Signal from dipole detcted at loop in free space

[Roodles01]

Homework Statement

A Hertzian dipole at origin generates a signal in empty space which is detected at a wire loop with position vector;
r=(50m)e_z

Homework Equations

Signal is detected by changing magnetic field;
B(t)=B₀ sin(2∏ft)e_x
Show it is consistent with the Maxwell's solution to a plane wave
B=iB0 exp[i(kz-wt)]e_x

The Attempt at a Solution

Hmmm! I'm not getting far enough manipulating Maxwell's equations to provide an answer. Could someone give a little help, please?

 

[rude man]

Homework Statement

A Hertzian dipole at origin generates a signal in empty space which is detected at a wire loop with position vector;
r=(50m)e_z

Homework Equations

Signal is detected by changing magnetic field;
B(t)=B₀ sin(2∏ft)e_x
Show it is consistent with the Maxwell's solution to a plane wave
B=iB0 exp[i(kz-wt)]e_x

The Attempt at a Solution

Hmmm! I'm not getting far enough manipulating Maxwell's equations to provide an answer. Could someone give a little help, please?

A few hints only; you need to grab the bull by the horns yourself:

1. what is really meant by B=iB0 exp[i(kz-wt)]e_x?

2. know the complex Euler relation betw. exponent and sine/cos?

3. Keep in mind that, at the receiving antenna, time is defined as t = 0, which is delayed from that of the transmitting antenna, so there will be a phase lag between transmitter and receiver.

This is mostly math.

 

[Roodles01]

Thank you.
1. I think that B=iB0 exp[i(kz-wt)]ex is the propagation of the signal/wave in the x direction

2. Euler relation betw. exponent and sine/cos
e^ix = cos x + i sin x (circular wave [e^ix] split into 2 planes . . . . . .

3. I realize I may be doing things the wrong way, but doing things by yourself when you have no time & tired is sooo hard to follow the right path . . . .
I'm trying.

 

[rude man]

Thank you.
1. I think that B=iB0 exp[i(kz-wt)]ex is the propagation of the signal/wave in the x direction

How can a signal be imaginary?

2. Euler relation betw. exponent and sine/cos
e^ix = cos x + i sin x (circular wave [e^ix] split into 2 planes . . . . . .

No circulraly polarized wave here. This hint means little until you figure out hint #1.

3. I realize I may be doing things the wrong way, but doing things by yourself when you have no time & tired is sooo hard to follow the right path . . . .
I'm trying.

I understand, but we are strictly prohibited from doing more than giving you hints and telling you if your approach is correct or not.

 

[Roodles01]

. . . . . "what is really meant by B=iB0 exp[i(kz-wt)]ex?"

This is a solution to the wave equation - a combination of sin(kz-wt) & cos(kz-wt).

"The electric field of a sinusoidal plane wave that travels in the direction of the
propagation vector k and is polarized transverse to that direction" . . . . is represented by this equation.

However, for this I have the general solution
E(r; t) = E0 sin (k.r - wt).
This is similar to the equation I have stated at the beginning, however I'm not sure where the "i" in front of Bo comes from.

I really do appreciate the hints. I'm not moaning, honest!

 

[rude man]

. . . . . "what is really meant by B=iB0 exp[i(kz-wt)]ex?"

This is a solution to the wave equation - a combination of sin(kz-wt) & cos(kz-wt).

"The electric field of a sinusoidal plane wave that travels in the direction of the
propagation vector k and is polarized transverse to that direction" . . . . is represented by this equation.

However, for this I have the general solution
E(r; t) = E0 sin (k.r - wt).
This is similar to the equation I have stated at the beginning, however I'm not sure where the "i" in front of Bo comes from.

I really do appreciate the hints. I'm not moaning, honest!

No offense taken, mate! :-)

But you didn't answer my question: how can an electric wave be imaginary? If you answer that question correctly you will see why the "i" is needed ahead of the expression for the transmitted wave.