Orientation of Hertzian dipole and Plane wave approximation - when is it valid?

[tomwilliam2]

Homework Statement

I have a long EM question in which there is a Hertzian dipole at a point (0,0,-100), (unknown orientation) and I am told the equation of the physical magnetic field detected 100m away at the origin of Cartesian coordinates. $$(B_0 \sin (2 \pi f t)\mathbf{e}_x$$, and $$B_0 = 0.1 \mu T$$, $$f=30 MHz$$.
I have to deduce the possible orientations of the Hertzian dipole and explain qualitatively why the plane wave approximation is valid at +/- 1m in any direction from the origin.

Homework Equations

Maxwell's equations.

The Attempt at a Solution

On the possible directions of the Hertzian dipole: using the Biot-Savart law, I think I can show that the vector product of the current element and the position vector have to produce positive e_x, which is the detected field. So I'm thinking that the dipole has to be pointing in the -e_y direction, because:
$$\mathbf{e}_{?} \times -\mathbf{e}_z = \mathbf{e}_x$$ gives me the -y direction, although two problems: I don't know how to write that argument mathematically (can't divide by a vector) and it only gives me one possible direction, where the question suggests there are more. The question also has a lot of marks attributed to it, and I'm wondering whether I need to talk about possible angular dependence of the dipole.
On the approximation part: I'm guessing that the plane wave approximation is only valid under certain conditions (maybe when the distance is much larger than the wavelength), but I'm unsure as to exactly how this comes in and whether or not the frequency plays a part.

I would gratefully appreciate any pointers.
Best wishes

P.S. A thought just occurred to me that the Hertzian dipole must have a -y component to explain the e_x component of the magnetic field, but any z component could also exist and would simply disappear under the vector product operation...that gives me a new range of direction possibilities...

 

[resurgance2001]

Are you working in Cartesian coordinates or polar coordinates?

When you have a dipole doesn't the magnetic field consist of concentric circles centered on the axis of the dipole. How then can a dipole produce a magnetic field that is only dependent on one axis? Is it a trick question? Or could the wave be polarized in some way?

Cheers

Pete

 

[D1688]

I'm not an expert - just passing some thoughts here.

I guess, we need to talk about the magnetic field in spherical polar coordinates rather than Cartesian? I don't think this is a trick question but would you measure anything in the z-axis? My thought is that a EM wave is propagates perpendicularly so perhaps not

I'd be interested though in getting some more thoughts on this as I've never truly understood the concept of Hertzian dipoles.

D.

 

[D1688]

I've had a think about this again over the weekend so here are my thoughts which I hope help:

What does the Biot-Savart law tell us about the direction of the dipole in relation to the magnetic field?
Given that result, and spherical coordinates, it should be possible to deduce possible dipole field patterns that should show what happens to the strength of the signal depending on where it is located.

However, having said that; I still can't give any clue as to how it's well-approximated by the plane wave in the given region, as I'm not too sure about this one. Maybe someone else can help here

D.

 

[paranormal]

I would like to provide a response to the content provided in the homework statement regarding the orientation of Hertzian dipole and the validity of the plane wave approximation.

Firstly, let's start by defining what a Hertzian dipole is. A Hertzian dipole is a theoretical model of an antenna that is used to explain the behavior of electromagnetic radiation. It consists of two equal and opposite electric charges oscillating in a sinusoidal manner. The orientation of a Hertzian dipole refers to the direction in which the dipole is pointing.

In the given problem, we are provided with the equation of the physical magnetic field detected at a point 100m away from the origin, which is given by $B_0 \sin (2 \pi f t)\mathbf{e}_x$. From this equation, we can deduce that the Hertzian dipole must be pointing in the -y direction, as you have correctly pointed out. This is because the vector product of the current element and the position vector must produce a positive e_x component, which can only be achieved if the dipole is pointing in the -y direction.

However, it is important to note that there could be other possible orientations of the Hertzian dipole that can produce the same magnetic field detected at the origin. This is because the electric and magnetic fields produced by a Hertzian dipole are not only dependent on its orientation, but also on its position and the direction of propagation of the electromagnetic radiation.

Moving on to the second part of the question, the validity of the plane wave approximation. The plane wave approximation is a simplification of the full electromagnetic wave equation that is used when the distance from the source of the wave is much larger than the wavelength. In other words, the plane wave approximation is valid when the distance is much larger than the wavelength. This is because at such distances, the curvature of the wavefront becomes negligible, and the wave can be approximated as a plane wave.

In this problem, we are told that the distance from the origin to the point where the magnetic field is detected is 100m. The frequency of the wave is given to be 30 MHz, which corresponds to a wavelength of approximately 10m. Therefore, at a distance of 100m, the plane wave approximation is valid, and we can use it to explain the behavior of the electromagnetic wave.

In summary, the orientation of the H