Find the maximum amplitude of the induced EMF

[chocolatecake]

Homework Statement

[/B]
A radio transmitter radiates isotropically at the frequency of 90.8 MHz. The peak magnetic field at
a receiver, 9km from the transmitter, is $9x10^{-10}T$. Calculate the maximum amplitude of the induced emf in a 12 turn coil with area A = $90cm^{2}$ at the receiver. Hint: choose the optimal angle.

Homework Equations

The amplitude of induced emf: $ε=BAω$
$ε=-N\frac{dΦ}{dt}=-NABωcos(ωt)$
$Φ=NBA$
$Φ=NABsin(ωt)$
$Φ=BAcosθ$
All the equations for Φ are in the lecture slides but how can I know which one is the correct one? Why are there several equations for Φ?

The Attempt at a Solution

First, I thought I have to find the first derivative etc. to find the absolute maximum since the question is asking about the maximum amplitude. But in one of the lectures we had a similar example (the question was to find the amplitude of the electromotive force induced in the loop by the signal. And the numbers were different, of course). In the example, they used this equation: $ε=BAω$
If I use the same equation for my problem, I get this:
$ω=(2)(π)(f)$
$ω=(2)(π)(9.08x10^7Hz)=5.7x10^8rad/s$
Plug in:
$ε=BAω$
$ε=(9x10^{-10}T)(0.009m^2)(5.7x10^8rad/s)$
$ε=4.6x10^{-3}V$

However, since the problem includes the hint about the angle, and my solution doesn't include an angle, I'm guessing that my solution is wrong. So how should I approach the problem instead? Any help is much appreciated!
Many thanks! :)

 

[kuruman]

Plug in:
$ε=BAω$

This is what you should use
$ε=-N\frac{dΦ}{dt}=-NABωcos(ωt)$
What happened to $N$? What happened to $\cos(\omega t)$?

 

[chocolatecake]

Ok, but why do I have to use this equation?

 

[chocolatecake]

Another problem is, is that t is not given, so I can't really solve the equation.

 

[BvU]

Ok, but why do I have to use this equation?

It's called Faraday's law (one of the Maxwell equations thta govern electromagnetism).

Another problem is, is that t is not given, so I can't really solve the equation.

You don't have to. Check the definition of amplitude.

 

[BvU]

my solution doesn't include an angle

In the last of your equations there is a factor $\cos\theta$ that you can optimize.

Do you understand your own relevant formulas ? If not, you have to go back to where they come from and find out what the variables are and what the equation describes.

 

[chocolatecake]

You don't have to. Check the definition of amplitude.

Is the amplitude is the maximum displacement in the y direction? So I have to find the derivative?

It's called Faraday's law (one of the Maxwell equations thta govern electromagnetism).

That I do understand, but I found many different equations for Φ and I don't know which one is the correct one. For example, here it says that $Φ=NBAcosθ$ , on this website $Φ=NBA$ and according to our professor $Φ=NABsin(ωt)$. I don't think that my professor is wrong but I don't understand why there are so many different equations for the same thing. Since there are so many, can they be used interchangeably?

 

[chocolatecake]

If not, you have to go back to where they come from and find out what the variables are and what the equation describes.

Well N is the number of turns, Φ is the magnetic flux, B is the magnetic field and A the area.

 

[chocolatecake]

And ω is the angular frequency in rad/s

 

[BvU]

The amplitude of induced emf:

1. $\varepsilon =BA\omega$
2. $\varepsilon =−N\displaystyle {d\Phi\over dt} =−NAB\omega \cos(\omega t)$
3. $\Phi =NBA$
4. $\Phi =NAB\sin (\omega t)$
5. $\Phi =BA\cos\theta$

All the equations for Φ are in the lecture slides but how can I know which one is the correct one?
Why are there several equations for Φ?

They are all one and the same relationship:

1. Is for the amplitude of $\varepsilon$ in a rotating one turn loop OR a loop in a time-dependent B(t) = |B| $\cos(\omega t$)
2. Is for $\varepsilon(t)$ of a rotating N turn coil OR a fixed coil with a time-dependent B = |B| $\cos(\omega t)$
3. Is a flux -- can be time dependent if B is time dependent or the coil rotates
4. Is 3 if the coil rotates OR the B as in 2
5. has a $\theta$ for the orientation of the coil.

Once you understand the material you won't have a problem picking the right expression.