

Find the distance d at which reception is lost from a 569 kW transmitter operating at 0.16 GHz.

Answer in units of km

Homework Equations

dE/dx = -dB/dt
E_induced = -N(dMFlux/dt)
E = E_0 cos(k(x - vt)) (where v = c)
k = 2Pi / lambda
S = P_tran / (2Pir^2) (not 4Pir^2 because its a hemisphere, so only half)
S = E_0^2 /(mu_naught * c * 2)
lambda = c/f

The Attempt at a Solution

I found -dB/dt with the wave function, took the derivative of E = E_0 cos(k(x - vt)), got dE/dx to be -.5*k*E_0.
Set them equal, solved for E_0, and got (-dB/dt * lambda) / Pi.
I then set the poynting vectors equal so P_tran / (2Pid^2) = E_0^2 /(mu_naught * c * 2), which when solved for r is

d = ((P_tran * mu_naught * c)/(Pi * E_0^2))^(1/2)

I ended up with a value or arbitrary units, no where near what it should be.
What am i doing wrong?

 

[mfb]

got dE/dx to be -.5*k*E_0.

Where does the .5 come from?

What did you get for E₀?

Where did you use the size of the antenna?

Where did the units start to get inconsistent? It would help if you write down the actual equations instead of describing them.

 

[Shostakovich]

Where does the .5 come from?

What did you get for E₀?

Where did you use the size of the antenna?

Where did the units start to get inconsistent? It would help if you write down the actual equations instead of describing them.

.5 gives the average of the sinusoidal function.

E_0 = (V*lambda)/(Pi^2*r_ant^2) (according the my calculation of using the wave equation to derivate E_0).

The size of the antenna is used in E_0, because you're calculating the induced voltage through the loop.

The units are consistent until the very end when i set there Poynting values equal to each other and solve for d.

 

[mfb]

.5 gives the average of the sinusoidal function.

It does not. The average first derivative of a sinus function is zero. And why do you want to take the average?

The size of the antenna is used in E_0

Sure, but that is not clear from the first post.

The units are consistent until the very end when i set there Poynting values equal to each other and solve for d.

How do you get inconsistent units there? Looks like power/area in both formulas.

 

[Shostakovich]

Sorry about it not being clear.
But I'm obviously lost then!
That's what I mean, I don't know WHY the units are wrong, I just know that when I plugged them into Wolfram Alpha to double check, they were very wrong!

Where would you recommend I start? Did I start on the right path and veer off? If so, where?

 

[mfb]

Where would you recommend I start?

Start with showing your actual work here (every step), otherwise it is impossible to tell what went wrong.

I just know that when I plugged them into Wolfram Alpha to double check, they were very wrong!

Why don't you copy the WolframAlpha query to this thread?

 

[Shostakovich]

Is my 4th step correct, before I continue? Hopefully everything is clearer now though!

 

[Shostakovich]

Sorry, that came out a lot less clear than I wanted.
If you sill can't read it I'll upload it to imgur or drop box or something.

 

[Shostakovich]

Here is the wolfram alpha interpretation,

 

[mfb]

I don't know where the Ohms come from. If you plug in the correct numbers it will work, here the result with a dummy electric field strength: http://www.wolframalpha.com/input/?i=+sqrt(569000W*mu_0+*+(speed+of+light)/(pi+*+(1V/m)^2))

 

[Shostakovich]

Technically mu_0 * c gives you 377 Ohms, i thought wolfram alpha would deal with it, but it did not.
I see what i did, I didn't square V/m. Ok, back to business then, is my 4th step correct, or no?

 

[mfb]

Hmm, nice relation with $\mu_0 c$.

Where did you square E_0?

 

[Shostakovich]

Where did you square E_0?

The units were stupid on my part! The problem is still knowing what E_0 is though.

 

[mfb]

dE/dx is right, the relationship to E_0 is the last missing step. And it is just a constant factor you have to fix.

 

[Shostakovich]

dE/dx is right, the relationship to E_0 is the last missing step. And it is just a constant factor you have to fix.

So, are you saying dE/dx = -κ* E_o is incorrect?

 

[Shostakovich]

So, are you saying dE/dx = -κ* E_o is incorrect?

For all of those in the future who come seeking a solution, this is it.
keep going from where i left off, plug and chug by setting energy flux of the transmitter (P_tran / 2 Pi d^2 ) equal to (E_0^2 / 2 mu_0 c).
Then solve for d!