Relating Values in Equations: Finding the Role of L

[LCSphysicist]

Homework Statement

A radio station AM transmit isotropically with a power avarage of 4kW. An dipole antenna of reception with 65cm of length is 4 miles away from the transmitter. Calculate the amplitude of the EMF induced by the signal between the ends of the antenna

Relevant Equations

.

I don't know how to relate the values given by the problem. I am trying to find something that relates these equations:
$$\epsilon = -d \phi_{B}/dt$$
$$\langle S \rangle = cB²/(2 \mu); (\vec J = 0)$$
$$\langle S \rangle = I = P_{pow}/A_{area}$$

But, i don't know where does the length of the antenna L comes in! Why is it important and where should i insert it? Please give me a tip.

 

[Delta2]

Your first equation doesn't apply for this problem, you rather must use the equation $$\mathcal{E_0}=E_0l$$ where $\mathcal{E_0}$ the amplitude of the induced emf, $E_0$ the amplitude of the electric field, and $l$ the length of the antenna.
So towards finding the amplitude of the electric field $E_0$ at distance 4miles from the antenna, how would you proceed? Your 2nd and 3rd equation can help you get the amplitude of the magnetic field $B_0$ if you know what you doing...

 

[rude man]

Homework Statement:: A radio station AM transmit isotropically with a power avarage of 4kW. An dipole antenna of reception with 65cm of length is 4 miles away from the transmitter. Calculate the amplitude of the EMF induced by the signal between the ends of the antenna
Relevant Equations:: .

I don't know how to relate the values given by the problem. I am trying to find something that relates these equations:
$$\epsilon = -d \phi_{B}/dt$$
$$\langle S \rangle = cB²/(2 \mu); (\vec J = 0)$$
$$\langle S \rangle = I = P_{pow}/A_{area}$$

But, i don't know where does the length of the antenna L comes in! Why is it important and where should i insert it? Please give me a tip.

You need to know the frequency of the transmitted signal. If it's f = c/2L with L=65cm that makesit a "half-wavelength" antenna, an efficient receiver/transmitter. There are formulas available for this casein the literature. Roughly the emf is as posted by @Delta2.

 

[Delta2]

You need to know the frequency of the transmitted signal. If it's f = c/2L with L=65cm that makesit a "half-wavelength" antenna, an efficient receiver/transmitter. There are formulas available for this casein the literature. Roughly the emf is as posted by @Delta2.

Yes I assumed that the frequency is such that we have a halfwave dipole antenna.

 

[tech99]

Homework Statement:: A radio station AM transmit isotropically with a power avarage of 4kW. An dipole antenna of reception with 65cm of length is 4 miles away from the transmitter. Calculate the amplitude of the EMF induced by the signal between the ends of the antenna
Relevant Equations:: .

I don't know how to relate the values given by the problem. I am trying to find something that relates these equations:
$$\epsilon = -d \phi_{B}/dt$$
$$\langle S \rangle = cB²/(2 \mu); (\vec J = 0)$$
$$\langle S \rangle = I = P_{pow}/A_{area}$$

But, i don't know where does the length of the antenna L comes in! Why is it important and where should i insert it? Please give me a tip.

Let us assume that the ground plays no part in this (I can explain why if wanted).
Then the electric field (e) at a distance D metres from a dipole antenna radiating P watts is 7 (SQRT P) /D in Volts per metre.
If the antenna is isotropic, as in this case, we need to reduce the field by 2.1 dB, the gain of a dipole over isotropic. -2.1 dB is a voltage ratio of 0.62.
Having found the field strength at the receiver, we find the EMF, E, across the antenna terminals by using E = L x e.

 

[Steve4Physics]

The question is open to interpretation. We are specifically told 'AM' This suggests (to me, anyway) that the signal is in the AM brodacast band. That would mean the wavelength is typically 100m to 2000m. If that's the case, the antenna is very short compared to the wavelength; this could be intentional.

We are also not given a specific wavelngth or the orientation of the antenna. So unless @Herculi is on an anntenna design course then I'd guess @Delta's approach (Post #2) is what is needed,

 

[tech99]

Let us assume that the ground plays no part in this (I can explain why if wanted).
Then the electric field (e) at a distance D metres from a dipole antenna radiating P watts is 7 (SQRT P) /D in Volts per metre.
If the antenna is isotropic, as in this case, we need to reduce the field by 2.1 dB, the gain of a dipole over isotropic. -2.1 dB is a voltage ratio of 0.62.
Having found the field strength at the receiver, we find the EMF, E, across the antenna terminals by using E = L x e.

Alternatively, if the transmitting antenna is a ground mounted monopole, as used for Medium Wave broadcasting, the field will be twice as great for a given power. A convenient formula for the received field in such a case is, e = 377 (SQRT p) / d, where d is in km and p is in kilowatts. In this case, e is in millivolts per metre.

 

[tech99]

Alternatively, if the transmitting antenna is a ground mounted monopole, as used for Medium Wave broadcasting, the field will be twice as great for a given power. A convenient formula for the received field in such a case is, e = 377 (SQRT p) / d, where d is in km and p is in kilowatts.

Correction: 2.1 dB reduces the field by a factor of 0.78, apologies.

 

[Delta2]

If that's the case, the antenna is very short compared to the wavelength; this could be intentional.

Ah yes I agree it must be intentional. If we use the definition of the EMF as the integral $$\int_C (\vec{E}+\vec{v}\times\vec{B})\cdot d\vec{l}$$ and we discard the term $\vec{v}\times\vec{B}$ which is the motional EMF, we simply left with $$\int_C \vec{E}\cdot d\vec{l}$$ which integral, if the wavelength of E is big in comparison with the length of the antenna and the antenna is oriented properly (aligned with the transmitting antenna), reduces to the simple formula $|\vec{E}|l$.

 

[Delta2]

@tech99 it is not given if the transmitting antenna is a dipole or a grounded monopole but it is given that it transmits isotropic and I think the formulas given by the OP are sufficient to find $E_0$ and $B_0$, no need to use specific formulas for dipole or monopole antennas.

 

[alan123hk]

But, i don't know where does the length of the antenna L comes in! Why is it important and where should i insert it? Please give me a tip.

I think you should use the effective length of the antenna to find the open circuit voltage appearing on the antenna terminals.

https://en.wikipedia.org/wiki/Antenna_aperture#Effective_length

 

[alan123hk]

I think that in general, multiplying the electric field strength by the physical length of the dipole antenna can already get a reasonable approximation, but if we want to look at it from another angle, the following calculation method is also worthy of reference (I hope I am not mistaken).