Electromagnetic waves & induced EMF

[sskakam]

Homework Statement

A circular loop of wire can be used as a radio antenna. If an antenna with a diameter of 20.0 cm is located a distance of 2.40 km away from a from a source with a total power of 45.0 kW at a frequency of 101 MHz, what is the maximum emf induced in the loop? (Assume that the plane of the antenna loop is perpendicular to the direction of the radiation's magnetic field and that the source radiates uniformly in all directions.)

Homework Equations

P = IA = cB^2/(2u) * 4 Pi r^2
Emf = -d(flux)/dt

The Attempt at a Solution

I looked at the solutions and I had a questions about that.

From P = IA, it solved for B = 2.42*10^-9, which I got. But then it says dB/dt = wB, w being angular frequency. Why?

 

[marcusl]

It is reasonable to assume that the transmitted signal is sinusoidal, so

$$B(t) = Bsin(\omega t + \phi_0),$$

where $$\phi_0$$ is a phase offset. If you differentiate this with respect to time, you get

$$\frac{dB(t)}{dt} = \omega B(t)$$, for an amplitude of $$\omega B$$.

 

[sskakam]

Wow, that was simple. Thanks.

 

[mtt]

I have the following doubts. Suppose we have a square loop(side length L) instead of the circular loop such that the sides are parallel to the propagation direction(k) of the plane electromagnetic wave and to the electric field(E).
1) in order to calculate the flux you have to integrate B from one side to the other side(B(t,r)) at a fixed time. A similar calculation to the problem with the circular loop and we should get the same result as for the circular loop given the area of the loops is the same.
2) Does the Electric field E also contribute to the emf? (only the two sides of the square loop that are parallel to E are relevant) The result of this calculation depends on the shape of the loop.

Is this correct or is there a flaw?